††thanks: Publication of the U.S. government, not subject to U.S. copyright.

#  Electromagnetically induced transparency based Rydberg-atom sensor for
quantum voltage measurements

Christopher L. Holloway  National Institute of Standards and Technology,
Boulder, CO 80305, USA<EMAIL_ADDRESS>Nikunjkumar Prajapati
National Institute of Standards and Technology, Boulder, CO 80305, USA
Department of Physics, University of Colorado, Boulder, CO 80305, USA  John
Kitching  Jeffery A. Sherman  Carson Teale  Alain Rüfenacht  Alexandra B.
Artusio-Glimpse  Matthew T. Simons  National Institute of Standards and
Technology, Boulder, CO 80305, USA  Amy K. Robinson  Department of Electrical
Engineering, University of Colorado, Boulder, CO 80305, USA  Eric B. Norrgard
National Institute of Standards and Technology, Gaithersburg, Maryland 20899,
USA

###### Abstract

We investigate the Stark shift in Rydberg rubidium atoms through
electromagnetically induced transparency for the measurement of direct current
(dc) and 60 Hz alternating current (ac) voltages. This technique has direct
applications to atom-based measurements of dc and ac voltage and the
calibration of voltage instrumentation. We present experimental results for
different atomic states that allow for dc and ac voltage measurements ranging
from 0 V to 12 V. A Rydberg atom-based voltage standard could become an
alternative calibration method with more favorable size, weight, power
consumption, and cost compared to the more precise Josephson voltage standard.
In this study, we also demonstrate how the voltage measurements can be
utilized to determine the atomic polarizability for the Rydberg states. The
Rydberg atom-based voltage measurement technology would become a complimentary
method for dissemination of the voltage scale directly to the end user.

##  I Introduction

Rydberg atoms (atoms with one or more electrons excited to a very high
principal quantum number $n$ gal ) in conjunction with electromagnetically-
induced transparency (EIT) techniques have been used to successfully detect
and fully characterize radio frequency (RF) electric (E) fields r14 ; gor1 ;
sed1 ; holl1 ; holl2 . In these applications, the E-fields are detected using
EIT both on resonance as Autler-Townes (AT) splitting and off-resonance as ac
Stark shifts. This approach has the capability of measuring amplitude gor1 ;
sed1 ; holl1 ; holl2 ; tan1 ; r5 ; gor2 ; gor3 , polarization sed2 ; access ,
and phase sim3 ; jing1  of the RF field and various applications are beginning
to emerge. These include E-field probes holl1 ; holl2 ; gor2  traceable to the
International System of units (SI), power-sensors holl5 , spectrum analyzers
army , angle-of-arrival detectionaoa , and receivers for communication signals
(AM/FM modulated and digital phase modulation signals) song1 ; meyer1 ; holl6
; cox1 ; holl4 ; anderson2 ; deb3 ; holl7 .

In this manuscript, we investigate the use of Rydberg-atom sensors to develop
a quantum-based voltage standard. Here, we measure the voltage induced between
two parallel plates embedded in an atomic vapor cell (see Fig. 1) by measuring
the Stark shifts in the atomic spectra of Rydberg atoms. By collecting a
series of measurements of the Stark shift for different voltages, we can make
a calibration curve and demonstrate a voltage standard. We discuss various
issues that must be considered in order for Rydberg-atom-based-sensors to
accurately and reliably function as voltage measurement standards.

Figure 1: Cylindrical vapor cell with stainless-steel parallel plates. The
vapor cell is 50 mm in length and has a outside diameter of 25 mm. The plates
are rectangular in shape with width of 18 mm, length of 45 mm, and placed
$\approx 2$ mm apart.

The primary voltage standard implemented for the realization of the SI volt is
based on the Josephson effect. Arrays of Josephson junctions, when cooled to
cryogenic temperatures and biased on their quantum locking range transduce an
accurately synthesized microwave frequency source into fixed or digitally
programmable voltages jose . Comparisons between such voltage standards
demonstrate accuracies at the level of 1 part in $10^{10}$ with about 100 s of
averaging (see, e.g. refa ). Scaling up the voltage output of Josephson-
junction based standards requires either increasing the number of junctions in
the array, or increasing the microwave driving frequency, or both refb . With
some exceptions (see, e.g. refc ), 10 V is the commonly adopted dc reference
for the primary realization of the voltage scale. Commercial instruments (like
voltage calibrators) used for the routine voltage calibration process and
voltage dividers implemented to expand the range to high voltages typically
have an accuracy on the order of 1 part in $10^{6}$ refd . Due to their large
cost ($\gtrsim~{}$$$300$k) Josephson voltage standards limit their deployment
to national metrological institutes (NMIs) or primary calibration
laboratories.

Zener diodes are the most commonly used voltage references that are
implemented with commercial instrumentation, ranging from high performance
secondary voltage standard ($>~{}$$10k) z1 ; z2  to the 0.1 % accuracy
handheld multimeters ($\approx~{}$$100). All Zener-diode voltage references
drift with time and require periodic calibration, that involves a comparison
with a more accurate voltage reference or another verification of the
manufacturer specifications. Zener-diode references are sensitive to
temperature, humidity, mechanical stress and shock, and power supply noise z1
.

Rydberg atom sensors, that rely on the atom’s predictable response to electric
field to measure voltage could become an alternative — in terms of size,
weight, power and cost (SWaP-C) — to Zener voltage references and, with lesser
accuracy, to Josephson voltage standards. For example, polarizability
measurements with low-lying excited states of alkaline-earth atoms show total
inaccuracy at the level of 2 parts in $10^{5}$ refe ; reff . Due to a large
dipole moment, Rydberg atomic states have high sensitivity to electric fields,
high intrinsic response bandwidth, and can be produced by laser excitation in
large quantities within simple vapor cells at or near room temperature. In
principle, the atomic response can be made free from environmental coupling,
which would make Rydberg atom sensors intrinsically stable, eliminating the
need of subsequent periodic external calibrations.

In this paper, we discuss the various nuances of Rydberg-atom-based voltage
measurements [e.g., the effects of energy level crossing and non-parallel
electrodes (causing inhomogeneous fields)]. While similar measurements have
been performed before das1 ; das2 ; adams2 ; oster , our study focuses on the
aspects that will need to be addressed and controlled for a Rydberg-atom
sensor to be used as a voltage measurement standard. We also demonstrate two
methods for measuring ac voltage amplitude at 60 Hz. AC Stark effect
measurements have been performed in Ref. das1 , where it was shown that indium
tin oxide electrodes where limited to 40 kHz and nickel electrodes operated
down to dc. Our cells achieve dc and low frequency sensing with stainless
steel plates. Finally, we discuss how accurate measurements of the Stark
shifts and the plate separation can lead to accurate estimates of the
polarizability for Rydberg states.

##  II Voltage Measurements

A Rydberg atom-based dc (or 60 Hz) voltage measurement is performed by
observing the dc (quasi-dc) differential Stark shifts in Rydberg states, which
is given by friedrich

$\Delta=\frac{\alpha(f)}{2}\,\,E^{2}\,\,\,,$ (1)

where $\Delta$ is the Stark shift (in units of Hz), $E$ is the applied
electric field (in units of V/m) with frequency $f$ (in units of Hz), and
$\alpha(f)$ is the polarizability [in units of Hz/(V2/m2)]. For Rydberg state
$i$, the polarizabiltiy may be expressed as

$\alpha_{i}(f)=\sum_{j}\frac{|\mu_{ij}^{z}|^{2}}{h}\Big{(}\frac{1}{f_{ij}-f-\frac{i}{2}\Gamma_{j}}+\frac{1}{f_{ij}+f+\frac{i}{2}\Gamma_{j}}\Big{)}.$
(2)

Here, the summation includes all states $j$ which are coupled to $i$ by
electric field, $\mu_{ij}^{z}$ is the $z$ component of the dipole matrix
element, $f_{ij}$ is the transition frequency, and $\Gamma_{j}$ is the
spontaneous decay rate (in units of Hz) of state $j$, respectively. Table 1
gives the calculated dc polarizability $\alpha(0)$ for the atomic states used
in the experiments presented in this paper using Eq. (2) and dipole matrix
elements $\mu_{ij}$ taken from the Alkali Rydberg Calculator Python package
ARC2020 . This package calculates dipole matrix elements using the Coulomb
approximation (CA) and the quantum defects given in Ref. han2006 . Reference
yerokhin2016  compared the accuracy of polarizablities calculated using the CA
to calculations with the more detailed Dirac-Fock with core potential
approach. For the 28S to 47S Rydberg states considered here, the theoretical
polarizability values calculated using the CA are accurate to better than 1 %.
The accuracy of the theoretical values exceed the experimental accuracy for
these states, which is typically on the level of a few percent osullivan1985 ;
osullivan1986 . Because $f_{ij}\,\gg\,60\,$Hz,
$\alpha(f=60\,\rm{Hz})\approx\alpha_{f}(0)$. To the precision given in Table 1
($<0.1$ %), $\alpha$ is identical for $f=0$ and $f=60$ Hz.

If the E-field is generated by applying a voltage across two parallel plates
with separation $d$ (neglecting fringing field, see discussion in section VI),
then the E-field is $E=V/d$. By substituting this into equation Eq. (1), the
voltage can be found by measurement of the Stark shift:

$V=d\,\,\sqrt{\frac{2\,\Delta}{\alpha}}\,\,\,.$ (3)

From this expression, we see that there are three sources of uncertainties in
this measurement. There are two measured quantities in this expression (the
plate separation $d$ and the Stark shift $\Delta$) and one calculated quantity
(the atomic polarizability, $\alpha$). $\alpha$ can also be determined
experimentally.

The measurement of $\Delta$ can be related to in situ laser spectroscopy of
the hyperfine structure, contributing a relative precision of $10^{-7}$ or
better refhh . Relative uncertainty in $d$ of $10^{-4}$ is possible with
gauge-block construction refi  or at $<2\times 10^{-5}$ with interferometric
techniques refe ; reff . Typically, poor knowledge of $\mu_{ij}$ limits the
calculation of $\alpha$ to the $10^{-3}$ level, though in some cases $\alpha$
is experimentally determinable by other means refj . Even lacking absolute
knowledge, the quantity $d/\sqrt{\alpha}$ can be calibrated to the extent that
$d$ remains stable. For a given device, Equation 3 is cast as:

$V=C_{\text{cal}}\sqrt{\Delta}$ (4)

where $C_{\text{cal}}=2d/\sqrt{\alpha}$ is determined by comparison to a
higher accuracy voltage standard, or by use of several Rydberg states with
various $\alpha$ of lower uncertainty.

The dc and 60 Hz measurements will share a common $C_{\text{cal}}$. Other
sources of uncertainty arise from fringing fields near the perimeter of the
electrode plates, non-idealities in plate geometry and orientation, and
spectroscopic features due to interfering transitions.

Table 1: Theoretical calculation of dc and 60 Hz ac polarizabilities for 85Rb.
Note that the dc and 60 Hz ac polarizabilities are essentially the same for
the digits shown Calculated Polarizability [Hz/(V/m)2]
---
Rydberg state |  $\alpha$: dc and 60 Hz ac
$28S_{1/2}$ | 84.2
$40S_{1/2}$ | 1058.1
$47S_{1/2}$ | 3275.7

##  III Experimental Setup

In these experiments, we generate EIT in rubidium (85Rb) atomic vapor and
measure the frequency shift in the EIT signal as a function of an applied
voltage across two parallel plates. The experimental apparatus is shown in
Fig. 1. It consists of a cylindrical-vapor cell of length 50 mm and diameter
of 25 mm. Inside the cell are two stainless-steel parallel plates with a
nominal separation of $d=2$ mm. We had two of the same type of cells
manufactured and we are able to determine that the plate separation for the
two cells were approximately 2.19 mm (referred to as “Cell 1”) and 2.11 mm
(referred to as “Cell 2”), which is within the uncertainty for the plate
separation stated by the manufacturer. While the data shown in this paper are
for Cell 1, we do comment on the data from Cell 2. The plates in both cells
are not perfectly parallel to one another, which causes field inhomogeneities
across the plates. We discuss the effect this has on the EIT spectra and
voltage measurement in Section VI.

Using a Rydberg-atom EIT measurement scheme for measuring Stark shifts
involves exciting the atoms to some Rydberg state. States with zero orbital
angular momentum ($S$ states) exhibit a less complicated spectrum as a
function of applied electric field. Fig. 2 shows the calcualted Stark shift of
the 85Rb $40S_{1/2}$ state (Fig. 2a) and $40D_{m_{j}}$ state (Fig. 2b).
Compared to $S$ states, the Stark spectrum of $D$ states is typically
complicated by the fine structure, resulting in many more possible energy
level crossings (ELC) than seen in the $S$ states. We therefore focus on $S$
states as they are typically well described by Eq. 1. However, as we will see
below, for large applied voltages, ELCs do appear in the $S$-state spectra and
can influence the measurements.

Figure 2: Calculated dc Stark shift $\Delta_{f}$ in 85Rb as a function of
electric field $E$ for $40S$ and $40D$.

Figure 3: (a) Level diagram depicting EIT coupling the 5S1/2,F=3 state to a
nS1/2 Rydberg state through the 5P3/2,F’=4 intermediate state. (b)
Experimental setup for the voltage measurement and the three-level EIT scheme.

The experimental setup and the atomic levels used are depicted in Fig. 3,
which consists of a 780 nm probe laser, a 480 nm coupling laser, a
photodetector connected to an oscilloscope, a voltage source, a voltage meter,
and the vapor cell shown in Fig. 1 filled with 85Rb atomic vapor. We use a
three-level EIT scheme to generate Rydberg atoms [see Fig. 3(a)] which
corresponds to the 85Rb $5S_{1/2}$ as the ground state, $5P_{3/2}$ as the
intermediate state, and a Rydberg state of n$S_{1/2}$ state. In these
experiments we use $n=47$, $n=40$, and $n=28$. The probe laser is locked to
the D2 transition ($5S_{1/2}(F=3)$ – $5P_{3/2}(F=4)$ or wavelength of
$\lambda_{p}=780.24$ nm stackrb ). To produce an EIT signal, we apply a
counter-propagating coupling laser with $\lambda_{c}\approx 480$ nm and scan
it across the $5P_{3/2}$-$nS_{1/2}$ Rydberg transition. We use a lock-in
amplifier to enhance the EIT signal-to-noise ratio by modulating the coupling
laser amplitude with a 37 kHz square wave. This removes the background and
isolates the EIT signal.

A voltage source is connected to the electrodes on the outside of the vapor
cell shown in Fig. 1. These electrodes penetrate the cell and are connected to
the two parallel plates. The EIT signal frequency shift (the Stark shift) is
measured for different applied voltages. The voltage meter is also connected
to the electrodes in order to monitor the applied voltage. Both the voltage
sources and voltage meter were calibrated before the experiments and the
absolute voltage accuracy of any reading is 0.5 mV. In these experiments, the
optical beams and the electric fields between the two plates are co-linearly
polarized. The probe laser was focused to a full-width at half maximum (FWHM)
of 80 $\mu$m with a power of 3.6 $\mu$W, and the coupling laser was focused to
a FWHM of 110 $\mu$m with a power of 70 mW.

##  IV Experimental Data

The EIT signals for two different applied voltages are shown in Fig. 4. These
results are for the 40$S_{1/2}$ state as the coupling laser is scanned. To
increase the signal to noise ratio, we gather 20 oscilloscope traces and
average them after accounting for laser drift using an external reference EIT
signal. The two different traces correspond to the zero voltage case (black-
solid trace) and the case with 2.4 V applied (red-dashed trace). For no
applied voltage, we see two peaks, the main EIT peak at $\Delta_{c}/2\pi=0$
MHz and another at $\Delta_{c}/2\pi=-75.6$ MHz, which corresponds to the
hyper-fine structure transition $5P_{3/2}(F=3)\rightarrow 40S_{1/2}$. Since
the coupling laser is scanned, the separation between these two peaks is 75.63
MHz. The detuning of the observed peak is the adjusted hyper-fine splitting
determined by accounting for the doppler mismatch between the probe and
coupling lasers with $120.96(\frac{\lambda_{p}}{\lambda_{c}}-1)$ MHz. Here,
$\lambda_{p}$ and $\lambda_{c}$ are the wavelengths of the probe and coupling
lasers, and 120.96 MHz is the hyper-fine structure separation between
5P3/2(F=3) and 5P3/2(F=4) stackrb . The separation between the main EIT peak
and the peak corresponding to the hyper-fine structure is used to calibrate
the laser detuning and the frequency for the measured Stark shift ($\Delta$).

Figure 4: Plot of % probe EIT transmission relative to the probe absorption
plotted against the coupling laser detuning. The black-solid curve is the
transmission trace for the 40S Rydberg state when no voltage is applied across
the plates. The red-dashed curve is the transmission trace for the 40S Rydberg
state when a potential of 2.4 V is placed on the plates.

When a voltage is applied, the EIT peak shifts according to Eq. (3). When 2.4
V is applied, we see the EIT peak shift to around -800 MHz (the red trace). We
see that the width of the EIT signal is broader than when no voltage is
applied. The broadening is due to inhomogeneous fields that occur across the
two laser propagation paths. This is discussed in more detail later. We also
see asymmetries and two peaks in the observed EIT signal. These two peaks and
asymmetries are a result of both ELCs and inhomogeneous fields.

Fig. 5 shows several EIT signals for several applied voltages. This series of
EIT signals allow us to observe several interesting features. Besides Stark
shifts and line broadening, we see other features that appear in the EIT
lines, including additional structure in the EIT signal that start to appear.
The structure becomes complicated for higher applied voltages. These
structures stem from two sources. First, these structures are due to the ELC
beginning to appear in the spectra. Secondly, the source of these structures
(that eventually develops into double peaks for higher voltages) are a result
of inhomogeneities in the field across the laser beam propagation path. The
inhomogeneities in the field are mainly caused by (1) non-parallel plates, (2)
imperfections in the surface of the electrodes due to attaching lead to the
plates [as our current plates have a dimple (and discontinuities) in the very
center where the leads are mounted], and (3) fringing fields at the edge of
the plates. To confirm the plates are not parallel, we used a microscope to
measure the plate separation at the four corners. We measured the electrode
separation in each cell at each of the four corners ten times. For the cell,
these measurements (and standard uncertainties) were: 2.306(11) mm, 2.105(7)
mm, 2.278(7) mm, and 2.103(14) mm. From this data, we see that the plates are
tilted (uniformly along the length of the cell) with respect to another.

We see the EIT signals broaden with applied voltage as seen in Fig. 5. This
broadening is likely due to a non-uniform field across the plates along the
laser propagation path. Furthermore, additional broadening may be present from
local charge distributions. These are likely from ionization of the Rydberg
atoms through collision das1 . Additionally, these E-field in-homogeneities
are amplified by the fact that the Stark effect is quadratic for non-
degenerate energy levels oster . Imperfect plate geometries have other
ramifications as we discuss below. The second cell (Cell 2 mentioned above)
had not only tilts in the plate, but had a twist in the plate. While not shown
here, the measurements from the second cell showed even more broadening and
the appearance of double peaks in the EIT lines.

Figure 5: EIT signals (for scanning the coupling laser) showing Stark shifts
for various applied voltages for 40S1/2.

Figure 6: (a)-(c) correspond to the measured Stark shift for 28S, 40S, and 47S
Rydberg states with a dc source (blue errorbars). Also shown are the curve
fits using the calibration factor found by eq. (5). The dashed-red lines are
from theory and when these lines become vertical indicate the position of the
ELCs.

We extract the measured Stark shifts (corresponding to the location of the
maximum of the shifted EIT peak) and plot them against the applied voltage.
This was done for the three Rydberg states mentioned previously (28S, 40S, and
47S), shown by Fig. 6. The error bars in these plots correspond to the
standard deviation from six sets of data. In this data, we can also identify
that the Stark shift is not zero for zero applied voltage. This offset is
possibly due to excess charge in both vapor cells. Grounding the two plates
did not remove this excess charge. The voltage offset comes from various
sources, including (1) impurities in the metal causing a build up of charge
which cannot be removed and (2) ionization caused by the 480 nm coupling
laser. It is also possible that this is a result of a galvanic potential.
Whatever the case may be this offset can be accounted for.

In order to determine the offset, we fit to the equation

$\Delta=\frac{1}{C^{2}_{cal}}\left(V-V_{o}\right)^{2}\,\,\,.$ (5)

where $V_{o}$ is the offset voltage caused by the excess charge. We found the
voltage offset is 235$~{}\pm$ 6 mV for this cell. These fits were within a
$95~{}\%$ confidence interval. Through the fit, we are also able to find the
calibration factor for each cell as measured using the three Rydberg states,
given in Table. 2. Fig. 6 shows these fits for the three states. In
determining the calibration factor $C_{cal}$ care must be taken to ensure the
ELCs do not influence its value. The effect of the ELCs are discussed in the
next two sections.

Table 2: Experimentally obtained calibration constants. The uncertainty is the
standard deviation of the calibration constant obtained from the 6 sets of
data. Calibration Factors
---
Rydberg state | CCal (V/$\sqrt{Hz}$)
$28S_{1/2}$ Cell 1 | $(339\pm 2.7)\cdot 10^{-6}$
$40S_{1/2}$ Cell 1 | $(96.7\pm 0.2)\cdot 10^{-6}$
$47S_{1/2}$ Cell 1 | $(53.8\pm 0.3)\cdot 10^{-6}$

In addition to this, $C_{cal}$ can be used to either estimate the
polarizability $\alpha(0)$ or to determine the plate separation. In the first
case, the calibration factor found through the fit of the Rydberg atom Stark
shift measurements and the independently measured plate separations can be
used to quantify the polarizability $\alpha_{e}$. In the second case, the
calibration factor and theoretically calculated polarizability can be used to
find the plate separation. In the first case, if the plate separation can be
independently determined, then $C_{cal}$ can be used to experimentally
determine $\alpha_{e}$ by the following [obtained by comparing eq. (3) to eq.
(4)]:

$\alpha_{e}=2d^{2}/C_{cal}^{2}\,\,\,.$ (6)

Using this expression and the average plate separations for the four corners
(measurement with a microscope) given above ($d=2.194$ mm for the cell), the
estimated $\alpha_{e}(0)$ for each state and each cell is given in Table 3. In
this table we also show the percent error
($\Delta\%$=100*[$\alpha_{e}(0)-\alpha(0)]/\alpha(0)$], where $\alpha(0)$ is
given in Table 1). Better control of the plate separation would allow for more
accurate determination of $\alpha_{e}(0)$.

Table 3: Experimentally obtained $\alpha_{e}(0)$, percentage error from calculated $\alpha(0)$, and experimentally obtained plate separations. The uncertainty is the standard deviation from 6 sets of data.. Rydberg state | $\alpha_{e}(0)$ | $\Delta\%$ | Plate separation (mm)
---|---|---|---
$28S_{1/2}$ Cell 1 | $83.95\pm 1.4$ | 0.29 | $2.20\pm 0.019$
$40S_{1/2}$ Cell 1 | $1032.43\pm 4.4$ | 2.4 | $2.21\pm 0.005$
$47S_{1/2}$ Cell 1 | $3328.59\pm 33.2$ | 1.6 | $2.16\pm 0.013$

##  V Energy Level Crossings

The measured data in Fig. 6 were corrected for the voltage offset and compared
to theoretical calculations for the Stark shift. The theoretical Stark maps
were obtain by using the Alkali Rydberg Calculator Python package ARC2020 .
These comparisons are shown in Fig. 6, where the theoretical model accounts
for ELCs from nearby Rydberg states. We show plots for the Stark shift as a
function of applied $V$. In the theoretical curves, the $E$ field was
determined using $E=Vd$ (where we use $d=2.19$ mm). We can see good consensus
between the experimental data and theoretical model. In particular, on closer
inspection, the locations of the ELCs can be seen in the measured data. This
is further illustrated in Fig.7, where we show an expanded view of the ELC for
the 28$S_{1/2}$ state spectra, shown in Fig. 6 (a). In this figure, the level
crossings are easily seen as a splitting in the spectra around -11.2 V and
-12.5 V. On this note, the Stark shifts corresponding to the ELCs can be used
to find the calibration factor as well.

Figure 7: Expanded view of the ELC’s for the 28$S_{1/2}$ state. We can observe
the measurement (blue error bars) deviations from the quadratic nature (solid
black line) of the dc Stark effect at these locations. Also shown is the
theory (red dashed line) that accounts for the ELC’s and matched the data much
better.

After obtaining a consensus on the calibration constant through multiple
measurements, we can utilize the cells for measuring dc voltages. However, the
ELCs can limit the maximum voltage that can be accurately measured since they
cause deviations from a perfect quadratic dependence of the frequency on the
applied voltage. For a given state, it would be best to avoid voltage levels
where the first ELCs appear. For example, this occurs at approximately 12 V,
1.5 V, and 0.6 V, for $28S_{1/2}$, $40S_{1/2}$, and $47S_{1/2}$ respectively.
The quadratic behavior depicted in eqs. (1) and (3) fails near the ELCs. As
such, using the eq. (4) and measured Stark shifts will need to be limited to
voltages well below the first ELC. So for large voltage measurements Rydberg
states with low $n$ should be used, while for small voltages Rydberg states
with high $n$ are more suitable (due to the higher sensitivity to weak fields
for high $n$). Alternatively, choosing the electrode plate separation, or
manufacturing a variety of plate separations in a cell (with care and
attention to fringing fields and geometrical accuracy) allows a given voltage
to produce a field appropriate for accurate measurement by a particular
Rydberg state $n$.

##  VI Subtleties to Calibration

In practice, a calibration must account for or correct three non-idealities:
non-uniformity (spatial regions where the electric field between electrodes
departs from the relationship E=V/d, from fringing fields or imperfections in
plate manufacturing), spectroscopic features due to ELC interference (or “line
pulling”), and a voltage offset caused by stray charge accumulation, galvanic
potentials, or other stray electric field sources. We discuss each of these
below.

The fringing fields at the edge of the plate can cause a broadening of the EIT
line. Fig. 8 shows the EIT signal for an applied voltage of 0.3 V as the
optical beams are moved from the center of the plates ($x$=7.6 mm) to the edge
of plates ($x$=0 mm). Recall the optical beams are propagating along the the
long dimension of the plates. In this figure, we see the EIT signal becomes
broader as the beams approach the edge of the plate and we see that the EIT
line shape begins to change. Such effects are due to E-field inhomogeneity in
the region near the edge where the fringing fields are present (in this case
the beams are approaching two sharp corners of the plates). The angle of the
cell relative to the incident angle of the beam will play a large role in the
broadening as well. With that said, such effects can be accounted for in a
calibration. In such calibrations, one needs to be careful when double peaks
appear. These double peaks can be the result of imperfections in the plate
manufacturing.

Figure 8: EIT signal as the optical beams approach the edge of the plate. The
plate has a width of 18 mm, the edge of the plate is define at $x$=0 mm and
the center of the plate is at 9 mm from the edge.

In the case of the ELCs, we analyze their effect on the fit of eq. (5).
Ideally, the quadratic Stark effect should fit the data well, but the ELCs
introduce higher order terms. This can introduce errors in the fit of the data
used to find the calibration constant $C_{cal}$. Therefore, we analyzed how
the calibration factors and fits change if we included different voltage
ranges (max voltage-min voltage, centered at 0 V) during the fitting process.
Fig. 9 shows the difference between the data and the fit [i.e., eq. (5)] for
different voltage ranges given by the separate traces. Fig. 9(a) shows the
difference (i.e., residuals), Fig. 9(b) shows the studentized residuals
($|{\rm residual}|$/standard deviation), and Fig. 9(c) shows the calibration
factors obtained from the fits for different voltage ranges. Here, we plot the
percent difference between the data and the fit for cases where we include
different voltage ranges for the different fit traces for the $28S_{1/2}$
state. Note that the large deviations for $<2$ V in Fig. 9(b) are a result of
the smaller error for the smaller voltages applied. This is due in part to the
increase in width of the EIT peak as a voltage is increased. For smaller
voltages, the standard deviation was less than 1 mV for the peaks with smaller
widths, leading to increased fluctuations. It can be seen that the fit with
the least deviation is for the case where we use voltages from -9 V to 9 V,
corresponding to a range of 18 V. In any other case, the fit breaks from the
data very quickly. The fits shown in Fig. 6 and the calibration factor
$C_{cal}$ given above were obtained by optimizing the voltage range in this
manner.

Figure 9: (a) Difference (residuals) between the experimental data and the fit
in Fig. 6. (b) Studentized residuals from (a). Each trace is for a fit
incorporating data from the negative voltage out to the positive voltage, as
labeled. (c) The calibration factor obtained from fits using different voltage
ranges plotted against half the voltage range. This data is for the 28S
Rydberg state.

Also shown in Fig. 9(b) are the effects of the ELCs which result in a higher
order dependence and do not simply follow the quadratic Stark effect. This is
further demonstrated in Fig. 9(c), where we see how the calibration factor
changes for the different voltage ranges used in the fit. As expected, we see
under fitting if not enough data is used (magenta region), a nearly flat line
with-in error for the region of good fit (green region), and then a skew of
the slope as the voltage range is increased into the region of the ELCs (red
region). To overcome the effects of the ELCs at higher voltages, we can use a
more precise model to account for their effects, shown by Fig. 6. The other
option is to tune to different Rydberg states to tune the sensitivity or
adjust the plate separation. The latter requires the use of several different
cells manufactured for specific operating conditions. By tuning the plates, we
change the necessary voltage to produce a given electric field.

In the case of the voltage offset, we found that a potential cause is the
combination of fringing fields and the charge density present on the plates.
As our parallel plates are not infinite, there are contributions which arise
from edge effects. These effects have a particularly strong response if the
cell is translated horizontally, shown by Fig. 10. In this case, we mean
perpendicular to the optical beams and parallel to the plates. As the cell
position is moved so that the optical beams are at the edge of the plates (0
mm trace in Fig. 10), we observe that the voltage offset decreases.

Figure 10: Stark shift measurements for different horizontal beam positions
parallel to the plates. The edge of the cell is at 0 mm and 7.62 mm is near
the center of the cell.

While the positioning of the cell changes the voltage offset, it does not
change the shape of the curve. With the exception of the beam position at the
very edge of the cell, the calibration factor changed by less than 2 % for the
other measurements. So long as the the calibration curve is acquired and the
voltage offset is accounted for, this method has potential as a standard of
measurement. However, at the edge of the cell, there is an increased
uncertainty in measurement.

Finally, non-uniform plate separation can be an issue for these Rydberg atom
sensors. One remedy is to use a microfabricated cell john2  to insure plate
uniformity, where microfabrication of a vapor cell allows for better control
of the plate or electrode separation as described in Ref. john1 . The use of
similar vapor cells with internal electrodes are currently being investigated
for voltage measurements and other applications das2 . However, in this type
of cell, care needs to be taken to ensure that any coating placed on the
electrodes does not cause shielding of the applied voltage and in turn the
E-field seen by the atoms das1 .

##  VII ac Voltage Measurements

In addition to dc measurements, this same system can be used to determine 60
Hz ac field strengths. However, 60 Hz ac voltage measurement require different
read-out methods. The ac Stark effect would result in a linear shift to the
EIT peak, but we instead rely on the slowly changing dc Stark effect in the
adiabatic limit where the frequency is slow enough such that the atoms can
respond. This limits the measurements to the range of frequencies from dc to 1
MHz; we work with 60 Hz. In this application, we observed that the EIT peak
shifts from the 0-voltage location to the peak-voltage location, as expected.
Since, the calculated polarizability for the dc and ac fields are nearly
identical, this peak voltage location also corresponds to the equivalent dc
voltage. So by tracking the maximum EIT peak shift, we can make a calibration
curve that matches the curve for a dc source, shown in Fig. 11.

Figure 11: Figure shows the Stark shift plotted against the peak voltage of an
60 Hz ac source for the three states 28S (a), 40S (b), and 47S (c). Also
included for comparison are the dc experimental data from Figs. 6(a)-(c)

For detecting 60 Hz ac volatge, one might consider sweeping a laser fast
enough to track the EIT as the 60 Hz source caused it to oscillate from peak-
voltage location to zero-voltage location. Unfortunately, such a method
requires the laser to scan over a GHz in frequency at a scan frequency much
larger than the source bandwidth. Such a scan speed for CW systems results in
mode instability and loss in lasing. Here, we demonstrate two alternative
methods for determining the peak to peak voltage of the source from the motion
of the EIT peak. The first method (Stark shift tracking) relies on similar
scans to the ones in the dc voltage measurements. The coupling laser is
scanned over a frequency range while 20 traces of the transmission spectrum
are gathered. During this time, the EIT peak is oscillating between the
0-voltage location and the peak-voltage location. We take the 20 traces and
find the variance at each detuning, shown in Fig. 12. Using this data, we find
the maximum shift at the location where the data drops off below a threshold
value that lies above the noise. The extracted drop-off locations are plotted
in Fig. 11 for the three different states also used for the dc measurements.
Also plotted are the dc data to show the agreement in the calibration between
the dc data and the ac data taken with this method.

Figure 12: Sample of variance of 20 traces plotted against the coupling laser
detuning for a 1.2 V AC source for the 40S state. As the EIT peak shifts
around, the variance increases in the regions where it has passed.

Unfortunately, there is more noise in the ac data than in the dc data. This
noise is due to the accuracy of determining the drop-off location, which
becomes increasingly more difficult for higher ac voltages. At higher
voltages, the drop-off almost smooths to the level of the noise. The dc data
did have a similar issue for the higher voltages, but because of the averaging
of the peak at one location, it was mitigated. With an ac source, this option
is not available and for the interest of time and as a demonstration, we only
used 20 traces. However, by collecting more traces, we should be able to bring
the noise down. Further sources of error or deviation from the calibration
could also be present from the imperfections in the plates of the cell. The
EIT peak widths broadens with voltage and this could potentially result in a
voltage dependent shift in the drop-off location. However, this is not
apparent here within the bounds of error.

The second method (dc biased ac sensing) relies on the use of a dc calibrated
source. As discussed previously, if we have an ac source on the plates, the
EIT peak will oscillate between the zero-voltage location and the peak-voltage
location. We can observe this shifting if we scan the coupling laser. However,
if we lock the coupling laser to the zero-voltage location, we will instead
observe transmission only when the ac voltage passes through the zero-voltage
location, as shown by the computer generated plot in Fig. 13 (a). This
produces a transmission peak that we can monitor that is dependant on the ac
voltage source. Now, if we apply a calibrated dc voltage along with the ac
voltage the transmission peak will shift since the ac voltage at which the
voltage sum crosses zero is no longer when the ac voltage passes zero, shown
in Figs. 13 (b) and (c). However, if the dc voltage is increased to a value
higher than the peak voltage, the transmission will begin to vanish, as shown
in Fig. 13 (d).

Figure 13: Sample showing expected EIT transmission (blue) in the presence of
a 60 Hz ac field (red) plotted against the phase of the applied ac field. The
plots correspond to offset voltages of (a) 0 V, (b) 0.25 V, (c) 0.50 V, and
(d) 0.75 V, as labeled.

In this method, we apply a calibrated dc voltage as an offset to an ac voltage
source that we wish to determine. We track the height of the transmission
peaks in Fig. 13. As we scan the dc voltage source from -2 V to 2 V, we can
see the transmission appear and then vanish, as shown in Fig. 14 (a). The
voltage difference in the two drop-off locations in the dc voltage scan
defines the peak to peak voltage of the applied ac voltage. Fig. 14 (a) shows
the traces of dc voltage scan for several different ac voltages and Fig. 14
(b) shows the extracted peak to peak voltages plotted against the applied
voltage.

DC biased ac sensing has certain advantages when compared to Stark shift
tracking. In the case of Stark shift tracking, we must average for a long time
to improve the statistics enough to measure larger voltages. However, for dc
biased ac sensing, we do not have this same limitation since the measurement
occurs at the zero-voltage location with maximum EIT peak height.
Unfortunately, the maximum ac voltage we can sense is limited by the range of
our calibrated dc source. While this voltage is higher than measurements with
the Stark shift tracking method, it is still limited. Furthermore, this method
requires a laser lock that is stable to less than 1 MHz for the probe and
coupling lasers.

Figure 14: (a) Sample of traces for different ac voltages where the peak
transmission is plotted against the calibrated dc voltage. (b) The width of
the traces in (a) that correspond to the peak to peak voltage, plotted against
the applied peak to peak voltage (V). This data is for the 47 S state. (black)
is the one-to-one line as a guide and the (blue) is the extracted ac voltage.

##  VIII Conclusion and Discussion

In this manuscript, we have demonstrated a means to an alternative voltage
standard based on the inherent and known dipole strengths of Rydberg atoms and
have filled the SWAP-C gap between Zener diode and Josephson junction based
standards in doing so. By measuring the dc Stark shift for various voltages at
different states, we generate a calibration standard which can be utilized as
an alternative to two established voltage measurement technologies. In
addition to this, we discuss how the calibration factor $C_{cal}$ can be used
in two ways. First, if the plate separation is known, the calibration factor
can be used to determine the polarizaibility for the Rydberg states. Secondly,
we discuss how calibration factors can be used to find the plate separation to
within 10s of microns. We also demonstrate two methods to measure 60 Hz ac
sources with little to no modification to the apparatus.

In the case of dc sources, we were able to find Stark shifts ($\Delta$) within
an average error under $\pm 5$ MHz over 6 sets of data. Even though there was
broadening of the EIT peak and contributions from inhomogeneities at larger
voltages, the uncertainty in the voltage did not change since the sensitivity
increases with larger voltages. The fractional uncertainty in the voltage is
given by

$\left(\frac{\delta V}{V}\right)^{2}=\left(\frac{\delta
d}{d}\right)^{2}+\left(\frac{\delta\alpha}{2\cdot\alpha}\right)^{2}+\left(\frac{\delta\Delta}{2\cdot\Delta}\right)^{2}.$
(7)

The error in the Stark shift measurement translated to a fractional
uncertainty in the voltage is on the order of .01 V/V on average over the
voltage range of the 47S Rydberg state. The fractional uncertainty is constant
due to the broadening of the EIT peak at higher applied voltages, as discussed
previously. The 20 sets averaged to find a given Stark shift have scan times
of 25 ms. This corresponds to a sensitivity in the voltage measurement on the
order of 0.007 V/V$~{}Hz^{-1/2}$ using the 47S state. While this level of
sensitivity seems limited, different Rydberg states can be used for increased
sensitivity. By simply choosing to work at the 28S state, the uncertainty in
the measurement decreases to 0.003 V/V. This is due to the stronger EIT peak
at the 28S Rydberg state. The sensitivities of these voltage measurements are
ultimately limited by several aspects. One is that the polarizability can only
be calculated to an uncertainty of 1%. However, certain experimental
techniques can determine the polarizabilities of atoms to a much better
accuracy bai_2020 . Another key factor is in the time domain. The laser sweep
must be able to span several GHz for these measurements and thus sets a limit
on how fast data can be collected. Finally, the ultimate determination of how
well the Stark shift can be measured depends on the linewidth and amplitude of
the EIT line. This is currently 15 MHz in our system. Similar Rydberg
techniques have demonstrated linewidths down to 2 MHz kumar1 . This is nearly
an order of magnitude improvement that can be achieved. Furthermore, new
three-photon configurations may offer linewidths under 100 kHz which could
offer over an order of magnitude improvement shaffer_three . In addition to
this, for increased voltages, we begin to see effects from energy level
crossings and plate inhomogeneities. The prior can be addressed by adjustments
to the quadratic fit and the latter by the manufacturing of a more uniform
plate geometry, as our current plates have a dimple (and discontinuities) in
the very center where the leads are mounted. These improvements are
complementary and would increase the range of voltages accessible and improve
our sensitivity to better than 10${}^{-5}V/V~{}Hz^{-1/2}$.

We analyzed various sources of uncertainty that arose from non-uniform
electric fields, ELCs, and plate imperfections. The combination of these
effects initially point to the infeasibility of the Rydberg atom as a voltage
standard, but through careful consideration and calibration, these problems
can be accounted for. We discussed the broadening of the EIT lines due to non-
uniform fields, EIT line perturbations from fringing fields at the edge of the
plates, and the effects of the ELCs on the EIT spectra. These considerations
are key to developing a portable Rydberg standard and will affect the
sensitivity, range, and reliability for future atom based voltage sensors. In
conclusion, this study on the dc and 60 Hz ac fields will provide insight for
future advancements for the realization of a self-calibrating, deployable
voltage standard. Future work will include developing vapor cells with more
precise plate separation, including microfabricated cells.

We emphasize that a future device could be made where instead of being
calibrated, the polarizability and plate spacing could be accurately measured
and then the device would be based on fundamental standards. Other questions
to be answered in future work is how the resolution, stability and fractional
uncertainties compare to Zener diodes systems.

##  IX Acknowledgement

The authors thank Dr. Yuan-Yu Jau with Sandia National Labs, Albuquerque, NM
87123, USA for his useful, insightful technology discussions, and for suppling
one of the vapor cells used in our experiments.

##  X Data Availability Statement

Data is available upon request.

## References

  * (1) T.F. Gallagher, Rydberg Atoms. Cambridge Univer. Press:Cambridge, 1994.
  * (2) A.K. Mohapatra, T.R. Jackson, and C.S. Adams, Phys. Rev. Lett., 98, 113003, 2007.
  * (3) J.A. Gordon, C.L. Holloway, S. Jefferts, T. Heavner, “Quantum-Based SI Traceable Electric-Field Probe,” Proc of 2010 IEEE International Symposium on Electromagnetic Compatibility, July 25-30, 321-324, July 2010.
  * (4) J.A. Sedlacek, A. Schwettmann, H. Kubler, R. Low, T. Pfau and J.P. Shaffer, Nature Phys., 8, 819, 2012.
  * (5) C.L. Holloway, J.A. Gordon, A. Schwarzkopf, D.A. Anderson, S.A. Miller, N. Thaicharoen, and G. Raithel, IEEE Trans. on Antenna and Propag., 62(12), 6169-6182, 2014.
  * (6) C.L. Holloway, M.T. Simons, J.A. Gordon, P.F. Wilson, C.M. Cooke, D.A. Anderson, and G. Raithel, IEEE Trans. on Electromagnetic Compat., 59(2), 717-728, 2017.
  * (7) Kumar, S., Fan, H., Kübler, H. et al.,Sci Rep, 7, 42981 (2017).
  * (8) Tanasittikosol, M., Pritchard, J.D., Maxwell, D., Gauguet, A., Weatherill, K.J., Potvliege, R.M., and Adams, C.S., J. Phys B, 44, 184020, 2011.
  * (9) C.L. Holloway, J.A. Gordon, A. Schwarzkopf, D.A. Anderson, S.A. Miller, N. Thaicharoen, and G. Raithel, Applied Phys. Lett., 104, 244102-1-5, 2014.
  * (10) J. A. Gordon, C. L. Holloway, A. Schwarzkop, D. A. Anderson, S. Miller, N. Thaicharoen, G. Raithel, Applied Physics Letters, 105, 024104, 2014.
  * (11) Gordon, J.A., et al., AIP Advances, 9, 045030, 2019.
  * (12) J.A. Sedlacek, A. Schwettmann, H. Kubler, and J.P. Shaffer, Phys. Rev. Lett., 111, 063001, 2013.
  * (13) M.T. Simons, A.H. Haddab, J.A. Gordon, and C.L. Holloway, IEEE Access, 7, 164975-164985, 2019.
  * (14) M.T. Simons, A.H. Haddab, J.A. Gordon, and C.L. Holloway, Applied Physics Letters, 114, 114101 2019.
  * (15) Jing, M., Hu, Y., Ma, J., Zhang, H., Zhang, L., Xiao, L., and Jia, S. Nat. Phys., 16, 911–915, 2020.
  * (16) C.L. Holloway, M.T. Simons, M.D. Kautz, A.H. Haddab, J.A. Gordon, T.P. Crowley, Applied Phys. Letters, 113, 094101, 2018.
  * (17) Meyer, D.H., Kunz, P.D., and Cox, K.C., Physical Review Applied, 15, 014053, 2021.
  * (18) Robinson, A., Simons, M.T., and C.L. Holloway, Applied Phys. Lett., 118(11), 114001, doi.org/10.1063/5.0045601, 2021.
  * (19) Z. Song,H. Liu, X. Liu, W. Zhang, H. Zou, J. Zhang, and J. Qu, ‘Optics Express 27(6), 2019.
  * (20) D.H. Meyer, K.C. Cox, F.K. Fatemi, and P.D. Kunz, Appl. Phys. Lett., 12, 211108, 2018.
  * (21) C.L. Holloway, M.T. Simons, A.H. Haddab, J.A. Gordon, D. Novotny, IEEE Antenna and Wireless Propag. Lett. 18(9), 1853-1857, 2019.
  * (22) K.C. Cox, D.H. Meyer, F.K. Fatemi, and P.D. Kunz, “Quantum-Limited Atomic Receiver in the Electrically Small Regime”, Phys. Rev. Lett. 121, 110502, 2018.
  * (23) C.L. Holloway, M.T. Simons, A.H. Haddab, J.A. Gordon, and S. Voran, “A Multi-Band Rydberg-Atom Based Receiver: AM/FM Stereo Reception”, accepted to appear in IEEE Antenna and Propogation Magazine, 2021.
  * (24) D.A. Anderson, R.E. Sapiro, and G. Raithel, “An atomic receiver for AM and FM radio communication”, arXiv:1808.08589v1, Aug. 26, 2018.
  * (25) S. Ottoa, M.K. Hunter, N. Kjærgaard, and A.B. Debb, “Data capacity scaling of a distributed Rydberg atomic receiver array”, Journal of Applied Physics, 129, 154503, 2021.
  * (26) Holloway, C.L., Simons, M.T., Haddab, A.H., Williams, C.J., and Holloway, M.W., AIP Advanced, 9(6), 065110, 2019.
  * (27) Rüfenacht, A., Flowers-Jacobs, N.E., and Benz, S.P., Metrologia 55, S152-S173, 2018.
  * (28) Y. Tang, V.N. Ojha, S. Schlamminger, A. Ruefenacht, C.J. Burroughs, P.D. Dresselhaus, S.P. Benz, ”A 10 V programmable Josephson voltage standard and its applications for voltage metrology”, Metrologia, 49, 635-43, 2012.
  * (29) P.D. Dresselhaus, S.P. Benz, C.J. Burroughs, N.F. Bergen, Y. Chong, ”Design of SNS Josephson arrays for high voltage application”, IEEE Transactions on Applied Superconductivity, 17(2), 173-176, 2007.
  * (30) T. Yamada, H. Yamamori, H. Sasaki, A. Shoji, Japanese Journal of Applied Physics, 48(7R), 076510, 2009.
  * (31) O. Rest, D. Winzen, S. Bauer, R. Berendes, J. Meisner, T. Thuemmler, S. Wuestling, C. Weinheimer, ”A novel ppm-precise absolute calibration method for precision high-voltage dividers”, Metrologia, 56, 045007, 2019.
  * (32) Witt, T.J., Reymann, D., and Avrons, D., IEEE. Trans. on Instrumentation and Measurement, 44(2), 226-229, 1995.
  * (33) Witt, T.J., Proc. IEE Sci. Meas. Technol., 149(6), 305-312, 2002.
  * (34) T. Middelmann, S. Falke, C. Lisdat, U. Sterr, ”High accuracy correction of blackbody radiation shift in an optical lattice clock”,T Middelmann, S Falke, C Lisdat, U Sterr, ”High accuracy correction of blackbody radiation shift in an optical lattice clock”, Physical Review Letters, 109, 263004, 2012.
  * (35) J.A. Sherman, N.D. Lemke, N. Hinkley, M. Pizzocaro, R.W. Fox, A.D. Ludlow, C.W. Oates, ”High-accuracy measurement of atomic polarizability in an optical lattice clock”, Physical Review Letters, 108, 153002, 2012.
  * (36) J.M. Amini, H. Gould, ”High precision measurement of the static dipole polarizability of cesium”, PRL, 91(15), 153001, 2003.
  * (37) M.D. Gregoire, I. Hromada, W.F. Holmgren, R. Trubko, A.D. Cronin, ”Measurements of the ground-state polarizabilities of Cs, Rb, and K using atom interferometry”, PRA 92, 052513, 2015.
  * (38) Daschner, R., Ritter, R., Kübler, H. Frühauf, N., Kurz, E., Löw, R., and Pfau, T., Optics Letters, 37(12), 2271-2273, 2012.
  * (39) Daschner, R., Kübler, H., Löw, R., Baur, H. Frühauf, N., and Pfau, T., Applied Phys. Letters, 105, 041107, 2014.
  * (40) M.G. Bason, M. Tanasittlosol, A. Sargsyan, A.K. Mohapatra, D. Sarkisyan, R.M. Potvliege, and C.S. Adams, N. Journal of Physics, 12, 065015, 2010.
  * (41) Osterwalder A., and Merkt, F., Phys Review Letters, 82(9), 1831-1834, 1999.
  * (42) H. Friedrich. Theoretical Atomic Physics. $4^{th}$ Edition, Springer, 2017.
  * (43) Elizabeth J. Robertson and Nikola Šibalić and Robert M. Potvliege and Matthew P. A. Jones, ARC 3.0: An expanded Python toolbox for atomic physics calculations, arXiv eprint 2007.12016 (2020).
  * (44) Jianing Han, Yasir Jamil, D. V. L. Norum, Paul J. Tanner, and T. F. Gallagher, Phys. Rev. A, 74, 054502 (2006)
  * (45) V. A. Yerokhin, S. Y. Buhmann, S. Fritzsche, and A. Surzhykov, Phys. Rev. A, 94, 032503 2016.
  * (46) M. S. O’Sullivan and B. P. Stoicheff, Phys. Rev. A 31, 2718 1985.
  * (47) M. S. O’Sullivan and B. P. Stoicheff, Phys. Rev. A 33, 1640 1986.
  * (48) D. Das and V. Nataranjan, “High-precision measurement of hyperfine structure in the D lines of alkalai atoms”, J Phys B, 41, 035001, 2008.
  * (49) D. A. Steck, “Rubidium 85 D line data”, revision 2.1.6 Sep. 20, 2013 [Online]. Available: http://steck.us/alkalidata.
  * (50) M.T. Simons, A.H. Haddab, J.A. Gordon, and C.L. Holloway, “Applications with a Rydberg Atom-based Radio Frequency Antenna/Receiver,” Procc in EMC EUrope 2019, Barcelona, Spain, Sept. 2019.
  * (51) M. T. Simons, M. Kautz, J. A. Gordon, and C. L. Holloway, “Uncertainties in Rydberg atom-based RF E-field measurements,” in Proc. EMC Europe, Amsterdam, The Netherlands, Aug. 2018, pp. 376-380.
  * (52) H. Fan, S. Kumar, J. Sheng, J.P. Shaffer, C.L. Holloway and J.A. Gordon, Physical Review Applied, 4, 044015, November, 2015.
  * (53) J. P. Shaffer and H. Kübler, “A read-out enhancement for microwave electric field sensing with Rydberg atoms,” Proc of SPIE Photonics Europe, July, 39-49, July 2018.
  * (54) P.R. Berman and V.S. Malinovsky, Priciples of Laser Spectroscopy and Quantum Optics. Princeton Univerity Press, 2011.
  * (55) M.G. Bason, Coherent atom-light interactions in multi-level systems. PhD. Thesis, Department of Physics, Durham University, chapter 7, Oct 20, 2009.
  * (56) L.A. Liew, S. Knappe, J. Moreland, H. Robinson, L. Hollberg, and J. Kitching, Applied Phys. Letters, 84(14), 2694-2996, 2004.
  * (57) Jingxu Bai and Suying Bai and Xiaoxuan Han and Yuechun Jiao and Jianming Zhao and Suotang Jia, Precise measurements of polarizabilities of cesium nS Rydberg states in an ultra-cold atomic ensemble, New Journal of Physics,22(9), 093032, 2020.
  * (58) J. Kitching, E.A. Donley, S. Knappe, M. Hummon, A.T. Dellis, J. Sherman, K. Srinivasan, V. A. Aksyuk, Q. Li2, D. Westly, B. Roxworthy A.Lal, , Journal of Physics: Conference Series, 723, 012056, 2016.