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# Detection of HF and VHF Fields through Floquet Sideband Gaps by ‘Rabi
Matching’ Dressed Rydberg Atoms

Andrew P. Rotunno  Samuel Berweger  Nikunjkumar Prajapati  Matthew T. Simons
Alexandra B. Artusio-Glimpse  Christopher L. Holloway  National Institute of
Standards and Technology, Boulder, CO 80305, USA  Maitreyi Jayaseelan
Department of Physics, University of Colorado, Boulder, Colorado 80302, USA
R. M. Potvliege  C. S. Adams  Department of Physics, Durham University, South
Road, Durham DH1 3LE, United Kingdom

###### Abstract

Radio frequencies in the HF and VHF ($3$ MHz to $300$ MHz) bands are
challenging for Rydberg atom-based detection schemes, as resonant detection
requires exciting the atoms to extremely high energy states. We demonstrate a
method for detecting and measuring radio frequency (RF) carriers in the HF and
VHF bands via a controlled Autler-Townes line splitting. Using a resonant,
high-frequency (GHz) RF field, the absorption signal from Townes-Merrit
sidebands created by a low frequency, non-resonant RF field can be enhanced.
Notably, this technique uses a measurement of the optical frequency separation
of an avoided crossing to determine the amplitude of a non-resonant, low
frequency RF field. This technique also provides frequency-selective
measurements of low frequency RF electric fields. To show this, we demonstrate
amplitude modulated signal transduction on a low frequency VHF carrier. We
further demonstrate reception of multiple tones simultaneously, creating a
Rydberg ‘spectrum analyzer’ over the VHF range.

\begin{overpic}[width=117.07924pt]{fig1a.pdf}
\put(70.0,15.0){\large(a)}\end{overpic}

\begin{overpic}[width=182.1196pt]{fig1b.jpg}
\put(17.0,40.0){\large\color[rgb]{1,1,1} HF Plates}
\put(37.0,50.0){\large\color[rgb]{1,1,1} Microwave}
\put(50.0,45.0){\large\color[rgb]{1,1,1} horn}
\put(50.0,16.0){\large{\color[rgb]{0,0,0}$\leftarrow$} \color[rgb]{1,1,1} Cs
Vapor Cell} \end{overpic}

\begin{overpic}[width=121.41306pt]{fig1cde.pdf} \put(91.0,82.0){\large(c)}
\put(91.0,50.0){\large(d)} \put(91.0,20.0){\large(e)} \put(17.0,20.0){ $-2$}
\put(32.0,23.0){ $-1$} \put(54.0,20.0){ $+1$} \put(70.0,20.0){ $+2$}
\end{overpic}

Figure 1:  Overview of the experiment. (a) Atomic level diagram, (b)
Experimental setup, and sample transmittance curves comparing (c) unperturbed
EIT, (d) the dressed AT splitting with $\Omega_{dr}\approx 2\pi\times 150$
MHz, and (e) with the HF field ($E_{HF}\approx 0.14$ V/cm) at 100 MHz (blue).
The two pairs of observed line splittings are labelled $\pm 1$ and $\pm 2$.
The Stark offset has been removed for illustration.

##  I Introduction

Highly excited Rydberg atomic states [1, 2] have been used as self-calibrated
sensors for both dc [3, 4, 5] and resonant radio frequency (RF) [6, 7, 8, 9]
electric fields, due to the large dc polarizability of the higher orbital
angular momentum states [10, 11] and the large transition dipole moments of
transitions between nearby Rydberg states. Atomic energy levels are typically
probed spectroscopically via electromagnetically-induced transparency (EIT).
RF and dc electric fields are measured by changes in the EIT spectrum, with
values and uncertainties based on physical atomic quantities. Measurements of
laser transmittance alone are subject to many experimental sources of noise,
but frequency-space measurements of atomic states using a scanning laser
provide a more robust measurement.

Autler-Townes (AT) splitting caused by resonant RF fields provides measurement
of GHz to THz fields, traceable to the International System of Units (SI)[12],
often with excellent accuracy. One advantage to using Rydberg atoms for RF
field sensing is their broad frequency range of operation. This is enabled by
the spectral density of Rydberg states, with many large electric dipole moment
transitions with resonant frequencies from MHz to THz. However, Rydberg
transitions in the HF and VHF range of 3 MHz to 300 MHz require either very
high principal quantum number states ($n>100$), or high orbital angular
momentum states [13, 14]. Either method suffers from weaker optical couplings
for EIT, as well as contamination of the signal from the very large density of
nearby atomic states. Off-resonant generalized Rabi measurements are possible
using nearby existing resonances [15], or farther away using ac Stark shifts
with a strong local oscillator (LO) [16]. A dc field can also be used to shift
the resonance, though this may require a very strong and spatially homogeneous
dc field. Power and frequency tuning methods [17, 18, 19] exist for shifting
resonances to meet an arbitrary field, but these require signals near existing
resonances. The ability to detect low frequency or long-wavelength signals
would enable long-distance reception in a compact form factor, with the active
receiver volume only a few cm3, compared to meters or longer classical dipole
antennas [20, 9, 21], enabling compact long-distance reception for over-the-
horizon, underground, or underwater applications.

Here we present a method for the detection of arbitrary electric fields in the
3 MHz to 300 MHz range (we will henceforth use HF and VHF interchangeably for
the purpose of this paper) using a Rydberg state with no nearby HF or VHF
transitions. We use a high-frequency RF field that is resonant with a nearby
Rydberg transition to ‘match’ the Rabi frequency with an applied HF signal
field. Tuning the Rabi frequency of the ‘matching’ field effectively probes
the Floquet sideband structure generated by the HF field. Absorption dips
within the AT peaks reveal an avoided energy crossing, at the mid-point of the
Floquet quasi-energy sidebands and the central EIT line. The generation of the
Floquet sidebands can be referred to as the Townes-Merritt (TM) effect [22,
23, 24, 25], where cyclic modulation of state’s Stark shift energy with a MHz-
range field will produce quasi-energy sidebands on the observed EIT line [26,
27, 28, 29, 30, 31]. This effect has been analyzed in other quantum systems as
well as Rydberg EIT [32, 33, 34].

To probe the TM or ‘Floquet’ sidebands caused by an HF field, we drive a
simultaneous AT splitting, which splits both the center EIT peak and the
sidebands. When we ‘match’ the dressing field’s Rabi frequency to the applied
HF, an avoided level crossing between the splitting EIT line and its sidebands
is observed. The frequency gap grows non-linearly with the HF field strength.
We briefly illustrate the effect in Fig. 1, showing (a) an atomic energy level
diagram, and (b) the experimental setup, and in (c-e) we show transmittance
spectra demonstrating the effect of a 100 MHz HF field on microwave-driven
EIT-AT spectrum. We observe both first and second order sidebands in
transmittance signal at $\pm$50 MHz and $\pm$100 MHz detuning from the central
EIT line. With appropriate atomic analysis, the non-linear HF field splitting
gives measurements of electric field ac amplitude and dc offset [35], as well
as the frequency via the spectrum, and phase can be measured with an LO. The
splitting allows for a measurement range continuous from a low end of the EIT
Doppler linewidth to an upper roll-off with broadening caused by field non-
uniformity [36], roughly 5 MHz to 500 MHz in our demonstration. In principle,
the range can be extended up to GHz with uniform electric fields, and sub-MHz
with Doppler cancellation or cold atoms.

We discuss our experimental setup in Sec. II, and present a combination of
Floquet and dressed atom theory in Sec. III and Appx. A. We show results of
experimental data against theory and the effect of multiple simultaneous HF
fields in Sec. IV. Looking toward applications, we demonstrate the ability to
perform ‘pseudo-resonant’ HF detection in Sec. V.2, and conclude in Sec. VI.
In Appx. C, we list the non-standard symbols used in this paper.

##  II Experimental Setup

The Rydberg EIT-AT setup is shown in the level diagram of Fig. 1(a) and image
of the experiment Fig. 1(b), including two counter-propagating lasers, a
cesium vapor cell (75 mm $\times$ 25.4 mm $\oslash$ ) between two parallel
plates, and a K-band microwave horn placed approximately 15 cm from the atoms.
The 852 nm probe laser is held near resonance on the D2 transition
$\bra{6S_{1/2},F=4}\leftrightarrow\ket{6P_{3/2},F=5}$. The probe transmittance
is measured through differential detection, with and without the overlapping
510 nm coupling laser. The coupling laser frequency detuning $\delta_{C}$ from
the $\ket{56D_{5/2}}$ state EIT resonance is scanned for an EIT spectrum
measurement. The probe beam had a ($1/e^{2}$) waist diameter of 1.03 mm and a
power of $0.5$ mW, yielding a probe Rabi frequency of
$\approx~{}2\pi\times~{}31.5$ MHz. The coupling beam had a waist diameter of
1.7 mm and a power of 33 mW, yielding a coupling Rabi frequency of
$\approx~{}2\pi\times~{}0.9$ MHz [37]. In this letter we plot the differential
probe transmittance intensity as a false-color axis while coupling laser
detuning $\delta_{C}/2\pi$ is scanned on the horizontal axis, averaged over 5
traces in the oscilloscope before read-out. Scanning either HF or RF power
along the vertical axis gives ‘waterfall’ spectrum plots of experimental data
that we compare against theory. Long-term drifting offsets of the coupling
laser are corrected using a simultaneous field-free reference cell, and the
frequency scale is given by the low-field limit of the fine energy gap between
the $\ket{56D_{3/2}}$ and $\ket{56D_{5/2}}$ states (392 MHz [37]).

The resonant GHz dressing field is applied using a horn antenna, also shown in
Fig. 1(b). Given the long wavelength of the HF and VHF bands, instead of using
a large antenna for propagation, we apply these RF fields across two parallel
plates (shown in Fig. 1(b)) [30, 27, 38, 31, 28, 29, 30]. Each plate is 102
mm$\times$45 mm in size, supplied electrically by stranded wires screwed down
at tapped holes near the corner, and separated by two 25.4 mm insulating
dowels on opposite corners. While the impedance of the glass cell is unknown
at low frequency, in theory the glass vapor cell insulates against external
dc-limit electric fields [39]. However, we observe a residual dc or root-mean-
squared (RMS) electric field in the following atomic measurements, possibly
due to internal residual or laser-induced charges [26], or background radio
waves.

The Rydberg transition driven by the dressing field is illustrated in Fig.
1(a), on
$\braket{56D_{5/2},m_{J}=\frac{1}{2}}{\leftrightarrow}{54F_{7/2},m_{J}=\frac{1}{2}}$,
which has a dipole transition strength of 1746 $ea_{0}$, and transition
frequency measured near 18.313(1) GHz, significantly different than the
expected resonance at 18.346 GHz [37], indicating a potential stray field in
the cell. This applied frequency $\omega_{dr}$ must compensate for the
differential Stark shifting of both states with the HF power, $P_{HF}$. This
effective resonance is determined by hand-tuning $\omega_{dr}$ at MHz
intervals to obtain equal AT peaks in the low-field limit, then the high-field
limit when the HF is applied. This shift was measured empirically at roughly
$-579(14)$ MHz per mW applied at 100 MHz, and using the power-to-field
conversion determined in Sec. IV, nearly $-3.62(9)$ GHz per (V/cm)2, about
$4\%$ below the theory value from the difference in polarizabilities (see
Appx. A). This power conversion ($\sqrt{P_{HF}}$ to $|E_{HF}|$) is measured at
100 MHz, and in general depends on the frequency-dependent coupling of the
plates and vapor cell. In order to make this conversion, we require an atomic
theory for the measurement of $E_{HF}$ using the observed splittings.

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{\color[rgb]{1,1,1}(a)} } \put(36.0,28.0){ {\color[rgb]{1,1,1}(c)} }
\put(56.0,28.0){ {\color[rgb]{1,1,1}(e)} } \put(75.0,28.0){
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\put(16.0,7.0){ {\color[rgb]{0,0,0}(b)} } \put(36.0,7.0){
{\color[rgb]{0,0,0}(d)} } \put(56.5,7.0){ {\color[rgb]{0,0,0}(f)} }
\put(75.0,7.0){ {\color[rgb]{0,0,0}(h)} } \put(95.0,7.0){
{\color[rgb]{0,0,0}(j)} } \put(44.4,9.5){ {\large${-2}$}{\Large$\square$}}
\put(46.0,7.0){{\large${-1}$}{\Large$\square$} }
\put(51.1,7.0){{\Large$\square$}{\large${+1}$}}
\put(52.2,9.5){{\Large$\square$}{\large${+2}$}}\end{overpic} Figure 2:
Comparison of experiment and theory, with the two pairs of avoided crossings
labelled $\pm 1,\pm 2$. Top: Experimentally observed transmittance spectra
showing a waterfall over applied dressing field vertically, with each plot
from left to right increasing amplitude ($\sqrt{P_{HF}}$) of a 100 MHz HF
field. Bottom: Calculated state quasi-energy $\varepsilon$, with state
projection of $\ket{56D_{5/2}}$ as marker size, plotted as a corresponding
waterfall over $\Omega_{dr}$. Theory plots use $E_{dc}=0.02$ V/cm,
$N_{max}=24$, $S_{\delta}=1.1$, scaling $\Omega_{dr}/\sqrt{P_{dr}}=85\times
2\pi$ MHz $/\sqrt{\textrm{mW}}$, varying $\Omega_{dr}$ and $E_{HF}$ as
labeled.

##  III Theory

We employ a model where the Rabi frequency $\Omega_{dr}$ of the dressing field
drives resonant transitions between two Floquet ‘ladders’ of quasi-energy
states, as illustrated in Fig. 1(a). We differentiate the method used here,
namely the Floquet expansion of a dressed two-level system, from other methods
including the use of a modulated RF field [40, 41, 42, 43, 44, 45], multiple
tones near Rydberg resonances [46, 47, 48, 49, 50, 51, 52], leverage off-
resonant ac Stark tuning using a resonance [43, 53], use continuous resonance
tuning methods [17, 18], or observe Floquet (TM) sidebands alone [22, 26, 27,
28, 54]. Analysis for two-level physics where the states are dressed and
driven have been performed [55, 33, 34, 56], and the effect can be extended to
other driven two-level quantum systems.

We focus on the dynamics of a two-state Rydberg system, leaving aside details
of the two-photon cascade EIT which populates Rydberg states and probes energy
levels spectrally [4]. We consider the states
$\ket{56D_{5/2},m_{J}=\frac{1}{2}}=\ket{D}$ and
$\ket{54F_{7/2},m_{J}=\frac{1}{2}}=\ket{F}$ in Cesium, which are subject to
three relevant electric fields at different rates, simultaneously, applied
perpendicular to the table: 1) the dressing field is nearly resonant to the
strong D-F Rydberg transition $\omega_{dr}\simeq\omega_{F}-\omega_{D}$,
performing controlled AT splitting, 2) an HF field of interest at a much lower
frequency $\omega_{HF}\ll\omega_{F}-\omega_{D}$ which populates Floquet
sidebands in the EIT spectrum, and 3) a dc or off-resonant effective RMS
background field which exists in the cell.

In the low-field limit, the Stark energy shift due to the HF field evolves
over time as $-\alpha_{D/F}\mathcal{E}(t)^{2}/2$, proportional to electric
field squared $\mathcal{E}(t)^{2}$, and dc polarizabilities $\alpha_{D/F}$,
where the subscript $(~{}\cdot~{})_{D/F}$ represents throughout the value for
either individual state. Proper Stark shift calculations are required at
higher fields, considering contributions from many allowed dipole transitions
which couple to the electric field, contributing to total energy shift and
population mixing. We show polarizability calculations for $\ket{D}$ and
$\ket{F}$ in Appx. B. We fit frequency shift over $\mathcal{E}^{2}$ for
effective polarizability in the field range used in this work. Throughout this
analysis we use $\alpha_{D}=-3003$ MHz per (V/cm)2 and $\alpha_{F}=12100$ MHz
per (V/cm)2. We utilize an effective polarizability for its computational
simplicity, although one could use a ‘multi-state’ computation using relevant
states from the Stark calculation for more accuracy, especially for high
fields. For instance, the $\ket{F}$ state represents $<90\%$ of the population
for only $0.008$ (V/cm)2, shown in Appx. B. We see good agreement between a
multi-state model and the one employed here over the modest field ranges used,
and we plan to investigate features which are only captured by a multi-state
model in future work.

The electric field of our HF sinusoid and a dc offset can be written
$\mathcal{E}(t)=E_{dc}+E_{\textrm{HF}}\cos(\omega_{HF}t)$. The $E_{dc}$ term
can be considered to contain broadband background ac electric fields,
effectively an RMS contribution to the average dc field value. Since the Stark
shift $-\alpha_{D/F}|\mathcal{E}(t)|^{2}/2\hbar$ is proportional to field
squared, we use the following abbreviations for the ac ($\sim$), dc ($-$), and
cross-term ($\times$) components:

$\begin{split}\Sigma_{D/F}^{\sim}=&\frac{-\alpha_{D/F}E_{HF}^{2}}{4\hbar}\\\
\Sigma_{D/F}^{-}=&\frac{-\alpha_{D/F}E_{dc}^{2}}{2\hbar}\\\
\Sigma_{D/F}^{\times}=&\frac{-\alpha_{D/F}E_{dc}E_{HF}}{\hbar}\end{split}$ (1)

letting us write the time-evolving Stark shifts:

$\begin{split}\frac{-\alpha_{D/F}|\mathcal{E}(t)|^{2}}{2\hbar}=\hskip
5.0pt&\Sigma_{D/F}^{-}+\Sigma_{D/F}^{\sim}\\\
+&\Sigma_{D/F}^{\times}\left(e^{i\omega_{HF}t}+e^{-i\omega_{HF}t}\right)/2\\\
+&\Sigma_{D/F}^{\sim}\left(e^{i2\omega_{HF}t}+e^{-i2\omega_{HF}t}\right)/2\end{split}$
(2)

When time-averaging, only the first two terms remain, the dc and HF RMS
fields, which define our time-averaged shifted resonance from
$\omega_{dr}\simeq\omega_{F}-\omega_{D}$. That is, due to a differential Stark
shift from the HF field intensity, an applied dressing field must shift in
order to observe equal AT splitting of the EIT line, with a new effective
detuning between the applied field $\omega_{dr}$ and the shifted resonance:
$\delta_{dr}\equiv\omega_{dr}-\left[\left(\omega_{F}+\Sigma_{F}^{-}+\Sigma_{F}^{\sim}\right)-\left(\omega_{D}+\Sigma_{D}^{-}+\Sigma_{D}^{\sim}\right)\right]$.
We adjust the applied value of $\omega_{dr}$ to keep $\delta_{dr}\simeq 0$ in
the high-field limit, by optimizing the symmetry of spectral peaks. We also
invoke a detuning scaling parameter to compensate for the difference in
experimental resonance shift and the theory values from the effective
polarizabilities:
$\omega_{dr}-(\omega_{F}-\omega_{D})=S_{\delta}\cdot(\Sigma_{F}^{-}+\Sigma_{F}^{\sim}-\Sigma_{D}^{-}-\Sigma_{D}^{\sim})$,
where the scaling parameter $S_{\delta}=1$ gives the theoretical resonance
($\delta_{dr}=0$).

Aside from constant terms, there are oscillating components: a dc-inclusive
cross term at $\pm 1\omega_{HF}$ with strength $\Sigma_{D/F}^{\times}/2\propto
E_{dc}E_{HF}$, and an ac ‘intensity’ term at $\pm 2\omega_{HF}$ with strength
$\Sigma_{D/F}^{\sim}/2\propto E^{2}_{HF}$. The rectified oscillating field
$\cos^{2}(\phi)=\frac{1}{2}\left(1+\cos(2\phi)\right)$ doubles the effective
modulation frequency, making even quasi-energy sidebands with multiples of
$2\omega_{HF}$ from the center EIT line [31]. The dc field breaks this
symmetry, enabling single $\omega_{HF}$ photon transitions, thus odd sidebands
in multiples of $\omega_{HF}$ [27]. For the case of two applied HF
frequencies, we expect ac components which oscillate at sum and difference
frequencies as
$\cos(\phi)\cos(\theta)=\frac{1}{2}\left[\cos(\phi-\theta)+\cos(\phi+\theta)\right]$,
which are demonstrated in Sec. IV.

We discuss the creation of a time-independent Hamiltonian using both Floquet
states and the dressed-atom approach in Appendix A. Diagonalizing this
Hamiltonian reveals an overall energy structure where states populate a ‘split
ladder’ of states with eigen-energies $\varepsilon_{N,\pm}\approx
N\hbar\omega_{HF}\pm\hbar\Omega_{dr}/2$ around the original state energy
$\varepsilon_{0}\approx\Sigma_{D}^{-}+\Sigma_{D}^{\sim}$, for integer $N$.
That is, on either side of the EIT resonance $\varepsilon_{0}$, we observe a
first order pair of sidebands $\varepsilon_{\pm 1}$ at
$\varepsilon_{0}\pm\omega_{HF}$, a second order pair $\varepsilon_{\pm 2}$ at
$\varepsilon_{0}\pm 2\omega_{HF}$, and so on. Each quasi-energy ladder state
splits into the ($\pm$) AT dressed states $\varepsilon_{N,\pm}$, approximately
separated by Rabi frequency $\Omega_{dr}=\wp_{DF}E_{dr}/\hbar$ with dipole
moment $\wp_{DF}=\bra{D}e\cdot\hat{z}\ket{F}$, as shown in Fig. 2(a,b).
However, when these splitting sidebands approach each other (moving up the
waterfall plot from the bottom), we observe an avoided energy level crossing
which alters the state energies, and depends on both the HF and dressing
fields, as seen in Fig. 2(c-j) over increasing $E_{HF}$. We find that the
observed effect cannot be interpreted as a simple linear second splitting of
the dressed Rydberg states, due to the apparent non-linear splitting by the HF
field, and the appearance of two pairs of HF resonances.

The characteristic feature of this method is observed as two pairs of avoided
energy crossings in the spectrum for $\Omega_{dr}\simeq\omega_{HF}$ and
$\Omega_{dr}\simeq 2\omega_{HF}$ in the presence of an HF field $\omega_{HF}$.
The case of $\Omega_{dr}\simeq\omega_{HF}$ we refer to as the $\pm 1$ feature
(or $\varepsilon_{0,\pm}\simeq\varepsilon_{\pm 1,\mp}$), and the case of
$\Omega_{dr}\simeq 2\omega_{HF}$ we call the $\pm 2$ feature (or
$\varepsilon_{0,\pm}\simeq\varepsilon_{\pm 2,\mp}$). These splittings grow
non-linearly with $\Sigma^{\times}_{D/F}/\omega_{HF}$ and
$\Sigma^{\sim}_{D/F}/\omega_{HF}$, respectively. The spectral location of the
avoided level crossing in the resonant case $\delta_{dr}=0$ (leaving aside the
shift of $\Sigma_{D}^{-}+\Sigma_{D}^{\sim}$) is at $\pm\omega_{HF}/2$ for the
first crossing and $\pm\omega_{HF}$ for the second crossing. These are at the
midpoints between $\varepsilon_{0}$ and $\varepsilon_{\pm 1}$, and
$\varepsilon_{0}$ and $\varepsilon_{\pm 2}$, respectively.

To explain these additional avoided level crossings, we find that we can
identify allowed transitions by considering the parity of the states involved
and counting photons. The $\pm 1\omega_{HF}$ modulation drives an allowed
transition between adjacent Floquet states of opposite (odd) parity, similarly
for the $\pm 2\omega_{HF}$ modulation allows a transition to the same (even)
parity sideband. We note only the first and second order sideband crossings
with the center $\varepsilon_{0}$ state are strongly avoided, while higher
order crossings ($>2$) do not have a significant gap.

Deriving a closed-form expression for field values from a splitting
measurement remains non-trivial, (i.e. not linear with field as in AT
splitting), as the combination of both dressing and HF fields shift the avoid
energy crossings, and effective state polarizability depends on the range of
$E_{HF}$. This leaves us to diagonalize the Hamiltonian in Eq. 7 to find
eigen-energies for various field values, and compare those to experimental
field waterfall spectra, as in Sec. IV. We did not attempt to calculate theory
transmittance spectrum curves, although proper fitting would require this, and
such theory could incorporate the range of Rabi frequencies sampled and
Doppler effects, but is computationally expensive.

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\end{overpic}\begin{overpic}[width=106.23477pt]{fig3cd.pdf}
\put(1.0,93.0){(c)} \put(1.0,46.0){(d)}
\end{overpic}\begin{overpic}[width=106.23477pt]{fig3ef.pdf}
\put(1.0,93.0){(e)} \put(1.0,46.0){(f)}
\end{overpic}\begin{overpic}[width=106.23477pt]{fig3gh.pdf}
\put(1.0,93.0){(g)} \put(1.0,46.0){(h)} \end{overpic}

Figure 3: Effect of increasing HF field, comparing experimental transmittance
data (top) with theory quasi-energy (bottom). We show $\omega_{HF}=$ (a,b) 56
MHz, (c,d) 100 MHz, (e-h) 200 MHz, using $\Omega_{dr}$ which demonstrate $\pm
2$ (a-f), and $\pm 1$ (g,h) splittings. These plots hold $\Omega_{dr}$
constant, keeping experimental $\delta_{dr}\simeq 0$ as $E_{HF}$ grows
($E_{dc}=20$ mV/cm), using detuning scaling $S_{\delta}=$ (b) 1.05, (d) 1.1,
(f) 0.8, and (h) 0.8.

##  IV Results

The primary finding here is the ability to detect far off-resonant HF and VHF
signals using an induced line splitting, in order to convert a relative
transmittance measurement to a more accurate frequency-space measurement. With
control over laser detuning $\delta_{C}$ and dressing strength $\Omega_{dr}$,
we can measure an arbitrary HF field’s frequency, dc and ac components in a
single spectrum. This provides a path to atomic measurements of field
amplitudes over decades of frequency range (3 MHz to 300 MHz here), with
around one decade of dynamic range demonstrated here. A sample measurement is
shown in Fig. 1, where transmittance spectra versus coupling laser detuning
show (c) Rydberg EIT, (d) resonant dressing AT splitting, and (e) two pairs of
absorption features, labelled $\pm 1$ and $\pm 2$. Spectra like Fig. 1(e) for
different RF powers are combined to create waterfall plots in following
figures.

In Fig. 2, we compare experimental waterfalls over AT splitting $\Omega_{dr}$
via controlled microwave power (top row), with calculated state quasi-energies
(bottom row), illustrating the ‘$\pm 1$’ ($\Omega_{dr}\approx\omega_{HF}$) and
‘$\pm 2$’ ($\Omega_{dr}\approx 2\omega_{HF}$) pairs of splittings. We vary
$P_{HF}$ so that $E_{HF}$ changes linearly between columns, then adjust
$\omega_{dr}$ so that $\delta_{dr}\simeq 0$ in the high $\Omega_{dr}$ limit.
We use the overlay theory points on transmittance data to ‘fit-by-eye’ to
determine an input power to applied field conversion for the HF, the
background field $E_{dc}$, and the dressing power-to-Rabi frequency
conversion. Comparing theory to experiment yields an input power to applied
field calibration factor $C_{PF}$ ($\sqrt{P_{HF}}=C_{PF}|E_{HF}|$) conversion
at 100 MHz of $C_{PF}\approx 0.40$ V/cm per $\sqrt{\textrm{mW}}$. Due to the
frequency-dependent coupling of the plates and the glass cell, we expect this
conversion to change across applied $\omega_{HF}$. We estimate a conservative
error of order 10%, given a combination of non-linear polarizability effects,
EIT linewidth, and non-uniformity of $\Omega_{dr}$. Additionally, the
existence of a $\pm 1$ sideband suggests residual dc field in the cell,
estimated at around $E_{dc}=20$ mV/cm, using the size of the $\pm 1$
splittings as $E_{HF}$ varies. We also use this method to obtain the Rabi
frequency conversion of $\Omega_{dr}\approx 2\pi\times 85$ MHz per
$\sqrt{\textrm{mW}}$ of dressing power $P_{dr}$, using the low-field limit of
HF in Fig. 2(a,b).

In Fig. 3, we show spectra in waterfalls over $E_{HF}$ with fixed
$\Omega_{dr}$, showing the non-linear induced splitting within each AT peak,
as well as the Stark shifting of the $\ket{D}$ state with $E_{HF}^{2}$.
Experimentally, $\Omega_{dr}$ is held constant ($\approx\omega_{HF}$ or
$2\omega_{HF}$), but $\omega_{dr}$ is varied to maintain $\delta_{dr}\simeq 0$
between rows. We do not compensate for frequency-dependent horn gain as we
maintain signal generator output power across frequencies. The comparison of
theory with experiment shows characteristic agreement, which we expect to
break down over the higher field values employed, as the real state energy
shifts differ from the quadratic polarizability model employed. The broadened
lines in experimental AT spectra represent a sample over many values of a
spatially non-uniform $\Omega_{dr}$ strength, while theory assumes a single
value of $\Omega_{dr}$. As well, a constant $\alpha_{D/F}$ is employed over
all sets, but in general this depends on $E_{HF}$ amplitude. There is
significant interplay in the level shifts due to both fields at high power,
causing difficulty in accurate modeling with simple two-level approximations,
or obtaining closed-form expressions for the gap. Nevertheless, this
measurement demonstrates a path toward making a spectral measurement of
$E_{HF}$ and $E_{dc}$, using a controlled microwave dressing field.

\begin{overpic}[width=433.62pt]{fig4.pdf} \put(14.0,89.0){(a)}
\put(14.0,66.0){(b)} \put(14.0,42.2){(c)} \put(14.0,20.0){(d)} \end{overpic}
Figure 4: Demonstration of multiple simultaneous tones. We average spectra
over a wide range of $\Omega_{dr}$, and remove the Stark offset. We compare
(a) no HF signal, (b) a 90 MHz, (c) 130 MHz HF field, and (d) both fields,
each at $P_{HF}=-10$ dBm($\approx 0.14$ V/cm at 100 MHz). We denote the $\pm
2$ and the $\pm 1$ absorption features by solid lines and dashed lines,
respectively, for the single tones (b,c). When both signals are applied in
(d), we denote additional features at half the sum (solid) and difference
(dashed) frequencies.

##  V Applications

We demonstrate two applications for HF field detection using the approach
presented here. The first example demonstrates the ability to detect multiple
low frequencies fields and the second example demonstrate the detection of a
amplitude modulation (AM) of a HF carrier.

###  V.1 Multiple Signal Detection

First, we demonstrate ability to detect multiple HF field simultaneously, and
the transmitted spectra for multiple simultaneous HF fields are shown Fig. 4.
This figure shows the spectra averaged over many values of $\Omega_{dr}$ (as
if collapsing the plots of Fig. 2 vertically). We compare (a) no HF field, (b)
a 90 MHz, and (c) a 130 MHz field separately, then (d) both simultaneously.
For illustration, we subtract the Stark shift so the avoided crossings are
symmetric around zero. Notably, we observe Floquet states at the sum and
difference frequencies, which then appear as absorption dips, noticeably
stronger than either single signal.

\begin{overpic}[width=247.16656pt]{fig5a.png} \put(-2.0,68.0){\large(a)}
\put(16.0,33.0){\large$-2$} \put(75.0,33.0){\large$+2$}
\put(20.0,60.0){\large$-1$} \put(75.0,60.0){\large$+1$} \end{overpic}

\begin{overpic}[width=173.44534pt]{fig5bc.pdf} \put(15.0,89.0){\large(b)}
\put(15.0,55.0){\large(c)} \end{overpic}

Figure 5:  Detection of AM signals across HF carrier frequency. (a) We plot
in-phase lock-in amplifier of the laser transmittance signal referenced to a
10 kHz modulation of $P_{HF}$, in a waterfall over $\omega_{HF}$, for fixed
$\Omega_{dr}\approx 2\pi\cdot 100$ MHz. (b) Expected relative field amplitude
from plate impedence measurements (c) Corresponding vertical slices of lock-in
signal, as if fixing $\Delta_{C}$ and $\Omega_{dr}$

###  V.2 HF Carrier AM Detection

The ability to continuously measure amplitude of an HF carrier via a laser
power transmittance allows for real-time, down-converted reception of
amplitude modulated (AM) signals. Notably, this cm-sized atom sensor package
is capable of simultaneous reception of meters-long wavelength carriers and
microwave frequencies. To demonstrate AM reception (modulating $E_{HF}$), we
apply at 10 kHz AM signal to a varying the $\omega_{HF}$ carrier frequency
while $\Omega_{dr}$ is providing a significant splitting. We then use a lock-
in amplifier on the laser transmittance signal, referenced to the AM signal,
while scanning $\delta_{C}$, tuning lock-in phase to optimize positive peaks.
In-phase peaks in the spectrum indicate resonant reception is possible by
tuning $\Omega_{dr}$ and $\delta_{C}$. In Fig. 5(a), we show in-phase lock-in
amplifier signal as the color axis, across scanning coupling laser frequency,
in a waterfall over $\omega_{HF}$, with $\Omega_{dr}/2\pi$ held fixed near
$200$ MHz. As a result of the induced splitting central to this work, we
observe in-phase modulation of the transmittance signal in both AT peaks, when
both $\pm 1\omega_{HF}$ and $\pm 2\omega_{HF}$ are brought resonant with
$\Omega_{dr}$. We observe signals using only $\approx$20 $\mu$W carrier power,
with 50% amplitude modulation, and notably, the Floquet sidebands alone are
not visible outside the parts of the spectrum populated by AT peaks. For
constant power, the electric field $E_{HF}$ from the plate’s impedance depends
on $\omega_{HF}$, and we plot $\sqrt{1-10^{S_{11}/10}}$, using reflection
coefficient of the plates $S_{11}$ in dB, measured by a SNA in Fig. 5(b). We
also observe the intensity-based Stark shifting [38] of all EIT peaks in the
absence of a resonant $\omega_{HF}$ carrier, visible in the broad in- and out-
of-phase coloring on the right and left sides of each AT peak, around
$\delta_{C}\approx\pm 100$.

In Fig. 5(c), we plot lock-in data slices to simulate the tuned in or ‘pseudo-
resonant’ case by fixed laser detuning $\delta_{C}$ (and $\Omega_{dr}$), and
scanning $\omega_{HF}$ with 5 MHz steps. Resonant peaks in the lock-in data
demonstrate the ability to pick out particular carrier frequencies using laser
detuning alone. From Fig. 5(c), we can estimate linewidth of order 10 MHz to
20 MHz, using a 5 MHz step size, and this will be the subject of future study.

##  VI Conclusion

We have demonstrated a method to detect relatively low frequency (3 MHz to 300
MHz) fields using Rydberg states of atoms, by driving AT splitting with Rabi
frequency approximately matching the HF frequency, and its second harmonic.
The low end of the frequency range is set by EIT linewidth, and the upper end
is set by dressing field uniformity. The demonstrated dynamic range is on par
with previous TM sideband measurements [26, 27, 30], although could be
improved with a LO, or other methods. The observation of a second non-linear
splitting due to $E_{HF}$ enables a path to measure field intensity in the HF
and VHF range, as well as dc field components, and the frequency of multiple
fields simultaneously, from atomic transmittance spectra. While the two-level
Floquet theory comparison was able to capture many of the spectral features
observed, we expect the precision of this style of field measurement can be
improved using a multi-state Stark-shifted approach, especially at higher
field values.

Looking toward applications, we demonstrated signal detection on HF carriers,
demonstrating its effective band-pass through Rabi matching, which is
unavailable to previous Stark-based reception. One apparent limitation of the
current apparatus is the non-uniform $\Omega_{dr}$, which will be improved
upon in future experiments. As a receiver, this compact cm-sized atomic vapor
cell is significantly smaller than wavelength-scale antennas, yet enables a
quantum receiver for HF and VHF signals.

##  Appendix A Two-level Floquet Theory

When the oscillating HF field modulates state energy faster than the linewidth
and away from atomic energy resonances with allowed transitions, one observes
induced quasi-energy states at spacing of $\hbar\omega_{HF}$, forming an
energy ladder $N\hbar\omega_{HF}$ for integer $N$. Floquet theory prescribes a
method to expand the basis states of a time-dependent Hamiltonian
($i\hbar\frac{\textrm{d}c}{\textrm{d}t}=H(t)c(t)$, where
$c(t)=\\{d(t),f(t)\\}$, the time-evolving state coefficients), into an
infinite ladder of states

$c(t)=\sum_{N=-\infty}^{\infty}c_{N}~{}\textrm{exp}\left(-i\varepsilon_{N}t/\hbar\right)$

Each new basis state has quasi-energy
$\varepsilon_{N}=\varepsilon_{0}+N\hbar\omega_{HF}$, relative to the time-
average state energies at
$\varepsilon_{D/F,0}/\hbar=\omega_{D/F}+\Sigma_{D/F}^{-}+\Sigma_{D/F}^{\sim}$.
Population probabilities are calculated by mapping the initial $|d_{0}|^{2}=1$
state into a new diagonalized basis, and normalizing the resulting
eigenvectors. The coefficients of the the $D$ and $F$ states’
$N^{\textrm{th}}$ sideband are given by $d_{N}$ and $f_{N}$ respectively,
which are rendered time-independent by this approach. In the absence of a dc
field, this ‘ladder’ of energy states has spacing $2N\omega_{HF}$, as has been
observed in Rydberg atoms [26, 27, 28, 29, 30, 31], well-predicted by this
Floquet theory. In the limiting case of $E_{dc}=0$ and $N\rightarrow\pm\infty$
Floquet states, the height of the $N^{\textrm{th}}$ sideband is given as
Bessel functions of order $N$: $J_{N}^{2}\left(\frac{-\alpha
E_{HF}^{2}}{8\omega}\right)$. Calculating a finite Floquet basis requires a
truncation of states (from $-N_{max}$ to $N_{max}$, we use $N_{max}=24$
throughout) for both the $D$ and $F$ ladders, and for our purposes must
include the rf dressing and the dc field.

The time-dependent terms of Eq. 2 will time-average to $0$, so we have
$\langle\mathcal{E}^{2}\rangle=E_{dc}^{2}+\frac{E_{HF}^{2}}{2}$, using the RMS
value for the ac part. With this field present, the $D$ and $F$ states shift
on average by $\Sigma_{D}^{-}+\Sigma_{D}^{\sim}>0$, and
$\Sigma_{F}^{-}+\Sigma_{F}^{\sim}<0$. The Stark shift in both states brings
the applied dressing frequency $\omega_{dr}$ down to the shifted resonance
with $\delta_{dr}$ (defined in Sec. III) kept near zero as the fields
increase. For these states, this differential shifting drops the observed
resonance by $(\alpha_{D}-\alpha_{F})/2=-7.552$ GHz per (V/cm)2.

Applying rf fields nearly resonant to an allowed atomic transition (here,
$\omega_{dr}\simeq\omega_{F}-\omega_{D}$), one can determine energy levels of
the atom-photon-interaction using dressed atom theory. The two-level coupling
between the $\ket{D}$ and $\ket{F}$ state has dipole strength
$\bra{56D_{5/2},m_{J}=\frac{1}{2}}e\cdot\hat{z}\ket{54F_{5/2},m_{J}=\frac{1}{2}}=\wp_{DF}\approx
1746~{}ea_{0}\approx 2234$ MHz per (V/cm). The time-dependent two-level
Hamiltonian is

$\begin{split}&H(t)/\hbar=\\\
&\left({\begin{array}[]{cc}\omega_{D}-\alpha_{D}\mathcal{E}^{2}(\omega_{HF}t)/2+\omega_{dr}&\Omega_{dr}\cos(\omega_{dr}t)\\\
\Omega_{dr}\cos(\omega_{dr}t)&\omega_{F}-\alpha_{F}\mathcal{E}^{2}(\omega_{HF}t)/2\end{array}}\right)\end{split}$
(3)

using shorthand
$\mathcal{E}^{2}(\omega_{HF}t)=\left(E_{dc}+E_{\textrm{HF}}\cos(\omega_{HF}t)\right)^{2}$.
Taking the time-average Stark shifts, we have a well-defined $\delta_{dr}$,
and using the rotating wave approximation (RWA) to shortcut another Floquet
basis, we write the typical dressed atom Hamiltonian for the two-level atom-
photon basis $\\{\ket{D,(N_{dr}+1)\omega_{dr}},\ket{F,N_{dr}\omega_{dr}}\\}$
and setting $\omega_{D}=0$:

$H_{atom-photon}=\hbar\left({\begin{array}[]{cc}0&\Omega_{dr}/2\\\
\Omega_{dr}/2&-\delta_{dr}\end{array}}\right)$ (4)

Driving this transition resonantly ($\delta_{dr}=0$), we induce AT splitting
linear with the field applied, given by the Rabi frequency:
$\Omega_{dr}=\wp_{DF}E_{dr}/\hbar$. Off resonance, eigen-energies are given by
$\varepsilon_{0,\pm}=\frac{\hbar}{2}\left(-\delta_{dr}\pm\sqrt{\delta^{2}_{dr}+\Omega_{dr}^{2}}\right)$
noting that $\delta_{dr}$ is defined including time-averaged shifts, and
therefore $\omega_{dr}$ must shift with $P_{HF}$ to maintain
$\delta_{dr}\simeq 0$. The dressing field forms an effective two-level system
for each of pair of quasi-energy states, connecting each $d_{N}$ component to
the same $N$’s $f_{N}$ state, splitting either state’s $\varepsilon_{N}$
quasi-energy into $\varepsilon_{N,\pm}$ dressed state components.

Looking to the time-dependent terms of Eq. 2, we have rotating components at
$\omega_{HF}$ with strength $\Sigma_{D/F}^{\times}/2$, the dc mixing term, and
at $2\omega_{HF}$ with strength $\Sigma_{D/F}^{\sim}/2$, the pure ac term.
These terms connect Floquet states with $\Delta N=\pm 1$, and $\Delta N=\pm
2$, in addition to the dressing between the $d_{N}$ and $f_{N}$ states. We can
write the the Schrödinger equation as a system of linear equations:

$\begin{split}\varepsilon
d_{N}=&(\Sigma_{D}^{-}+\Sigma_{D}^{\sim}+\omega_{dr}+N\omega_{HF})~{}d_{N}\\\
&+\Sigma_{D}^{\times}/2~{}(d_{N+1}+d_{N-1})\\\
&+\Sigma_{D}^{\sim}/2~{}(d_{N+2}+d_{N-2})\\\
&+\Omega_{dr}/2~{}f_{N}\end{split}$ (5)

and

$\begin{split}\varepsilon
f_{N}=&(\Sigma_{F}^{-}+\Sigma_{F}^{\sim}+N\omega_{HF})~{}f_{N}\\\
&+\Sigma_{F}^{\times}/2~{}(f_{N+1}+f_{N-1})\\\
&+\Sigma_{F}^{\sim}/2~{}(f_{N+2}+f_{N-2})\\\
&+\Omega_{dr}/2~{}d_{N}\end{split}$ (6)

The resulting Hamiltonian and basis state vector for this dressed-Floquet
system is given in Eq. 7, using $\omega_{D}+\Sigma_{D}^{-}+\Sigma_{D}^{\sim}$
as the 0 energy reference. The diagonal blocks include on the diagonal the
quasi-energy ladder, shifted by time-averaged fields and rendered near-
degenerate between diagonal blocks by the addition of $\hbar\omega_{dr}$,
included in $\delta_{dr}\simeq 0$. Just off the diagonal, we have the
$\omega_{HF}$ and $2\omega_{HF}$ interactions between (near-) neighboring
quasi-energies. In the off-diagonal blocks, we have the dressing coupling
$\Omega_{dr}$ between corresponding near-resonant states.

$\footnotesize\begin{split}&\frac{\hat{\mathcal{H}}}{\hbar}\left(\begin{array}[]{c}d_{N}\dots\\\
\hline\cr
f_{N}\dots\end{array}\right)=\varepsilon/\hbar\left(\begin{array}[]{c}d_{N}\dots\\\
\hline\cr f_{N}\dots\end{array}\right)=\\\
&\left(\begin{array}[]{ccccccc|ccccccc}\ddots&\Sigma^{\times}_{D}/2&\Sigma^{\sim}_{D}/2&0&0&0&\ddots&\ddots&0&0&0&0&0&\ddots\\\
\Sigma^{\times}_{D}/2&-2\omega_{HF}&\Sigma^{\times}_{D}/2&\Sigma^{\sim}_{D}&0&0&0&0&\Omega_{dr}/2&0&0&0&0&0\\\
\Sigma^{\sim}_{D}/2&\Sigma^{\times}_{D}/2&-\omega_{HF}&\Sigma^{\times}_{D}/2&\Sigma^{\sim}_{D}/2&0&0&0&0&\Omega_{dr}/2&0&0&0&0\\\
0&\Sigma^{\sim}_{D}/2&\Sigma^{\times}_{D}/2&0&\Sigma^{\times}_{D}/2&\Sigma^{\sim}_{D}/2&0&0&0&0&\Omega_{dr}/2&0&0&0\\\
0&0&\Sigma^{\sim}_{D}/2&\Sigma^{\times}_{D}/2&\omega_{HF}&\Sigma^{\times}_{D}/2&\Sigma^{\sim}_{D}/2&0&0&0&0&\Omega_{dr}/2&0&0\\\
0&0&0&\Sigma^{\sim}_{D}/2&\Sigma^{\times}_{D}/2&2\omega_{HF}&\Sigma^{\times}_{D}/2&0&0&0&0&0&\Omega_{dr}/2&0\\\
\ddots&0&0&0&\Sigma^{\sim}_{D}/2&\Sigma^{\times}_{D}/2&\ddots&\ddots&0&0&0&0&0&\ddots\\\
\hline\cr\ddots&0&0&0&0&0&0&\ddots&\Sigma^{\times}_{F}/2&\Sigma^{\sim}_{F}/2&0&0&0&\ddots\\\
0&\Omega_{dr}/2&0&0&0&0&0&\Sigma^{\times}_{F}/2&-2\omega_{HF}-\delta_{dr}&\Sigma^{\times}_{F}/2&\Sigma^{\sim}_{F}&0&0&0\\\
0&0&\Omega_{dr}/2&0&0&0&0&\Sigma^{\sim}_{F}/2&\Sigma^{\times}_{F}/2&-\omega_{HF}-\delta_{dr}&\Sigma^{\times}_{F}/2&\Sigma^{\sim}_{F}/2&0&0\\\
0&0&0&\Omega_{dr}/2&0&0&0&0&\Sigma^{\sim}_{F}/2&\Sigma^{\times}_{F}/2&-\delta_{dr}&\Sigma^{\times}_{F}/2&\Sigma^{\sim}_{F}/2&0\\\
0&0&0&0&\Omega_{dr}/2&0&0&0&0&\Sigma^{\sim}_{F}/2&\Sigma^{\times}_{F}/2&\omega_{HF}-\delta_{dr}&\Sigma^{\times}_{F}/2&\Sigma^{\sim}_{F}/2\\\
0&0&0&0&0&\Omega_{dr}/2&0&0&0&0&\Sigma^{\sim}_{F}/2&\Sigma^{\times}_{F}/2&2\omega_{HF}-\delta_{dr}&\Sigma^{\times}_{F}/2\\\
0&0&0&0&0&0&\ddots&\ddots&0&0&0&\Sigma^{\sim}_{F}/2&\Sigma^{\times}_{F}/2&\ddots\\\
\end{array}\right)\left(\begin{array}[]{c}\vdots\\\ d_{-2}\\\ d_{-1}\\\
d_{0}\\\ d_{1}\\\ d_{2}\\\ \vdots\\\ \hline\cr\vdots\\\ f_{-2}\\\ f_{-1}\\\
f_{0}\\\ f_{1}\\\ f_{2}\\\ \vdots\end{array}\right)\end{split}$ (7)

##  Appendix B Determining Effective Polarizability

A true Stark energy shift calculation [10] considers coupling to all other
atomic states, but in the low field limit, an effective quadratic polarization
coefficient is fit from this process. Rather than use the low-field limit for
polarizability, we calculate effective polarizabilities in-house, given
slightly non-quadratic behavior, for fields up to $\approx 0.4$ V/cm. In Fig.
6, we demonstrate an example of calculating the Stark shift from a dc electric
field when including all the states which significantly contribute to the
energy shift. In principle, this effective polarizability value changes for
different maximum fields, which alters the implied measurements of electric
field from a spectral gap.

Figure 6: Results from a many-level calculation of Stark energy shift. Fitting
across electric field values gives the effective polarizability used in this
paper for $\ket{D}$ in (a), and $\ket{F}$ in (b).

##  Appendix C List of Symbols

Symbol |  Meaning
---|---
EIT |  Electromagnetically-Induced Transparency
AT |  Autler-Townes effect, dressed state splitting
HF |  High Frequency radio band, 3 MHz to 30 MHz
VHF |  Very High Frequency radio band, 30 MHz to 300 MHz
TM |  Townes-Merritt effect, Floquet quasi-energy sidebands
$\delta_{C}$ |  Coupling laser detuning from low-field (reference cell) EIT resonance
$\omega_{dr}$ |  Dressing field angular frequency
$\Omega_{dr}$ |  HF field angular frequency
$P_{HF}$ |  Power applied to low frequency plates
$E_{HF}$ |  Low frequency electric field on plates
$E_{dc}$ |  Static electric field inside vapor cell
$\mathcal{E}(t)$ |  Time-evolving total electric field seen by the atoms
$(~{}\cdot~{})_{D/F}$ |  Two values represented, for 56D5/2 and 54F7/2
$\hbar\omega_{D/F}$ |  Energy of the unperturbed {D,F} state
$\alpha_{D/F}$ |  Polarizability of the {D,F} state
$\hbar\Sigma_{D/F}^{-}$ |  Energy shift from dc + RMS electric field of the {D,F} state. See Eq. 1
$\hbar\Sigma_{D/F}^{\sim}$ |  Energy shift from the HF RMS electric field of the {D,F} state. See Eq. 1
$\hbar\Sigma_{D/F}^{\times}$ |  Energy shift from ac/dc cross term of the {D,F} state. See Eq. 1
$\delta_{dr}$ |  Detuning of $\omega_{dr}$ from the shifted $\ket{D}$ and $\ket{F}$ state energy gap
$S_{\delta}$ |  Scaling parameter for $\delta_{dr}$, near 1.0
$\varepsilon_{N}$ |  N${}^{\textrm{th}}$ Floquet sideband quasi-energy
$(~{}\cdot~{})_{\pm}$ |  Two values represented, for the upper ($+$) and lower ($-$) AT dressed state
Table 1: Table of non-standard variables used in this paper.

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