Investigating electromagnetically induced transparency spectral lineshape distortion due to non-uniform fields in Rydberg-atom electrometry

In recent years, Rydberg-atom spectroscopy has been a fruitful method for making traceable measurements of radio-frequency (RF) electric (E) field amplitudes.¹ In these sensors, Rydberg-atom energies are observed via electromagnetically induced transparency (EIT). With on-resonance RF fields, spectral lines can be split with applied resonant electric fields via the Autler–Townes (AT) effect, and with off-resonance ones, lines are shifted in energy by AC Stark effects. By measuring these effects carefully, the RF field amplitude^2–8 can be measured traceable to the International System of Units (SI),⁹ as well as polarization,^10,11 and phase^12,13 of an RF field, yielding numerous applications.¹ 

Because of their large dipole moments, Rydberg atoms respond sensitively to an incident RF E-field. The typical method in Rydberg-atom electrometry is to utilize EIT/AT schemes to read out the response of the atoms. There are various EIT schemes that have been used. In this paper, we experimentally and theoretically study spatially inhomogeneous RF field sensing to comprehensively understand the RF field variations that frequently occur in experimental efforts. We examine both the two-optical photon and three-optical photon schemes shown in Fig. 1, where (a) corresponds to a cesium $133$Cs system and (b) corresponds to a rubidium $87$Rb system.

An example of an experimental EIT signal when no RF E-field is applied as a function of coupling laser detuning (⁠ $Δ c$⁠) is illustrated in Fig. 2 (black trace with a peak at $Δ c=0$⁠). These experimental data are for the three-photon scheme shown in Fig. 1(b), and details on this experiment are given in Sec. IV. When an on-resonant RF field is applied, AT splitting of the EIT signal occurs; see Fig. 2 (green trace with well defined peaks at $Δ c/2π≈±160$ MHz). In these types of measurements, the atom-interaction region (i.e., the region where the RF field is measured) is typically a long cylinder for the case of counter-propagating EIT schemes, where the cylinder has a diameter that is the laser beam size, and the length is the region where the beam encounters the atomic vapor. The data shown in Fig. 2 (labeled “Uniform field”) correspond to the case when the RF field is uniform across the laser beam propagation path (the atom-interaction region); as such, we see symmetric Gaussian/Lorentzian line shapes. When the RF field is non-uniform across the atom-interaction region, the EIT spectra become distorted. Figure 2 (the trace labeled as “Standing wave”) shows the spectra for a RF standing wave across the laser propagation path (i.e., a non-uniform E-field across the atom-interaction region). This standing-wave distribution is a result of how the RF is applied; see Sec. IV. For the non-uniform field case, the spectrum is distorted; i.e., it is broadened with additional peaks. These non-ideal line shapes affect the ability to make accurate E-field measurements. In fact, RF tuners (i.e., matching devices) have been used to remove standing-wave effects in waveguides for the purpose of improving and/or correcting the lineshape distortion of EIT signals for accurate RF power measurements in Rydberg sensors.¹⁴ 

The sensitivity and accuracy of an E-field measurement are directly related to the ability to precisely determine the AT splitting⁹ or the beat-note strength when a local oscillator is used in a Rydberg mixer scheme;⁸ for example, the difficulty in determining the AT peak location is part of the uncertainty budget in E-field measurements.¹⁵ The standing-wave result (red trace in Fig. 2) clearly indicates the difficulty in determining the AT splitting of an E-field measurement when compared to the uniform field case (green trace in Fig. 2). In particular, the applied E-field strength cannot be determined accurately with AT splitting when the two-peak structure is present as the RF standing-wave case illustrates.

A major thrust in Rydberg-atom electrometry is the development of deployable sensors with fundamental limits on accuracy and sensitivity for applications, such as metrology-grade measurements and for atom-based receivers. To achieve this, it is imperative to understand the sources of distortion of the EIT line shape under different operational conditions. While atomic spectra distortions can be caused from various sources, in this paper, we focus on the effects from non-uniform fields.

The non-uniformity in the E-field can manifest from several sources, ranging from the RF field interactions with the dielectric vapor cells to internal surface charges in the vapor cells. The RF fields can scatter off of local environmental features and even reflect within an atom vapor cell,^16,17 causing the RF field to be non-uniform over the measurement volume. This non-uniformity generally leads to frequency broadening and additional peaks in the spectrum, which results in a loss of accuracy and signal amplitude. Surface charges are a result of some manufacturing practices and materials and/or are caused by ionization of the atomic vapor inside the cell. While one would like to remove these sources of non-uniformity, this is not always possible. Consequently, understanding their effects on the EIT spectra is important. To this end, we develop a model to understand these effects and investigate some of the more common field non-uniformities experienced in practice; for example, we show results for (1) a linear gradient in the field across the laser beam propagation path (this type of non-uniformity is present when plate electrodes are used in voltage measurements¹⁸ and other applications^19–21), (2) an RF standing-wave distribution of the E-field (this type of non-uniformity results from vapor cell effects and for some non-ideal applied fields^14,17), and (3) a stepwise distribution in the RF field across the beam propagation path,²² as well as a few others.

We present a model for calculating EIT spectra through spatially varying electric-field amplitudes along the longitudinal beam propagation path by discretizing those changing fields and combining the consequent local transmission values into an aggregate spectrum. This discretization method can generally be applied to any optical parameter; it appears implicitly in many spectral curves and associated with various applications. In this paper, we consider spectral line broadening due to spatial non-uniformity in an RF electric-field amplitude as measured by the EIT/AT method. Our work suggests that prior works where EIT/AT broadening was observed^{7,15–18,23–29} might be reinterpreted in terms of spectral broadening caused by a spatially non-uniform E-field amplitude. In fact, the anomalies observed in Ref. 30 are attributed to non-uniform field effects. Here, we summarize several characteristic types of distortion and broadening of the EIT signals, each associated with a distinct electric-field inhomogeneity. These distorted spectra provide a diagnostic tool that can be used to quantify novel field sources or eliminate experimental imperfections. In particular, rather than losing precision due to EIT lineshape distortions, this technique can yield more information about a particular field’s spatial distribution.

In Sec. II, we present the background of typical absorption calculations and our modifications to account for a spatially varying field. In Sec. III, we illustrate the principle with a number of example spectra calculated for various field arrangements using this method. In Sec. IV, we compare the model to measured data observed in different vapor cell arrangements, i.e., a vapor cell embedded in a transmission-line feed and in a vapor embedded in a rectangular waveguide. These comparisons demonstrate the model’s ability to characterize field distributions and its accurate prediction of the EIT line shapes in non-uniform fields. We conclude in Sec. V. Furthermore, in the  Appendix, we describe the master equation underlying the optical calculations.

The Rydberg resonant field measurement scheme has been used previously to give traceable measurements of electric-field amplitudes.^3,9,15 The corresponding simulation of transmission for a four-level system is well-documented,^31,32 and more recently, five- and six-level systems^23,33–38 have been developed. Precision measurements using spectroscopy are often limited by the linewidths observed, which can become broadened by non-uniform energy shifts over the observation volume. Here, we focus on the adverse effects on the EIT line shapes due to variations of the RF electric-field amplitude $|E(x)|$ along the laser propagation path, i.e., the $x$ axis in our case as shown in Fig. 3.

To illustrate the model for both the two-photon $133$Cs scheme and the three-photon $87$Rb schemes shown in Fig. 1, we first concentrate on the former. The three-photon scheme follows very similarly. In principle, the effect of the E-field non-uniformity of the EIT line shape would be no different when utilizing either two-photon or three-photon schemes. Nevertheless, we show results (modeling and experiments) for both a two- and three-photon scheme because we have sensors that utilize both types. These two sensors exhibit different types of E-field non-uniformities due to their geometries and, as such, give us two different and independent types of data to compare our model against.

All of the parameters in this single-medium Beer’s law absorption model are assumed fixed over the entire length $L$⁠. When they vary across the sample region (such as the variability in the applied RF field), further analysis is required to capture the distortion in the EIT line shape: like the distorted EIT line shown in Fig. 2 for the standing-wave case. To model a non-uniform $|E(x)|$ field across the laser beam propagation path, e.g., as depicted in Fig. 3, we use a multi-layer media approach. In this approach, we have discretized space along the laser propagation path (the $x$ axis) into $N S$ segments of length $dL=L/ N S$⁠, and we calculated the transmission profile (the EIT signal) for the local $|E( x i)|$ amplitude over each of them.

We investigate different E-field distributions along the $x$ axis. An example of the EIT signal for each segment for a linear gradient in the field is shown in Fig. 4. In this example, the optical fields are propagating along the $x$ axis, and the E-field amplitude has a linear variation inside the vapor cell (as would be indicative for the case of non-parallel-plate electrodes placed across a vapor cell) along the $x$ axis as well. These results are for a linearly increasing $|E(x)|$ from 5 to 15 V/m over the laser propagation length of 75 mm, i.e., the length of the vapor cell. Such a linear variation can occur in some situations and could be produced by non-parallel plates when Rydberg sensors are being used for voltage measurements¹⁸ or when parallel-plate electrodes are used in other applications.^19–21 

It is important for the sampling density to converge to a smooth physical curve, particularly since sparse sampling leads to large jumps in the peak position from $Ω R F(x)$ to $Ω R F(x+dL)$ (i.e., the changes in the E-field strength across the vapor cell) will lead to errors in the calculated resultant curves. The sampling number $N S$ can be raised to an arbitrary spatial resolution, which is useful for the narrow linewidths typically used in precision experiments.

We note that this broadening gives some additional information about the field non-uniformity experienced by the atoms, i.e., giving many local measurements of the Rabi frequency.³⁹ These types of measurements might be especially useful as field monitors within other systems, such as our motivating case of low-loss power monitoring within a traditional waveguide. These new peak features can be fit for parameters, such as standing-wave amplitude, field gradient, etc., by fitting observed line shapes to ansatz functions. We also note that one potential optimized calculation would be to pre-calculate the spectral curves across a range of $|E|$ values (even interpolating curves). One can then invoke these ansatz functions to combine curves with varying weights rather than calculating an arbitrary function $E(x)$ each time.

We now provide samples of some common field distributions found in practice and use our simulation method outlined in Sec. II to illustrate the expected probe transmission curves, i.e., the EIT signals. As indicated in Fig. 3, the laser beams are propagating along the x axis. The E-field amplitude has a prescribed variation along the x axis inside the vapor cell over its length: $x=0$ to $x=L$⁠. The modeled results are shown in Fig. 5. Each sub-figure of Figs. 5(a)–5(f) shows the total transmission plot (top left) aggregated over the entire cell length. Its horizontal axis is the coupling detuning $Δ c$⁠, and the vertical axis is the probe transmission after propagating across a prescribed E-field amplitude variation inside the vapor cell. The heatmap surface plot (bottom left) gives the transmission spectrum as a function of the detuning (horizontal heatmap axis) and position along the axis of the vapor cell (vertical heatmap axis). The E-field amplitude along the cell is plotted as well (bottom right), where the horizontal axis is the E-field amplitude and the vertical axis specifies the x-axis position it is acquired in the vapor cell. Each field distribution has a maximum of 15 V/m for direct comparison.

The uniform or “ideal” case is shown in Fig. 5(a). This is synonymous with the single-segment Beer’s law approach; it is used as the reference linewidth for the other line shapes. The stepwise case is shown in Fig. 5(b). This case is physically realized in transverse waveguide probing,²² where atoms are observed in pinholes before and after the central waveguide region in which a constant field is present. We see a large unperturbed peak at the center from the section with no RF field, as well as the expected split AT peaks, with roughly half the height of the center peak (at 1/3 and 2/3 of the total length $L$⁠).

The linear gradient case is given in Fig. 5(c), with the net transmission spectrum of the model and curves being shown in Fig. 4(b). This linear field gradient case represents a first-order modification of the line broadening. It demonstrates that the loss in peak transmission, as well as broadening, characterizes effects from field non-uniformity. For experimental sources, which cause a slight field gradient, the gradient usually grows in proportion to the field amplitude. This effect causes the linewidths to grow with the applied power. This case illustrates, to a first-order approximation of a gradient in the field, that the total FWHM observed is approximately the sum of the total range of $Ω R F(x)$ induced by $|E(x)|$⁠, plus the non-broadened EIT linewidth.

The $1/x$ case is shown in Fig. 5(d). It represents the typical E-field falloff from a generic antenna, which is typically employed transverse to the sampling path, not along it. Note that the asymmetric peak shape is weighted to the lower end of the field range.

The sine- or standing-wave case is shown in Fig. 5(e). This field motivated the present investigation. It accounts for the line broadening as the E-field is measured longitudinally within an un-matched waveguide. This EIT signal is similar to that observed in the experimental data shown in Fig. 2. Note the outside “devil horn” characteristic; it is a result of the sine’s sampling density near the extrema.

The case of an arbitrary standing-wave pattern is shown in Fig. 5(f). This type of field distribution occurs, for instance, in a rectangular waveguide closed with two glass slab ends, which cause poor matching between the interior and exterior regions.^14,22 Significantly higher sampling density is required, owing to the spatial gradients involved.

The simulated EIT curves given in Fig. 5 can be used to understand what types of field non-uniformity are present when observing distorted EIT line shapes in experimental data. In fact, the choices of model parameters used in Fig. 5 are motivated by the experimental observed line shapes given in Sec. IV.

In order to validate the application of our theoretical model to experimental results, we show a few different experimental examples compared to EIT line shapes predicted by it.

As the first example, we investigate the effects of longitudinal (along the laser propagation direction) field non-uniformity in a vapor cell placed in a twin-lead waveguide, as depicted in Fig. 6(a). The vapor cell and waveguide are taped to a block of styrofoam in order to hold the device during the experiments. Note that this type of waveguide structure is one method in which a local oscillator field (LO) can be applied to the atoms.¹¹ This LO approach is a typical method used for the detection of the phase of an incident RF field and has been used for weak field detection via a Rydberg atom-mixer.^8,12,13 The twin-lead wires are attached at each end (defined as port 1 and port 2) to two channels of a single RF source through custom-designed baluns, which convert the twin-lead waveguide to an unbalanced 50  $Ω$ impedance. If the excitation of the twin-lead waveguide is imperfect, RF standing waves can develop along the vapor cell, which then cause a non-uniform field along the vapor cell. These effects are measured and described in detail next.

We use the three-photon scheme shown in Fig. 1(b) to generate EIT in the cell and to demonstrate the RF standing-wave effects on the EIT line shape. Figure 6(b) shows the diagram of the measurement setup. It indicates the propagation directions of its three lasers, which consist of a 780 nm (probe) laser beam counter-propagating with respect to the 776 nm (dressing) and 1266 nm (coupling) laser beams. We use a differential detection scheme to measure the transmission (the EIT signal) of the probe laser, where the difference measurement [from two photodiodes (PD)] is made between the signal 780 nm and a reference is separated in the cell as depicted in Fig. 6(b); further details can be found in Ref. 40. In this experiment, the full-width at half maximum (FWHM) beam diameters for the probe, dressing, and coupling lasers, respectively, are 1.38, 1.32, and 1.39 mm. The corresponding powers of the three lasers are 129.95  $μ$W, 12.67 mW, and 214.73 mW. These values correspond to Rabi frequencies of the probe, dress, and coupling lasers of $Ω p=2π⋅10.02$ MHz, $Ω d=2π⋅32.52$ MHz, and $Ω c=2π⋅7.28$ MHz, respectively. In these experiments, the RF E-field and all three optical fields are co-linear polarized, with the E-field vectors pointing from one wire to the other. The schematic in Fig. 6(c) illustrates the interfaces that can give rise to RF reflections that contribute to field inhomogeneities.

As discussed previously, Fig. 2 shows the EIT signal as a function of the coupling laser’s detuning (⁠ $Δ c$⁠) for the case of no RF signal (black curve) on either port of the twin-lead waveguide. The effects of longitudinal field non-uniformity resulting from the RF standing wave on the line can be seen readily by applying an RF signal to ports 1 and 2. We apply $V 1=A sin(ωt)$ and $V 2=B sin(ωt+ϕ)$ to ports 1 and 2, respectively, where $ω=2π⋅1.906$ GHz is at the $35 F 7 / 2⟺35 G 9 / 2$ resonant transition in $87$Rb.

We begin by considering the RF signal applied only at port 1, with $V 1$ chosen to produce an $E 1$ across the vapor cell to be large enough to well-resolve the resulting EIT/AT features. The measured EIT signal for this case is shown in Fig. 7(a) [the blue curve]. We observe cusped features, i.e., two peaks on both EIT lines, and attribute them to longitudinal field non-uniformity in the cell. In fact, upon comparing the shape to the modeled data in Fig. 5(e), we see that a sinusoidal field distribution gives similar EIT line shapes as those observed experimentally. The sinusoidal field distribution results from a standing wave within the twin-lead waveguide. The standing wave is caused by several factors as illustrated in Fig. 6(c). First, since the vapor cell is made of a dielectric, it will cause reflections of the $V 1$ signal as it propagates alone the waveguide and interacts with the vapor cell. Second, the impedance mismatches due to the termination of the twin-lead at the two ports will also cause reflections. Third, as seen in Fig. 6(a), the twin-leads are not parallel along the propagation path, and they are curved around the foam block. This leads to impedance mismatches along the transmission-line propagation path and results in additional reflections. All these sources of reflections and impedance mismatches cause a standing-wave voltage waveform and, more importantly, a corresponding $E$-field (⁠ $E 1$⁠) standing-wave distribution inside the vapor cell (i.e., the magnitude of the E-field seen by the atomic vapor). Furthermore, since the 75 mm vapor cell is near the $λ/2$ length for this RF-frequency (79 mm at 1.906 GHz), we attribute an element of these features as arising from the longitudinal non-uniformity due to partial backreflection(s) inside the vapor cell.

This standing-wave feature can be minimized by injecting a signal into port 2 (i.e., $V 2$⁠) in order to cancel the reflections of $V 1$⁠. This idea is similar to using the RF tuner to cancel the reflections in the Rydberg-atom waveguide-power measurements,¹⁴ where it is shown that RF tuners can cancel out standing-wave effects and eliminate the EIT lineshape distortion. As such, we find that by applying $V 2< V 1$ and varying $ϕ$⁠, we can dramatically modify the line shape to the extent that the standing-wave effects are eliminated. By iteratively tuning $V 2$ and the inserted phase (⁠ $ϕ i$⁠) of the RF signal at port 2, we found that we are able to minimize the AT lineshape distortion as illustrated by the solid green trace in Fig. 7(a). These optimal values correspond to $V 2= V 1/5.54$ and $ϕ= 290 °$⁠. For this case, we see that the EIT/AT peaks are non-distorted and have nearly the Gaussian/Lorentzian line shapes we expect in a uniform RF field. At this optimized $V 2$⁠, we found maximal distortion is obtained when a $180 °$ phase shift is added, i.e., when $ϕ i+ 180 °$⁠. This effect is illustrated with the solid red trace in Fig. 7(a). In effect, the added $180 °$ phase induces a larger standing wave of the field by coherently adding to $V 1$’s reflection. These results show that we can constructively or destructively cancel backreflections by introducing a smaller $V 2(t)$ at port 2 and, in effect, cause a non-uniform field to become uniform across the Rydberg sensor.

The E-field distribution inside the vapor obtained from the magnitude of Eqs. (4) and (5) forms a standing wave, shown in Fig. 7(b). These field distributions were used in the model presented in Sec. II, and the resultant EIT/AT signals are shown in Fig. 7(a). Upon comparing the modeled and experimental data, we see that the model predicts the EIT/AT lineshape distortion very well and indicates that the distortion of the measured EIT line is due to the standing wave present in the twin-lead structure.

The second example is related to using a Rydberg atom-based sensor to perform SI traceable measurements of RF power in a rectangular waveguide.^14,22 This waveguide used here was designed to allow laser propagation either along the $x$ axis or the $z$ axis; see Fig. 8. Details about the measurements described here are given in Ref. 22. The rectangular waveguide shown in Fig. 8(a) is filled with $133$Cs. The first set of data is obtained with the probe and coupling lasers propagating along the $x$ axis, and the E-field is measured along the $x$ axis inside the waveguide via windows on the top and bottom of the waveguide. This configuration is illustrated in Fig. 8(b). The laser beams are propagating orthogonal to the RF field propagation direction. For this case, the RF E-field is polarized along the $x$ axis, and both optical fields are polarized along the $z$ axis. Due to this configuration, we see from Fig. 8(b) that the atoms have a region that is exposed to a nearly constant RF field and two regions that are not exposed to the RF field. Thus, the atoms are exposed to a field distribution similar to the step-function distribution given in Fig. 5(b).

We use the two-photon $133$Cs scheme shown in Fig. 1(a) to generate EIT in the waveguide and to measure the RF field distribution effects on the EIT line shape. A 852 nm (probe) laser beam is counter-propagating with the 511 nm (coupling) laser beam.

In these waveguide experiments, the nominal beam parameters are FWHM beam diameters for the probe and coupling lasers are 80–120  $μ$m and 400–600  $μ$m, respectively. The powers in the lasers are 20–100  $μ$W and 50–100 mW for probe and coupling, respectively. These values correspond to Rabi frequencies for the probe laser in the few MHz range and for the coupling laser on the order of 10’s of MHz.

Figure 9 shows the measured EIT/AT signal. Some percentage of the atoms remain unaffected by the field in this case (caused by the zero-field region) and, thus, exhibit the resonant EIT peak at $Δ c=0$⁠. On the other hand, a portion of the atoms in the field are AT-split as usual. A stepwise field distribution is used with the model in Sec. II, and its predicted results are also given in Fig. 9. A good correlation between the two is observed. There are additional asymmetric effects that distort the spectrum beyond the non-uniformity in the E-field, including spurious EIT peaks resulting from optical reflections off of the viewports, i.e., the optical windows at the end of the vapor cell.

The last example considers a longitudinal standing wave along the $z$ axis in the waveguide shown in Fig. 8(c). In this case, the RF E-field and both optical fields are co-linear polarized along the $x$ axis, and the RF and optical fields are propagating along the $z$ axis. A comparison between experimental and modeled data for this case is shown in Fig. 10. The presence of the inserted glass windows in the waveguide structure holds the atoms and induces a standing wave in the traveling microwave signal. We simulate the field inside of the waveguide using an EM solver for three different power values, each exhibiting varying levels of broadening. We note that a constant offset of $≈11$ dB between the measured and modeled curves accounts for the significant insertion loss into the waveguide when directional couplers were used in the laboratory. In contrast, the model was fed by an appropriate waveguide mode. Again, we see a good correlation between the experimental and modeled data. Consequently, the distortions in the EIT signals are indeed due to an RF standing wave. Holloway et al.¹⁴ have demonstrated that these distortions of the EIT signals due to the standing wave caused by the end windows can be eliminated, or at least reduced, with RF stub-like tuners.

We have presented a computational method to approximate the distortions of Rydberg-EIT line shapes due to spatially non-uniform E-field amplitudes inside a vapor cell. We have segmented the optical path, calculated the local transmission values, and then combined these into a composite transmission spectrum. The calculation method is generic and applicable to any alteration in the electric susceptibility parameters when the total transmission or phase delay is monitored. It can, therefore, help bridge the gap between observed line shapes and fitting theory curves. We validate our modeling method by demonstrating good agreement between the simulated and experimental atomic spectra in several well defined RF environments. Rather than losing information from broadening, this method enables extracting additional information about the spatial field variations that can be fit to observed transmission spectra. The importance of the simulated EIT curves given in Fig. 5 is that they reveal characteristic lineshape effects based on the type of underlying field inhomogeneity present in the experimental EIT data. Understanding these observed types of distorted EIT line shapes can facilitate the ability to control and correct effects arising from such nonuniformities. Figure 5, thus, acts as a “Rosetta stone,” translating spatial RF field distributions to EIT/AT atomic spectra. Those results give us a means to understand the effect of different non-uniformities and in turn to achieve fundamental limits on the accuracy and sensitivity for applications, such as metrology-grade measurements and for atom-based sensors and receivers.

This work was partially funded by the NIST-on-a-Chip (NOAC) Program and was developed with funding from the Defense Advanced Research Projects Agency (DARPA). The views, opinions, and/or findings expressed are those of the authors and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government. This work is a contribution of the U.S. Government and is not subject to copyright in the United States.

The authors have no conflicts to disclose.

Andrew P. Rotunno: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Christopher L. Holloway: Conceptualization (lead); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Nikunjkumar Prajapati: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Samuel Berweger: Data curation (equal); Formal analysis (equal); Investigation (equal); Validation (equal); Writing – review & editing (equal). Alexandra B. Artusio-Glimpse: Data curation (equal); Formal analysis (equal); Methodology (equal); Visualization (equal); Writing – review & editing (equal). Roger Brown: Data curation (equal); Formal analysis (equal); Investigation (equal); Validation (equal); Writing – review & editing (equal). Matthew Simons: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Validation (equal); Writing – review & editing (equal). Amy K. Robinson: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Baran N. Kayim: Data curation (equal); Investigation (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Michael A. Viray: Data curation (equal); Formal analysis (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Jasmine F. Jones: Data curation (equal); Formal analysis (equal); Investigation (equal); Validation (equal); Writing – review & editing (equal). Brian C. Sawyer: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Visualization (equal); Writing – review & editing (equal). Robert Wyllie: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Thad Walker: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Richard W. Ziolkowski: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Steven R. Jefferts: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Steven Geibel: Data curation (equal); Formal analysis (equal); Investigation (equal); Validation (equal); Writing – review & editing (equal). Jonathan Wheeler: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Eric Imhof: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (supporting); Visualization (equal); Writing – review & editing (equal).

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