Accurate measurement of the loss rate of cold atoms due to background gas collisions for the quantum-based cold atom vacuum standard

Since the first magnetic trapping of laser-cooled neutral alkali-metal atoms, experiments performed in ultra-high vacuum (UHV) chambers,¹ it has been recognized that collisions of residual or background gas atoms and molecules with the trapped atoms establish a limit on the lifetime of cold atoms in their shallow magnetic trap. Inverting the problem—using the measured loss rate of cold atoms from a conservative magnetic trap to sense vacuum pressure in the ultra-high vacuum regime—has since been pursued in several experiments.^2–9 Such a conversion requires knowledge of gas-species-dependent loss rate coefficients L to determine the background-gas number densities n from measured trap loss rates Γ. In fact, $n = Γ / L$ and a value for pressure p then follows from the ideal gas law $p = nkT$⁠, where k is the Boltzmann constant and T is the background gas temperature, assuming that this gas is in thermal equilibrium with the walls of the vacuum chamber. The loss rate coefficients correspond to thermally averaged rate coefficients for elastic, momentum-changing collisions between a trapped atom and room-temperature background-gas atoms or molecules.

Many of the first attempts to measure pressure with laser-cooled atoms relied on semi-classical theory of elastic scattering¹⁰ to compute the loss rate coefficients.^2,3,5,6,8 Quantum universality of these elastic and small-angle, or diffractive, collisions, derived from this theory, has been put forward as a means to extract the loss rate coefficients.^9,11,12 The accuracy of the semi-classical model, however, is not well characterized. Analyses by Refs. 13 and 14, for example, have suggested that loss rate coefficients based on this theory can be in error by as much as 30%.

Here, we measure loss rate coefficients with high-accuracy in a model-independent way. Our measurements achieve one-standard-deviation combined statistical and systematic (k = 1) relative uncertainties better than 1.6% for ⁷Li and 2.7% for ⁸⁷Rb. We use two different cold atom vacuum sensors^15–18 that trap a relatively small number of either ⁷Li or ⁸⁷Rb sensor atoms in a weak magnetic trap with energy depth W, typically $W / k ≲ 1$ mK, connected to a dynamic expansion (DE) system, which sets a known number density of background atoms or molecules. We compare our findings to recent fully quantum mechanical theoretical results¹⁴ and, in the case of ⁸⁷Rb, the results from experiments utilizing the theory of universality of quantum diffractive collisions (UQDC).^9,11,12 For the former, we find excellent agreement; for the latter, we find more nuanced agreement.

Quantum-mechanical scattering calculations of K(T) and $a gl ( T )$⁠, including an analysis of their theoretical uncertainties, have been conducted for a few systems. The first to be characterized was $6 , 7$Li + H₂,^19,20 followed by $6 , 7$Li + ⁴He.^21,22 Recently, Ref. 14 presented comprehensive calculations of ⁷Li and ⁸⁷Rb colliding with H₂, ¹⁴N₂, and all the noble gases and provides tables for the coefficients $K i , A i$⁠, and $B i$ with i = 0 or 1.

Our principal results for natural abundance noble gas species and N₂ are given in Fig. 1. The figure compares experimentally determined values of L(W, T) for ⁷Li and K(T) and $a gl ( T )$ for ⁸⁷Rb with the corresponding theoretical values from Ref. 14. For ⁷Li data, $W / k = 0.95 ( 14 )$ mK and $T = 300.2 ( 2.9 )$ K. For ⁸⁷Rb data, $T = 295.2 ( 3 )$ K and W ranges between $k × 0.3986 ( 4 )$ and $k × 1.594 ( 1 )$ mK in order to extract K(T), $a gl ( T )$⁠, and $b gl ( T )$⁠. Most values of $b gl ( T )$ are consistent with zero at the two-standard-deviation (k = 2) level and are thus omitted from Fig. 1. The experimental and theoretical values for L(W, T) and K(T) are consistent at the two-standard deviation combined statistical and systematic (k = 2) uncertainty level, except for ⁷Li–Ar and ⁸⁷Rb–Ar.

The agreement observed in Fig. 1 has a second but equally valid interpretation. Specifically, the pressure measured by a cold atom pressure sensor, when using the values of L from Ref. 14, agrees with the pressure set by a classical dynamic expansion system. When used to measure pressure in this way, the cold atom pressure sensor is traceable only to the SI second and kelvin, making it a primary standard. We thus refer to our two sensors as cold atom vacuum standards (CAVSs). Agreement between a CAVSs and the DE system in our direct comparison validates a CAVS as a standard of vacuum pressure.

Given that a CAVS can easily measure loss rates between 0.01 and 10 s⁻¹ and typical values of K(T) are on the order of $10 − 9$ cm³/s at T = 300 K, we therefore estimate that a CAVS's range of operation spans background-gas number densities (pressures) from of the order of 10⁷ cm⁻³ (⁠ $4 × 10 − 8$ Pa) to $∼ 10 10$ cm⁻³ (⁠ $4 × 10 − 5$ Pa). Indeed, similar devices have been operated up to $6 × 10 − 5$ Pa.¹¹ These pressures correspond to most of the ultra-high vacuum and part of the high-vacuum regimes.

A significant difference between a CAVS based on ⁷Li and ⁸⁷Rb sensor atoms is the value of $a gl ( T )$⁠. ⁸⁷Rb with its larger mass has typical values $a gl ( T ) k ∼ 10 − 7$ cm³/(s K), while ⁷Li has typical values $a gl ( T ) k ∼ 10 − 8$ cm³/(s K).¹⁴ In a trap with depth $W ∼ k × 1$ mK, roughly one of every ten collisions between the background gas and a ⁸⁷Rb sensor atom is a “glancing” collision. As shown in Fig. 1, measured values of $a gl ( T )$ are consistent at the two-standard deviation combined statistical and systematic (k = 2) uncertainty level, except for ⁸⁷Rb–Kr and ⁸⁷Rb–Xe. For ⁷Li confined in a trap with the same depth, the fractional rate of glancing collisions is an order of magnitude smaller. Given current fractional measurement uncertainties of order of 1%, glancing collisions are thus not detectable for ⁷Li.

The remainder of the paper is divided as follows. Section II describes the salient features of our two types of apparatuses. In Sec. III, we analyze our observed sensor atom loss curves as a function of background gas pressure or, equivalently, number density produced by the dynamic expansion system. Section IV presents our measured total and glancing rate coefficients along with a description of uncertainty budgets. We conclude in Sec. V.  Appendixes A and  B provide additional details on the dynamic expansion standard and sensor atom imaging, respectively.

While a known partial number density is generated, either the l-CAVS or the p-CAVS measures the loss rate Γ of sensor atoms held in a quadrupole magnetic trap. Simultaneous operation of both CAVSs was not possible because operation of the l-CAVS interferes with the stability of the p-CAVS. Preparation of the sensor atom cloud in either CAVS involves several steps (see Refs. 17 and 18). First, a magneto-optical trap (MOT) is loaded with atoms. Complementary metal-oxide semiconductor (CMOS) cameras record fluorescence images of the MOT during the loading process and we determine the final number of atoms in the MOT, N₀, using these images. For both the l- and p-CAVSs, N₀ is of the order of 10⁶.

Next, the atoms are transferred into the quadrupole magnetic trap. For both the l- and p-CAVSs, the transfer process involves optical pumping to the F = 1 hyperfine ground state and, for the l-CAVS, subsequent heating and removal of any remaining F = 2 hyperfine states. See Refs. 17 and 25 for details. All trapped atoms are then in the F = 1, $m F = − 1$ hyperfine state.

Radio frequency (RF) radiation with a frequency $ν RF$ between 5 and 40 MHz induces spatially localized transitions between magnetic Zeeman states of the sensor atom and sets the energy depth of the magnetic trap to $W = h ν RF ( 1 − M g / μ B g F B ′ )$⁠, where M is the mass of a sensor atom, g is the local gravitational acceleration, g_F is the Landé g-factor, and $μ B$ is the Bohr magneton. In practice, after loading the l-CAVS magnetic trap, this so-called RF knife is applied with an initial frequency of 40 MHz. The RF frequency is then linearly decreased to $ν RF = 5$ MHz in 1 s. The end of this RF frequency ramp corresponds to t = 0 for the l-CAVS loss rate measurement. At t = 0, the remaining 10^{5 87}Rb atoms have a temperature between 50 and 200 μK. The former estimate comes from fitting an in situ image of the atoms in the magnetic trap to the expected distribution for a thermal cloud; the latter comes from time-of-flight expansion of similarly prepared clouds with 10 times the atom number to achieve good signal-to-noise. For t > 0, the RF frequency is changed to a final, constant $ν RF$ between 10 and 40 MHz and is applied for the remainder of the time the atoms are in the magnetic trap. This controllably sets the trap depth to values between $k × 0.3986 ( 4 )$ and $k × 1.594 ( 1 )$ mK. We have verified the effectiveness with which our RF knife removes atoms with $E > h ν RF$ by extending the RF knife ramp down to $ν RF = 100$ kHz, which removes all the atoms.

For the p-CAVS, approximately $1 × 10 5$ ⁷Li atoms are transferred from a grating MOT²⁶ into a magnetic quadrupole trap with axial magnetic field gradient $B ′ = 4.59 ( 17 )$ mT/cm. No RF knife is used in the p-CAVS; instead, the trap depth is set by the distance between the center of the trap and the nearest in-vacuum surface, the magneto-optical trap's diffraction grating.¹⁶ We calculate a trap depth of $W / k = 0.95 ( 14 )$ mK, where the uncertainty comes from the uncertainty in the distance. The temperature of the magnetically trapped ⁷Li cloud could not be measured. It can be as high as 0.75 mK based on temperatures observed in other Li grating MOTs.^25,26 Loading atoms into the magnetic trap marks t = 0 for the p-CAVS loss rate measurement.

For both l- and p-CAVSs, sensor atoms are held in the magnetic trap for a variable amount of time t > 0, after which the atoms are recaptured into a MOT. Fluorescence from the MOT is imaged onto CMOS cameras to determine sensor atom number $N S ( t )$ as function of time. The atom-number measurement is destructive, so the atom cloud preparation described above is repeated for each t. For the l-CAVS, we also repeat the cloud preparation process for each trap depth W. In practice, we measure the ratio $η S ( t ) = N S ( t ) / N 0$⁠, which reduces our statistical noise by eliminating fluctuations in the atom number loaded into the MOT N₀ from one cloud preparation to the next. Once a full decay curve is measured, taking between 0.25 and 3 h, the background gas density n is changed and another decay curve is taken. We do not require an absolute measurement of sensor atom number, so properties of our imaging system, such as the quantum efficiencies of the cameras, do not contribute to our uncertainty budgets, provided such properties do not vary with time. Details about our imaging system, including its stability and non-linearity can be found in  Appendix B. The instability and lack of linearity add a small systematic uncertainty in our final uncertainty budget for the rate coefficients.

We find that satisfactory fits to the decays curves can be found by adjusting the initial $η S ( t = 0 )$⁠, the parameters in $Γ ( W )$ and $β ( W )$ as well as $σ η$ and σ₀ in noise function $u ( η S ) = ( σ η η S ) 2 + σ 0 2$⁠. The noise function is a model for the uncertainty in the sensor atom number and is an implicit function of time t. The first component, proportional to $η S$⁠, is related to random fluctuations in the initial sensor atom number in the magnetic trap and the fluctuating detuning of the MOT laser beams. The second component σ₀ reflects the minimum number of sensor atoms that is detectable by our imaging system. The parameters $σ η$ and σ₀ are different for ⁷Li and ⁸⁷Rb but should be independent of background species, n, and W.

For ⁷Li in the p-CAVS with its fixed W, we fit all values for $η S ( t )$ to Eq. (3) and, in this manner, determine Γ and β and their covariances. For ⁸⁷Rb in the l-CAVS with its variable W, we simultaneously fit the time traces $η S ( t )$ at all W to the combination of Eqs. (3) and (4). This procedure gives us reliable values for the two parameters in the noise function, as a single time trace at a single W does not contain enough data. This simultaneous fit determines $Γ 0 , Γ 1$⁠, β₀, and β₁ and their covariances.

Figure 2 also shows the quality of our fits. The residuals normalized by the noise function do not have recognizable patterns. A cumulative distribution function (CDF) constructed from the residuals is well described by the CDF for a Gaussian distribution. For our p-CAVS with ⁷Li atoms $σ η ≲ 0.03$⁠, while for our l-CAVS with ⁸⁷Rb atoms, $σ η ≲ 0.08$⁠. The minimum detectable atom number σ₀ is about 500 for the p-CAVS and is 300 for the l-CAVS.

The best fit values of $Γ ( W p − CAVS )$ for ⁷Li and $Γ 0$ for ⁸⁷Rb are 0.003 88(6) and 0.0119(8) s⁻¹, respectively. Here, $W p − CAVS$ is the fixed trap depth of the p-CAVS. Assuming that H₂ is our dominant background gas and using the theoretical values of rate coefficients $K Li – H 2 = 3.18 ( 6 ) × 10 − 9$ cm³/s at $T = 301.7 ( 3.3 )$ K and $K Rb – H 2 = 3.9 ( 1 ) × 10 − 9$ cm³/s at $T = 295.8 ( 3 )$ K, we find pressures of 5.19(3) and 14.2(1.4) nPa, according to ⁷Li p-CAVS and ⁸⁷Rb l-CAVS, respectively. Here, the uncertainty is dominated by the uncertainty in the theoretical rate coefficients. The factor of nearly three difference in the base pressure readings may be due to a variety of factors, including pressure gradients (see  Appendix A), the difference in Majorana loss of the two species, and the inability to accurately separate Γ from two- or even three-body losses in the fits. We note that a previous experiment with two p-CAVSs closely connected to each other on a different vacuum chamber than used here measured the same, higher pressure [42.2(1.0) nPa] within their respective uncertainties.¹⁷ 

For ⁸⁷Rb, we find $Γ 1 = d Γ / d W = 1.56 ( 81 )$ s⁻¹/K. This value is consistent with zero at two standard deviations (k = 2). The ratio of $Γ 1 / Γ 0 = 142 ( 85 )$ K⁻¹ is likewise consistent with the theoretical prediction of 36.7(1.8) K⁻¹ for ⁸⁷Rb + H₂ and a recent measurement¹³ of $43 ( 5 )$ K⁻¹.

We convert the fitted values of $β ( W p − CAVS )$ for ⁷Li and β₀ for ⁸⁷Rb from the data in Fig. 2 to rate coefficients K₂ defined through the differential equation $n ̇ S = − Γ n S − K 2 n S 2$ for the sensor atom number density $n S ( t )$⁠.^18,27 The fitted β₁ is consistent with zero. For ⁸⁷Rb, the derived $K 2 ≈ 2 × 10 − 10$ cm³/s is remarkably close to the known elastic scattering rate coefficient of $1.2 × 10 − 10$ cm³/s among ⁸⁷Rb atoms using the in situ rubidium temperature estimate of 50 μK.¹⁸ Elastic collisions only change the momenta of the atoms and thus should not lead to sensor atom loss when the sensor atom temperature is much less than the trap depth W as is the case in our ⁸⁷Rb experiments. We observe no difference in the two-body loss rate when we reduce the efficiency of the RF knife by halving the amplitude of the RF radiation, further indicating that our RF knife is efficient at removing highly energetic atoms, which, if left behind, could increase the observed two-body loss rate. For ⁷Li, the derived K₂ is inconsistent and much larger than the known elastic scattering rate coefficient at a lithium temperature of roughly 750 μK. The origin of the non-zero values for β in both CAVSs remains a mystery.

We are now ready to study the readings of the CAVSs when a known background number density n of a gas species X is set by the combined flowmeter and dynamic expansion system. A sampling of the available data for ⁷Li with natural abundance Ar gas, taken with the p-CAVS, and for ⁸⁷Rb also with natural abundance Ar gas, taken with the l-CAVS, are shown in Figs. 3(a) and 3(b), respectively. The figure shows $η S ( t )$ as functions of time for several values of n between $0.2 × 10 8$ and $4 × 10 8$ cm⁻³. For ⁸⁷Rb, Fig. 3(b) also shows time traces for several trap depths W. We observe that for roughly the same Ar gas density, the observed lifetimes for ⁷Li are about 60% longer than those of ⁸⁷Rb, consistent with the observation that the rate coefficients K for ⁷Li are about 60% smaller than those of ⁸⁷Rb. We have similar quality data for the other noble gases as well as for N₂. In all cases, we use gases containing a natural abundance distribution of the stable isotopes.

For ⁷Li, we fit all values for $η S ( t )$ taken at number density n to Eq. (3), even though the non-exponential decay is not always apparent. In this manner, we determine Γ and β and their covariances for each n and each background gas species X. For ⁸⁷Rb, we fit all values of $η S ( t )$ at all values of W at a single background-gas number density n to the combination of Eqs. (4) and (3), even though, again, the non-exponential decay is not always apparent. This simultaneous fit determines $Γ 0 , Γ 1$⁠, β₀, and β₁ and their covariances for each n and each background gas species X. We find that within their uncertainties, the fitted values for $σ η$ and σ₀ are consistent for all n and all background species, as expected, for both ⁷Li and ⁸⁷Rb.

Our values of $χ 2 / ν$⁠, where ν is the number of degrees of freedom, are 0.41 (⁷Li + He, ν = 5), 0.59 (⁷Li + Ne, ν = 11), and 0.57 (⁷Li + N₂, ν = 8), 0.62 (⁷Li + Ar, ν = 16), 0.47 (⁷Li + Kr, ν = 4), and 1.61 (⁷Li + Xe, ν = 5). In fact, no fits fail the $χ 2$ test,²⁸ where the probability of a hypothetical repeated realization of the experiment with the same uncertainties yielding a larger $χ 2 / ν$ is less than 5%. Fitted values of $Γ base$ range from 0.0042(6) to 0.066(4) s⁻¹, consistent with the long-term fluctuations observed in repeated measurements of the decay rate at base pressure (such as that shown in Fig. 2).

Table I shows the complete uncertainty budget for the experimental value of L for ⁷Li + Ar. Its statistical uncertainty follows from the linear least squares fit for L of the data in Fig. 4. We observe that the statistical and systematic uncertainties of the experimental L are approximately equal. The experimental uncertainty budgets for L of ⁷Li with other natural-abundance background species are similar.

Atom loss decay curves for ⁸⁷Rb sensor atoms described in Sec. III have resulted in values for $Γ 0$ and $Γ 1$ at approximately ten background number densities n for each background species. The values for $Γ 0$ and $Γ 1$ at the same n and background species are correlated. The approximately ten values of $Γ 0$ are then fit to $Γ 0 = K n + Γ 0 , base$ and we find values, uncertainties, and covariances for K and $Γ 0 , base$⁠. Finally, we fit all values for $Γ 1$ to $Γ 1 = a gl n + Γ 1 , base$ and all values for $Γ 2$ to $Γ 2 = b gl n + Γ 2 , base$ and obtain $a gl , Γ 1 , base , b gl , Γ 2 , base$⁠, respectively. As for ⁷Li, we do not include data taken at base pressure in the fits to determine these four parameters. We find that the values of $χ 2 / ν$ for the linear least squares fits to extract K are 1.32 (⁸⁷Rb + He, ν = 8), 1.20 (⁸⁷Rb + Ne, ν = 7), and 1.07 (⁸⁷Rb + N₂, ν = 8), 0.33 (⁸⁷Rb + Ar, ν = 7), 0.42 (⁸⁷Rb + Kr, ν = 6), and 1.34 (⁸⁷Rb + Xe, ν = 8). Again, no fits fail the $χ 2$ test.

Values of $Γ 0 , base$ for ⁸⁷Rb range from 0.017(2) to 0.034(4) s⁻¹, much larger than rate 0.0119(8) s⁻¹ determined from the fit to data shown in Fig. 2(b). This larger $Γ 0 , b ase$ suggests that we cannot sufficiently separate the effects from $β η S 2 ( t )$ and $Γ η S ( t )$ in decay curves. We have performed analyses of the experimental systematic uncertainties in the same manner as described for the ⁷Li data. Our systematic relative uncertainties are approximately the same as those for the ⁷Li experiments. The relative statistical uncertainties of K, however, are larger by a factor between 2 and 4 compared to those for L of ⁷Li.

In addition, we must adjust for the fact that we experimentally use natural isotope abundance background gases while the data in Ref. 14 is computed for a gas containing only the most abundant isotope. We scale the theoretical rate coefficients for one isotope to values for other isotopes by using the semi-classical dependence on the mass of the background gas species m and the mass of the sensor atom M. We then find “weighted” rate coefficients based on the natural abundance of each isotope. The relevant semi-classical mass dependencies are $K ∝ m − 3 / 10 , a gl ∝ m − 1 / 10$⁠, and $b gl ∝ m 1 / 10$⁠. This scaling matters most for neon and xenon, for which the isotope correction $δ K isotope$ represents a −0.28% and +0.14% shift in K, respectively. We take this scaling to be approximate with a 15% relative uncertainty that is $u ( K ) | isotope = 0.15 δ K isotope$ to be added in quadrature to all other uncertainties in the theoretical rate coefficients. The relative uncertainty due to isotopic abundance for $b gl$ is negligible, so we omit it from the uncertainty budget.

Table I shows the uncertainty budget in the theoretical value for L for the ⁷Li + Ar system. The relative uncertainty for the theoretical value is half that of the combined systematic and statistical uncertainties of the experimental value. Table II shows our final theoretical and experimental values of L for ⁷Li + X systems, along with the degree of equivalence E(L) for L defined by $E ( O ) = ( O thr − O exp ) / [ 2 u ( O thr − O exp ) ]$⁠, where $u ( O thr − O exp ) = u 2 ( O thr ) − 2 cov ( O thr , O exp ) + u 2 ( O exp )$ is the uncertainty of the difference between the correlated theoretical and experimental values for quantity O. As the temperature dependence of the theory and experiment values are correlated, E(L) is larger than the uncorrelated combination of the theoretical and experimental uncertainties would suggest. All values agree at three standard deviations, k = 3; all except ⁷Li–Ar agree at k = 2.

Table III shows the predicted and measured K, $a gl$⁠, and $b gl$ for ⁸⁷Rb colliding with He, Ne, N₂, Ar, Kr, and Xe as background species. We find k = 2 agreement between the theoretical and experimental K for all collision partners except ⁸⁷Rb + Ar. The theoretical and experimental values of $a gl$ and $b gl$ agree at k = 2 for all collision partners except ⁸⁷Rb + Kr, which agrees at k = 3. We constrained $Γ 2 = 0$ in our fits ⁸⁷Rb + He because the expected size of $b gl$ is two orders of magnitude lower than the uncertainty on the values for all other background species. The experimental relative uncertainties for $a gl$ are much larger than the corresponding theoretical uncertainties and experimental uncertainties observed in Refs. 9, 11, and 12 because the present experiment focused on taking data at many distinct pressures, rather than at many trap depths for each pressure.

We examined several other potential systematic effects. For the l-CAVS, we studied sensor atom loss rates after changing the laboratory temperature from 22.0(1) to 19.0(5) °C, the magnetic field gradient of the quadrupole trap from 18 to 9.0 mT/cm and 24 mT/cm, and the applied RF powers from 25 to 12 W, but saw no statistically significant dependence of K or $a gl$ on these parameters. We also tested an alternative application of the RF knife, that of Ref. 9. After loading the magnetic quadrupole trap and waiting for a time t, we apply an RF sweep such that the trap depth decreases from $k × 3.188 ( 3 )$ mK to final trap depth W to eject sensor atoms with kinetic energy E > W and, immediately afterward, measure the final atom number N. We observed no change of K or $a gl$ when using this alternative application of the RF knife. For the p-CAVS, we changed the power dissipated in the source from 2.7 to 2.0 W and 3.5 W, magnetic field gradient of the quadrupole trap from 4.59 to 7.53 mT/cm, and laboratory temperature from 22.0(1) to 19.0(5) °C and 25.0(1) °C, but again saw no statistically significant dependence of L on these parameters.

Finally, we compare our rate coefficients with those published in Refs. 12–14 in Table IV. Agreement is observed for ⁸⁷Rb–⁴He at the one-standard deviation (k = 1) level. For ⁸⁷Rb–N₂ and ⁸⁷Rb–Xe, rate coefficients based on universality of quantum diffractive collisions (UQDC) from Ref. 12 are smaller than our K by 12% and 7%, corresponding to more than four and two standard deviations, respectively. The data points for ⁸⁷Rb–Ar are discrepant. Further research for this system is needed. As discussed in Ref. 13 and reflected in the table, UQDC does not work well for the ⁸⁷Rb–H₂ system. In Ref. 13, the authors measure the ratio of loss rate coefficients for ⁸⁷Rb and ⁷Li with background H₂ and use the theoretical results for the ⁷Li + H₂ system from Refs. 19 and 20 to derive a loss rate coefficient for ⁸⁷Rb with H₂. The resulting loss rate coefficient is in agreement with Ref. 14. This scaling procedure was first suggested by Ref. 15.

We have measured total rate coefficients for room-temperature natural abundance gas species He, Ne, N₂, Ar, Kr, and Xe colliding with ultra-cold ⁷Li and ⁸⁷Rb sensor atoms using a flowmeter combined with a dynamic expansion system and two cold-atom vacuum sensors. Our measurements have an uncertainty of better than 1.6% for ⁷Li and 2.7% for ⁸⁷Rb. We find consistency at the two-standard-deviation combined statistical and systematic (k = 2) uncertainty level for all gas combinations except for ⁷Li–Ar and ⁸⁷Rb–Ar with recently published quantum-mechanical scattering calculations.¹⁴ We also compare the rate of glancing collisions for ⁸⁷Rb, collisions that do not impart enough energy to eject ⁸⁷Rb from its shallow magnetic quadrupole trap, and find consistency at the two-standard-deviation combined statistical and systematic (k = 2) uncertainty level with the calculations of Ref. 14 for all collisions except ⁸⁷Rb–Kr.

An equivalent interpretation of our results is that quantum-based measurement of vacuum pressure with cold atoms is consistent with that set by a combined flowmeter-dynamic expansion standard. Thus, cold-atom based vacuum pressure sensors are also cold atom vacuum standards, or CAVSs. Agreement between the dynamic expansion standard and the CAVS validates their operation as quantum-based standards for vacuum pressure.

This validation opens potential new opportunities in vacuum metrology at ultra-high vacuum (UHV) pressures. In particular, the quantum measurement of pressure by a CAVS is primary. It is not traceable to a measurement of like kind. Given the demonstrated consistency, the CAVS could now potentially replace the combined flowmeter and dynamic expansion systems in the calibration of other pressure gauges.

The portable CAVS (p-CAVS), in particular, can also replace common classical gauges, like the Bayard–Alpert ionization gauges.^16,17 The p-CAVS shows lower uncertainties than calibrated ionization gauges in the UHV.²⁹ The performance of our p-CAVS is comparable and complementary to that of the recently developed 20SIP01 ISO ionization gauge,³⁰ which has better than 1.5% relative uncertainties without calibration but operates at higher pressures from 10⁻⁶ to 10⁻²Pa. Both have absolute uncertainties that are independent of the individual gauge.

Another advantage over ion gauges is related to pressure sensing with unknown mixtures of background gases. Despite the range of masses and polarizabilities of the background gas species for which we have calculated and measured the loss rate coefficients, the maximum relative deviation of L from L for N₂ is roughly 40% for both ⁷Li and ⁸⁷Rb, as seen in from Table I of Ref. 14. We believe, based on semi-classical scattering theory,¹⁰ that the mean and variation will not significantly increase as data for other background gases become available. Thus, we can expect a pressure measurement of (mixtures of) unknown gases by a CAVS to have at most a 40% relative uncertainty if one simply used the value of L for N₂. The uncertainty is small compared to the factor of five difference in readings seen by an ionization gauge between N₂ and He at the same pressure.^31,32

If the background gas contains a single species with an unknown L, then the procedure outlined in Refs. 9, 11, and 12 can determine L from measurements at a single, unknown n. The procedure relies on the validity of semi-classical scattering theory¹⁰ and a measurement of the variation of the atom loss rate Γ on trap depth W. The procedure is known to fail when the colliding pair's reduced mass is small compared to the cold atom's mass; the discrepancy of K for ⁸⁷Rb + H₂ between Refs. 9, 11, and 12 and Refs. 13 and 14 is roughly 30%. However, disagreements between the K of Refs. 9, 11, and 12 and those of Ref. 14, mostly verified by this present work, can be between 5% and 9%, with these residual discrepancies not strongly dependent on the reduced mass. If we ignore ⁸⁷Rb + H₂, then, in the same spirit as the ionization gauge discussion above, we conclude that the maximum relative uncertainty for a cross section obtained using the procedure of Refs. 9, 11, and 12 is 9%. Further work is required to verify the uncertainty of the methods of Refs. 9, 11, and 12. Because it requires knowledge of the variation of Γ on W, however, the procedure will likely not be feasible for ⁷Li, given its light mass.¹² There are simply fewer glancing collisions with which to accurately measure this dependence compared to ⁸⁷Rb.

This decrease from 40% to 9% in relative uncertainty due to an unknown L is not the only motivating factor in choosing between ⁷Li and ⁸⁷Rb as the CAVS sensor atom. Another key difference between ⁷Li and ⁸⁷Rb is that ⁸⁷Rb exhibits significant non-exponential decay in the atom-loss decay curves at the lowest UHV pressures, as evidenced by the large, fitted β in Eq. (3) and shown in Fig. 2. We currently have no satisfactory explanation for this observation. This unexpected discovery suggests that ⁸⁷Rb-based CAVSs will probably not be as accurate as one based on ⁷Li in the low ultra-high vacuum and extreme-high vacuum regimes. Combined with the other advantages outlined in Ref. 16, we believe that ⁷Li offers superior performance.

To realize the low <2% uncertainty potential of a ⁷Li based p-CAVS, loss rate coefficients for other common gases found in vacuum chambers like CO, CO₂, O₂, and H₂ must be measured and compared to theoretical evaluations when available. Measurement of L with these more reactive gases requires an upgrade to our dynamic expansion system, which is under way. Theoretical calculations for CO, CO₂, and O₂ are also forthcoming; theoretical calculations for H₂ are already contained in Ref. 14.

Finally, we must further validate the pressure range of operation of the CAVSs. Currently, such devices have been operated as high as $6 × 10 − 5$ Pa,¹¹ where loss rates are of the order of 10 s⁻¹. The lowest detectable pressure of a CAVS is less well characterized; we are currently endeavoring to understand the physics behind the non-exponential behavior at low pressures.

The authors thank L. Ehinger, P. Elgee, and A. Sitaram for initial development of the p-CAVS; B. Acharya, E. Newsome, and R. Vest for technical assistance; N. Klimov for fabrication of the grating-MOT chip; E. Norrgard and W. Phillips for useful discussions; and K. Douglass and G. Fraser for a thorough reading of the manuscript.

D.S.B., J.A.F., J.S., and S.P.E. have U.S. patent 11,291,103 issued. D.S.B. and S.P.E. have U.S. provisional patent 63/338,047 field.

Daniel S. Barker: Data curation (equal); Formal analysis (equal); Investigation (lead); Methodology (equal); Project administration (equal); Software (equal); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). James Fedchak: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (lead); Methodology (equal); Writing – review & editing (equal). Jacek Klos: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Julia Scherschligt: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Abrar Sheikh: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Eite Tiesinga: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Writing – original draft (equal); Writing – review & editing (equal). Stephen Eckel: Conceptualization (equal); Data curation (equal); Formal analysis (lead); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (lead); Supervision (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

For cylindrical tubes, the transmission probability α is known analytically under reasonable gas flow assumptions and is only a function of l/r.³⁵ At our uncertainties for l and r, the transmission probability given by Eq. (16) of Ref. 35 is sufficiently accurate and gives $α An = 0.8157 ( 1 )$⁠. Here, the standard uncertainty $u ( α An )$ follows from uncertainty propagation of u(l) and u(r) ignoring correlations between the measurements of l and r.

We amend this analytical estimate of α using Monte Carlo simulations of particles in our dynamic expansion standard based on the actual orifice and chamber geometries and assuming that the temperature of the particles is that of chamber walls, $T DE$⁠.³⁶ In these simulations, particles only collide with the chamber walls, which is a good assumption at our UHV pressures as the mean free path for particle–particle collisions is orders of magnitude larger than the chamber sizes. Reflections from the walls are Lambertian: the particle is given a new random speed, sampled from the Maxwell–Boltzmann velocity distribution independent of its incoming velocity, and a random angle θ with respect to the surface normal sampled from a $cos θ$ probability distribution. Finally, particles colliding with vacuum pump surfaces have an absorption coefficient that, given the surface's area, yields the correct pumping speed.

From the Monte Carlo simulations, we find $α MC = 0.8160 ( 2 )$⁠, which is 0.03% larger than but consistent with $α An$⁠. This result confirms that the chamber geometry has a negligible impact on C₀. The standard uncertainty of $α MC$ is twice that of $α An$ as it combines two sources of (uncorrelated) uncertainty: (1) the counting uncertainty of the Monte Carlo simulations and (2) the uncertainty in the dimensions of our orifice. We use the more conservative $α MC$⁠.

We measure $T DE$ by averaging the time-series readings of four calibrated platinum resistance thermometers (PRTs). The thermometers are mounted to the exterior walls of the dynamic expansion standard and are placed in pairs. Each pair is placed on opposing sides of the standard. One pair is coplanar with the orifice while the other pair is mounted on the first chamber 18.9(4) cm away from the orifice plane. A reading $T i , DE ( t )$ of thermometer i = 1, 2, 3, or 4 at time t has a standard uncertainty of 36 mK. Self-heating of the PRTs, measured to be about 3 mK, is negligible. Temperature gradients of approximately 0.4 K combined with drifts of roughly 0.05 K over the time interval it takes to map out the decay of sensor atom number $N S ( t )$⁠, however, are observed in the dynamic expansion system. Hence, temperature gradients dominate the uncertainty of $T DE$ and thus $u ( T DE ) = s$ with sample variance $s = ∑ i = 1 4 ∑ j = 0 m ( T i , DE ( j Δ t ) − T DE ) 2 / ( 4 ( m + 1 ) − 1 )$⁠, where time step $Δ t = 30$ s, integer $m = ⌊ t tot / Δ t ⌋$⁠, and $t tot$ is the total time to acquire a measurement of a time trace $N S ( t )$⁠. $T DE$ tracks the stabilized air temperature $T lab$ in the laboratory well. For example, $T lab = 295.2 ( 1 )$ K and $T DE = 295.3 ( 3 )$ K for the data shown in Fig. 2.

The temperature of the l-CAVS vacuum chamber is found by averaging the readings of four PRTs, in a manner identical to that of $T DE$⁠. Oscillations in the cooling water temperature for the electromagnets²³ that generate the l-CAVS quadrupole magnetic field causes the temperature of the l-CAVS vacuum chamber to oscillate with an amplitude of up to 0.5 K. No temperature change is observed due to the application of current in the electromagnets. This leads to a standard uncertainty of $u ( T ) = 0.3$ K for the l-CAVS, while T and $T DE$ typically agree within their uncertainties.

The temperature of the p-CAVS vacuum chamber is found by averaging the readings of two PRTs, in a manner identical to that of $T DE$⁠. When the p-CAVS is turned on, we empirically observe that its temperature has a time dependence $T ( t ) = T 0 + Δ T [ 1 − exp ( − γ t ) ]$⁠, with $T 0 ≈ T DE , Δ T ≈ + 5$ K, and $1 / γ ≈ 1$ h. The temperature increase is caused by the effusive lithium source dissipating roughly 3 W of heat to evaporate lithium. Because the outside of the p-CAVS vacuum chamber is heated above the laboratory temperature, we reasonably assume that the inside is even warmer. Indeed, measurements with a separate, identical p-CAVS with an in-vacuum thermocouple suggest that the interior of the vacuum chamber is 1 K warmer than the exterior-mounted PRTs measure. We conservatively take $u ( T ) = | T − T DE | / 2 ≈ 2.5$ K for the p-CAVS.

We use a turbo-molecular pump attached to the second chamber with a finite pumping speed $S ≈ 1500$ l/s to evacuate the dynamic expansion system leaving a small residual pressure in this chamber and thus allowing some particles to return to the first chamber. Equation (A2) is derived under the assumption that particles do no return, i.e., assuming $S → ∞$⁠. We can correct for the finite S by measuring the pressure ratio $r p$ of the pressure in the first chamber to the pressure in the second chamber and using the substitution $C 0 → C 0 ( r p − 1 ) / r p$ in Eq. (A2). Our measurement of $r p$ is described in Ref. 18. We give a brief synopsis here. A spinning rotor gauge (SRG) is connected via pneumatically actuated valves to either the first or the second chamber. The SRG's decay rate, which is a proxy for the pressure, is measured sequentially as it is connected to the first and second chamber. The ratio of these decay rates corresponds to $r p$⁠. Accurate measurements of $r p$ require pressures in the first chamber between 0.1 and 0.6 Pa to obtain sufficient signal. At these pressures, the non-linear conductance of the orifice needs to be accounted for and we measure pressure ratios at several pressures and linearly extrapolate to zero pressure. The dominant uncertainty in this measurement is statistical and is typically $u ( r p ) / r p = 0.02$⁠.

Before we conclude this Appendix, let us consider the potential for pressure gradients within the DE system at base pressure. Differences in measured pressure at base pressure between the two CAVSs could be caused by local differences in the specific outgassing rate combined with differences of the effective vacuum conductance from each of the CAVS to the orifice. Considering solely the latter, Monte Carlo simulations assuming uniform specific outgassing throughout the first DE chamber and the two CAVSs show that the l-CAVS should be at a 25% higher pressure than the p-CAVS because of the former's slightly longer connection to the DE chamber. We note that there is no guarantee that the specific outgassing of chamber walls is uniform; factors of 3–5 difference in local outgassing rates are reasonable and might explain our observations at base pressure. Over the duration of our experiment, such imbalanced outgassing is stable. By contrast, Monte Carlo simulations of the added inert gasses, injected into the DE chamber at a specific point, show that their partial pressure is uniform to within the simulations' uncertainty when the chamber is at uniform temperature.

Our imaging system is a potential source of uncertainty in both the MOT atom number N₀ and the number of sensor atoms in the magnetic quadrupole trap $N S ( t )$ at hold time t. As described in the introduction to Sec. II, the experiment has several steps for each hold time t: An atom cloud is prepared in the MOT, subsequently held in the magnetic trap for a time t, and then atoms are recaptured into the MOT. We take images before we load the MOT, at the moment when the MOT is fully loaded, and then after the recapture of the atoms in the MOT. In the end, we store and analyze six images for each hold time t. Specifically, before the MOT loading stage, a first image with neither the atoms nor lasers present and a second image with the MOT lasers but no atoms present are taken. At the end of the MOT loading stage, the third image is taken and we turn off the MOT light. These three images determine N₀. This step is nondestructive. After the recapture of the sensor atoms at time t, we then take three more images, spaced in time about 0.3 s apart. The first is an image with the MOT lasers on and sensor atoms present, the second an image with the MOT lasers on and no atoms present, and finally, an image with neither laser nor atoms. The latter three images determine $N S ( t )$ and is destructive.

We process or combine each set of three images using a procedure similar to that described in Appendix A of Ref. 37, to account for “dark counts” and differences in MOT laser intensities, and construct sensor atom number densities. We then calculate N₀ or $N S ( t )$⁠. For mathematical convenience, we label an image with (1) neither the atoms nor lasers present, (2) an image with the MOT lasers on and no atoms present, and (3) an image with the MOT lasers on and sensor atoms present.

We finally consider correlations between sensor atom number density $n S ( x → )$ and t, or, equivalently, correlations between the shape of $n S ( x → )$ and $N S ( t )$⁠. Most easily inferred from Eq. (B2), the sensor atom number is proportional to a three-dimensional integral with an integrand that is the product of $n S ( x → )$ and scattering rate $R ( x → , I )$⁠. The spatial dependence of $R ( x → , I )$ can be found by generalizing Eq. (B6). We include spatially dependent Zeeman shifts in the detuning Δ and a spatially dependent laser intensity. Combined with the variation of the shape of $n S ( x → )$ with $N S$⁠, this produces a systematic relative uncertainty in our calculated $η S ( t )$ of <3%. This “imaging stability” uncertainty is propagated through the fitting described in Secs. III and IV.