††thanks: Present address: Physical Sciences Inc., Andover, MA 01810, USA

# High-angular-momentum Rydberg states in a room-temperature vapor cell for DC
electric-field sensing

Alisher Duspayev<EMAIL_ADDRESS>Department of Physics, University of
Michigan, Ann Arbor, MI 48109, USA  Ryan Cardman  Department of Physics,
University of Michigan, Ann Arbor, MI 48109, USA  David A. Anderson  Rydberg
Technologies Inc., Ann Arbor, MI 48103, USA  Georg Raithel  Department of
Physics, University of Michigan, Ann Arbor, MI 48109, USA

###### Abstract

We prepare and analyze Rydberg states with orbital quantum numbers $\ell\leq
6$ using three-optical-photon electromagnetically-induced transparency (EIT)
and radio-frequency (RF) dressing, and employ the high-$\ell$ states in
electric-field sensing. Rubidium-85 atoms in a room-temperature vapor cell are
first promoted into the $25F_{5/2}$ state via Rydberg-EIT with three infrared
laser beams. Two RF dressing fields then (near-)resonantly couple $25\ell$
Rydberg states with high $\ell$. The dependence of the RF-dressed Rydberg-
state level structure on RF powers, RF and laser frequencies is characterized
using EIT. Furthermore, we discuss the principles of DC-electric-field sensing
using high-$\ell$ Rydberg states, and experimentally demonstrate the method
using test electric fields of $\lesssim$ 50 V/m induced via photo-illumination
of the vapor-cell wall. We measure the highly nonlinear dependence of the DC-
electric-field strength on the power of the photo-illumination laser.
Numerical calculations, which reproduce our experimental observations well,
elucidate the underlying physics. Our study is relevant to high-precision
spectroscopy of high-$\ell$ Rydberg states, Rydberg-atom-based electric-field
sensing, and plasma electric-field diagnostics.

##  I Introduction

Rydberg-atom-based sensors for electromagnetic fields are a prominent example
how atomic-physics research is being translated into applications (for recent
reviews see, for example, Adams  _et al._ (2019); Anderson  _et al._ (2020);
Fancher  _et al._ (2021); Artusio-Glimpse  _et al._ (2022); Yuan  _et al._
(2023)), offering high sensitivities, large bandwidth, traceability and self-
calibration Holloway  _et al._ (2014); Anderson  _et al._ (2021a). Modern
Rydberg field sensors avoid complex apparatuses by the use of room-temperature
vapor cells and electromagnetically-induced transparency (EIT) Boller  _et
al._ (1991) for optical interrogation of field-sensitive Rydberg atoms
Mohapatra  _et al._ (2007), allowing hybridization with existing classical
technologies and the realization of portable instrumentation Simons  _et al._
(2018); Anderson  _et al._ (2018, 2021a); Meyer  _et al._ (2021); Mao  _et
al._ (2023). Beyond demonstrations of electric field sensing with atomic
Rydberg states Osterwalder and Merkt (1999); Sedlacek  _et al._ (2012);
Holloway  _et al._ (2014), Rydberg sensors have proven their potential utility
in modulated signal reception from long-wavelength radio-frequencies to
millimeter-wave Anderson  _et al._ (2021b); Holloway  _et al._ (2021);
Prajapati  _et al._ (2022); Legaie  _et al._ (2023), phase detection Anderson
_et al._ (2022); Berweger  _et al._ (2023); Vouras  _et al._ (2023),
polarization measurements Sedlacek  _et al._ (2013); Jiao  _et al._ (2017);
Anderson  _et al._ (2018, 2021b); Wang  _et al._ (2023), spatial field mapping
and RF imaging Holloway  _et al._ (2014); Downes  _et al._ (2020); Cardman
_et al._ (2020); Anderson  _et al._ (2023) and in the establishment of new
atomic measurement standards Sedlacek  _et al._ (2012, 2013); Holloway  _et
al._ (2018, 2022a); Duspayev and Raithel (2022).

More recently, two-photon Rydberg-EIT has been generalized to three optical
photons Carr  _et al._ (2012); Thaicharoen  _et al._ (2019); Prajapati  _et
al._ (2023a), which afford operation at all-infrared wavelengths Thoumany  _et
al._ (2009); Fahey and Noel (2011); Johnson  _et al._ (2012); You  _et al._
(2022), allow efficient Doppler-shift cancellation conducive to narrower EIT
linewidth Bohaichuk  _et al._ (2023), and offer inroads towards excitation of
high-angular-momentum ($\ell\geq 3$) Rydberg states for electric-field
detection in very-high and ultra-high frequency bands Brown  _et al._ (2023);
Elgee  _et al._ (2023); Prajapati  _et al._ (2023b). High-$\ell$ Rydberg
states exhibit large sensitivities to DC fields Ma  _et al._ (2022) due to
their DC electric polarizabilities, which scale as $n^{7}$ (with principal
quantum number $n$) and also increase with $\ell$ Gallagher (2005). Hence, $n$
and $\ell$ can be traded in favorable ways to achieve high DC-field
sensitivity at a comparatively low $n$, where optical excitation pathways are
more efficient than at high $n$, and where Rydberg-atom interactions (which
typically scale with high powers of $n$) are less limiting. These aspects are
expected to be particularly relevant to field sensing in plasmas Alexiou  _et
al._ (1995); Feldbaum  _et al._ (2002); Borghesi  _et al._ (2002); Park  _et
al._ (2010); Anderson  _et al._ (2017); Weller  _et al._ (2019); Chng  _et
al._ (2020); Goldberg  _et al._ (2022) and ion-beam sources Smith  _et al._
(2006); Bassim  _et al._ (2014); Bischoff  _et al._ (2016); Gierak  _et al._
(2018). For potential use in the latter Knuffman  _et al._ (2013); Claessens
_et al._ (2007); McCulloch  _et al._ (2016), methods for Rydberg EIT plasma
field measurement and diagnostics have been developed Anderson  _et al._
(2017) and microfield sensing using Rydberg atoms have recently been
demonstrated Duspayev and Raithel (2023).

Motivated by prospects in plasma-field sensing, here we investigate the
response of high-$\ell$ Rydberg atoms to electric fields in a vapor cell. We
prepare $25F$, $25G$, $25H$ and $25I$ ($\ell=3$ to 6) Rydberg states in
rubidium-85 (85Rb) using three-optical-photon EIT and up to two radio-
frequency (RF) dressing fields. The efficacy of high-$\ell$ Rydberg-state
preparation is verified by measurements of Autler-Townes (AT) splittings
induced by the RF dressing fields. DC electric fields are created by photo-
illumination of the dielectric vapor-cell wall using 453-nm light Ma  _et al._
(2020), and the response of the high-$\ell$ Rydberg states to the DC fields is
explored. The paper is organized as follows. In Sec. II we outline the
benefits of high-$\ell$ Rydberg states for DC-field sensing. In Sec. III we
describe our experimental setup and the utilized RF dressing methods. In Sec.
IV we explain our method of creating DC electric fields by illuminating the
cell wall with a $453$-nm laser. We then present our DC-field sensing results,
which include an approximate calibration of the weak, DC field in the vapor
cell versus $453$-nm illumination power. In the concluding Sec. V, possible
applications and research venues for future work are discussed.

##  II High-$\ell$ Rydberg states in static electric fields

FIG. 1: Computed Stark maps for 85Rb $n=25$ and two sample cases of $|m_{J}|$, and zoomed-in detail in the inset, illustrating the increasingly sensitive quadratic response of Rydberg atoms to weak DC electric fields as $\ell$ increases from $\ell=3$ (25$F$) to $\ell=7$ (25$J$). The fine structure is not resolved in this figure. Table 1:  Calculated scalar and tensor, and $|m_{J}|$-dependent (see Eq. 2) DC electric polarizabilities, $\alpha_{S}$, $\alpha_{T}$, and $\alpha_{DC}(|m_{J}|)$ respectively, [all in MHz/(V/m)2] for the high-$\ell$ Rydberg states used in our study. State | $\alpha_{S}$ | $\alpha_{T}$ | $|m_{J}|$ | $\alpha_{DC}$
---|---|---|---|---
$25F_{5/2}$ | 0.010037 | -0.003492 | 1/2 | 0.012830
3/2 | 0.010735
5/2 | 0.006544
$25G_{7/2}$ | 0.039466 | -0.014841 | 1/2 | 0.050067
3/2 | 0.045827
5/2 | 0.037346
7/2 | 0.024626
$25H_{9/2}$ | 0.106447 | -0.042547 | 1/2 | 0.134812
3/2 | 0.127720
5/2 | 0.113538
7/2 | 0.092264
9/2 | 0.063899
$25I_{11/2}$ | 0.247529 | -0.105721 | 1/2 | 0.314806
3/2 | 0.303273
5/2 | 0.280206
7/2 | 0.245606
9/2 | 0.199474
11/2 | 0.141808

Rydberg atoms are very sensitive to external electric fields Gallagher (2005).
Applications discussed in Sec. I include the measurement of electric fields in
plasmas and ion sources. In a recent cold-atom study Duspayev and Raithel
(2023), $\ell=1$ ($P$-) states were employed at relatively high-$n$ to measure
electric fields on the order of a few tens of V/m with $\ell=3$ ($F$-) states
to increase sensitivity to weaker electric fields. In this work, we increase
the Rydberg-state electric-field sensitivity by extending to higher $\ell$
Rydberg states beyond $\ell=3$, while also using considerably lower $n$
Rydberg states, which exhibit larger optical oscillator strengths, scaling as
$n^{-3}$ Duspayev and Raithel (2022); Gallagher (2005), affording stronger
spectroscopic signals. Lower-$n$ states also provide the benefit of being less
sensitive to Rydberg-atom interactions Han  _et al._ (2018, 2019); Gaj  _et
al._ (2014), making them suitable for Rydberg-atom applications in inert
buffer gases and in low-pressure discharge plasmas. The latter applications
benefit from a room-temperature vapor-cell platform Anderson  _et al._ (2017);
Ma  _et al._ (2020), which we employ in the present work. Electric
polarizabilities increase quickly with $\ell$, which is key to reducing $n$
while maintaining a desired level of field sensitivity. Lastly, electric-
dipole transition matrix elements between high-$\ell$ Rydberg states are large
($\sim n^{2}$) Duspayev and Raithel (2022); Gallagher (2005), allowing
efficient RF dressing to access higher-$\ell$ states with relatively weak RF
fields.

In Fig. 1 we have calculated Stark maps of Rb Rydberg states near $n=25$ for
two exemplary values of $|m_{J}|$, the conserved component of the electronic
angular momentum in the direction of the DC electric field, $E_{DC}$. The
Stark maps visualize the quadratic Stark effect of the levels
$|n=25,\ell,J,m_{J}\rangle$, indicating a large increase in polarizability
from $\ell=3$ to $7$. The quadratic Stark shifts,

$\Delta W=-\frac{1}{2}\alpha_{DC}|E_{DC}|^{2},$ (1)

are given by state-dependent electric polarizabilities, $\alpha_{DC}$. The
latter can be calculated as

$\alpha_{DC}(m_{J})=\alpha_{S}+\alpha_{T}\frac{3m_{J}^{2}-J(J+1)}{J(2J-1)},$
(2)

with $(n,\ell,J)$-dependent scalar and tensor polarizabilities $\alpha_{S}$
and $\alpha_{T}$, respectively. Table 1 lists calculated values of
$\alpha_{S}$, $\alpha_{T}$ and $\alpha_{DC}$ for the $n=25$ high-$\ell$
Rydberg states utilized in our study. As one can see, the tensor contributions
are substantial. In the experiments discussed in the following Sections, only
states with $|m_{J}|=1/2,3/2,5/2$ are coupled and relevant to electric-field
sensing. It is apparent that $\alpha_{DC}$ increases by a factor of $\approx$
3 with each increment in $\ell$. Thus, with our progression from $F$ to $I$
states Stark shifts increase by about a factor of 30. Here we are able to
measure Stark shifts as low as about 3 MHz, corresponding to $E_{DC}$
sensitivity limits of $\sim$20 V/m and $\sim$4 V/m for $25F$ and $25I$ states,
respectively. These sensitivities are suitable for measuring electric fields
in plasmas. The approximate $\sim n^{-7/2}$-scaling of the field sensitivity
as a function of $n$ may be used to match sensitivity requirements elsewhere.

##  III Excitation of $\ell\leq$ 6 Rydberg atoms in a room-temperature vapor
cell

We use a multi-photon scheme with up to six photons to access high-$\ell$
states for electric-field sensing in a vapor cell. We first describe the
optical three-photon Rydberg-EIT setup, which provides optical spectroscopic
access to the Rb $25F_{5/2}$ Rydberg states. Then, we describe the three modes
of strong RF drive using additional one-, two-, and three-RF-photon couplings,
which we employ to extend the state space to include $25G_{7/2}$, $25H_{9/2}$
and $25I_{11/2}$ states, respectively. These high-$\ell$ states are then
utilized to measure weak DC electric fields.

FIG. 2: Sketch of the experimental setup (a) and the energy-level diagram (b)
(not to scale). See text for details.

###  III.1 Experimental Setup

We concentrate on the 85Rb isotope, but analogous experiments could be
performed with other Rb isotopes and atomic species. Three laser beams are
aligned co-linearly into a room-temperature cell of length $L_{c}\approx 7.5$
cm filled with a natural Rb vapor, as illustrated in Fig. 2 (a). The energy-
level diagram in Fig. 2 (b) identifies the utilized atomic transitions. The
three lasers driving transitions have approximate wavelengths of 795 nm, 762
nm and 1280 nm, and are referred to as EIT probe, dressing, and coupler
lasers, respectively Carr  _et al._ (2012); Thaicharoen  _et al._ (2019). The
probe laser is an external-cavity diode laser (ECDL) that is locked to the
Doppler-free $5S_{1/2},F=3\rightarrow 5P_{1/2},F=3$ transition on the 85Rb D1
line using saturation spectroscopy. A small portion of the 795-nm light is
red-shifted by $\Delta_{r}=-42$ MHz and counter-aligned with a small portion
of the 762-nm dressing laser, which also is an ECDL, in a separate two-photon
EIT reference vapor cell. The 762-nm dressing laser is locked to the
$5P_{1/2},F=3\rightarrow 5D_{3/2},F=4$ EIT line in this separate cell,
resulting in a shift of
$\Delta_{d}=-\Delta_{r}\frac{795~{}{\rm{nm}}}{762~{}{\rm{nm}}}=44$ MHz. In the
measurement cell in Fig. 2 (a), the 795-nm laser probes the
$5S_{1/2},F=3\rightarrow 5P_{1/2},F=3$ resonance for atoms near zero velocity
along the $z$-axis, and the 762-nm laser is blue-detuned by $\Delta_{d}=44$
MHz from the zero-velocity $5P_{1/2},F=3\rightarrow 5D_{3/2},F=4$ resonance.

The tunable $\approx$ 1280-nm coupler laser drives the transition to the
$25F_{5/2}$ Rydberg state. While the coupler laser is scanned, the
transmission of the locked probe is detected using a photo-diode [PD in Fig. 2
(a)]. The PD current is amplified in a transimpedance amplifier, and the
resultant EIT signal is recorded on an oscilloscope. Each experimental
spectrum is an average of typically 300 scans. To calibrate the coupler-laser
frequency scan, a portion of the 1280-nm laser power is sent through a Fabry-
Pérot (FP) etalon with a free spectral range of 374 MHz whose transmission
resonances are recorded simultaneously with the spectroscopic measurements and
used as frequency test points in post-processing. For higher-resolution
coupler-laser frequency calibration, we also RF-modulate the beam sample in a
fiber electro-optic modulator before passing it through the FP, generating a
higher density set of frequency test points.

The probe and the coupler lasers are co-aligned in the measurement cell [$+z$
direction in Fig. 2 (a)], while the dressing beam is counter-aligned to them
($-z$ direction). We use various dichroic mirrors [DM in Fig. 2 (a), only one
is shown] to align the lasers of the different wavelengths through the cell.
The counter-alignment of the probe and dressing beams helps with preventing
leakage of dressing light onto the PD used to record the EIT spectra. Further,
by routing the beams through polarization-maintaining fibers we ensure that
all lasers have horizontal polarization [parallel to $x$ in Fig. 2 (a)]. The
beams are focused into the cell with full-widths-at-half-maxima (FWHM) of the
intensity of $\approx$ 80 $\mu$m for the probe and $\approx$ 250 $\mu$m for
the dressing beam and the coupler. Typical powers are $\approx$ 10 $\mu$W,
$\approx$ 8 mW and $\approx$ 15 mW for the probe, the dressing and the coupler
beams, respectively. To raise the three-photon EIT signal, we first find and
optimize the two-photon EIT signal on the $5S_{1/2}\rightarrow
5P_{1/2}\rightarrow 5D_{3/2}$ cascade [see Fig. 2 (b)] while scanning the
dressing beam (with the coupler blocked). With the dressing beam properly
aligned and locked, we un-block the coupler and scan it to search for the
three-photon EIT signal. Once the three-photon EIT signal has been located, it
is optimized further by fine-tuning the coupler alignment.

To drive RF Rydberg-Rydberg transitions from $25F_{5/2}$, we use two different
antennas to access different frequency ranges. The first is a log-periodic
planar printed-circuit-board (PCB) antenna with 7 dBi gain for frequencies
between 1 and 10 GHz, which emits an $x$-polarized field [see Fig. 2 (a)].
This RF line starts with a signal generator with power $P_{1}$, followed by a
coaxial cable (11 dB loss) connection to a 40-dB-gain amplifier and PCB
antenna. The antenna is mounted above the cell at a distance of $d\approx 15$
cm, which is marginally in the antenna far-field, and is oriented to co-align
the polarization of the microwave with the laser polarizations in the cell. In
this study, the PCB antenna is used to drive the $25F_{5/2}\leftrightarrow
25G_{7/2}$ and $25F_{5/2}\leftrightarrow 25H{9/2}$ transitions [field with
frequency $f_{1}$ and power $P_{1}$ in Fig. 2 (b)]. A second telescopic
rabbit-ears antenna is fed directly by a second signal generator to apply the
RF field driving the $25H_{9/2}\leftrightarrow 25I_{11/2}$ transition [field
with frequency $f_{2}$ and power $P_{2}$ in Fig. 2 (b)]. The rabbit-ears
antenna, shown in Fig. 2 (a), is folded such that it brackets the vapor cell
in the $xz$-plane with its ears parallel along $z$ to generate linear RF
polarization along $x$, and a small offset in $-y$ to avoid obstructing the
alignment of the laser beams. The optimal leg length, which is on the order of
$c/(4f_{2})$, where $c$ is the speed of light, is kept at $L_{2}\approx$ 22 cm
to maximize the RF field inside the cell.

###  III.2 $25F_{5/2}\longleftrightarrow 25H_{9/2}$ transition

FIG. 3: Spectroscopy of the $25F\longleftrightarrow 25H$ two-photon transition
vs coupler-laser detuning. Experimental frequency and power scans are shown in
(a) and (c) respectively, with the corresponding calculations in (b) and (d).
In the experimental data in (a) and (c), the $y$-axes show absolute frequency,
$f_{1}$, and power at the signal generator, $P_{1}$, respectively, and the
signals are displayed on a linear gray scale. In the theory plots the areas of
the symbols show approximate signal strength. In (b) the $y$-axis shows half
of the two-photon detuning from the low-field resonance, and in (d) intensity
in dBI (0 dBI = 1 W/m2).

Experimental scans of $f_{1}$ and $P_{1}$ for the log-periodic antenna driving
the two-photon $25F_{5/2}\leftrightarrow 25H_{9/2}$ transition are shown in
Figs. 3 (a) and (c) respectively (the rabbit-ears antenna is off). In (a)
$P_{1}=-8$ dBm at the signal generator, corresponding to about 21 dBm
injection into the antenna, after taking cable losses and amplifier gain into
account. With the 7 dBi antenna gain, we estimate an RF electric field at the
atoms of about 41 V/m. Corresponding calculations of dressed-state energy
levels for the optically accessed $m_{J}=1/2$, $3/2$ and $5/2$ manifolds are
shown in Fig. 3 (b) and (d), with symbol area proportional to the $F$-state
probability in the dressed states, which gives an approximate relative measure
for EIT signal strength. In the calculations we use the absolute intensity
unit dBI defined with reference to $I_{0}$=1 W/m2, i.e. the intensity
$P($dBI$)=10\log_{10}[I/(1~{}$W/m${}^{2})]$ with intensity $I$ in SI units.
The calculations include off-resonant AC shifts of all states in the RF field
and utilize computed two-photon matrix elements between
$|25F_{5/2},m_{J}\rangle$ and $|25H_{9/2},m_{J}\rangle$.

Generally, we obtain very good agreement between the experimental data and the
supporting computations, including measured two-photon Autler-Townes (AT)
splittings and electric-field signal strengths. There are three AT pairs, one
for each optically-coupled $m_{J}$. From Fig. 3 (a) one can conclude that the
two-photon resonance occurs at $\approx$ 3.158 GHz, i.e. the weak-field energy
level difference is $h\times 6.316$ GHz, which agrees with the calculated
energy-level difference to within 1 MHz. As $P_{1}$ is increased, the red-
shifted AT peak in Fig. 3 (c) exhibits larger shifts and broadening than the
blue-shifted one, and approaches the noise floor at $P_{1}\gtrsim$ 1 dBm. The
strong $m_{J}$-splitting of the lower AT branch, visible in (d) at an
intensity of $P_{1}>12$ dBI, is marginally resolved in the experimental data
in (c). The asymmetry of the pattern in Figs. 3 (c) and (d) results from the
combination of AT splitting and AC shifts. By comparing measured and
calculated spectra in Figs. 3 (c) and (d) one obtains an approximate atomic
power calibration of $P_{1}(\text{dBI})=P_{1}(\text{dBm})+$ 14.5 dB. Taking
into account the 1 dBm step-size in the experimental data in (c), the
calibration uncertainty on this is $\pm$ 1dB. Also, the RF electric-field
amplitude at $P_{1}=-8~{}$dBm is $E_{RF1}=58\pm 7$ V/m, which is sufficiently
precise for the present objective. The measured field exceeds the field
estimated from the RF transmission line by about 40%, which may be
attributable to the fact that the vapor cell is only borderline in the far
field.

###  III.3 $25F_{5/2}\longleftrightarrow 25H_{9/2}\longleftrightarrow
25I_{11/2}$ transitions

FIG. 4: Spectroscopy of the $25F\longleftrightarrow 25H\longleftrightarrow
25I$ transition with both RF fields on vs coupler-laser detuning. Experimental
frequency and power scans of the signal generator driving the
25H$\longleftrightarrow$25I transition are shown in (a) and (c), respectively.
A power scan of the signal generator driving the $25F\longleftrightarrow 25H$
transition is shown in (e). The $y$-axes show absolute frequency, $f_{1}$, in
(a), and powers at the respective signal generators in (c) and (e), and the
signals are displayed on a linear gray scale. Corresponding calculations are
presented in (b), (d) and (f). In the theory plots, the areas of the symbols
show approximate signal strength. The $y$-axis in (b) shows detuning from the
25H$\longleftrightarrow$25I resonance, and those in (d) and (f) RF field
intensities in units dBI.

We use the rabbit-ears antenna to drive the $25H_{9/2}\leftrightarrow
25I_{11/2}$ transition. In this case, there are three RF-coupled Rydberg
states (per $|m_{J}|$), leading to the three dressed-state peaks in the
experimental spectra in Fig. 4. The $|m_{J}|$-splittings are only resolved in
the calculations. The calculations include off-resonant AC shifts of all
states caused by both RF fields, and we use two-photon matrix elements for the
$|25F_{5/2},m_{J}\rangle$ $\leftrightarrow$ $|25H_{9/2},m_{J}\rangle$ and one-
photon matrix elements for the $|25H_{9/2},m_{J}\rangle$ $\leftrightarrow$
$|25I_{11/2},m_{J}\rangle$ transitions. As before, only the optically accessed
$m_{J}=1/2$, $3/2$ and $5/2$ manifolds are relevant. While $m_{J}$-splittings
occur in the calculations in Figs. 4 (b), (d) and (f), these are not resolved
in the present experimental data, leaving three broadened peaks in the data in
(a), (c) and (e). Regardless, overall good agreement between the experimental
data and the theory is observed.

For the scan of $f_{2}$ in Fig. 4 (a), it is $P_{1}=-5$ dBm and $f_{1}=3.158$
GHz, the resonance frequency found in Fig. 3, and the power into the rabbit-
ears antenna is set at $P_{2}=$ 0 dBm. The $25H_{9/2}\leftrightarrow
25I_{11/2}$ resonance is observed at $\approx$ 340 MHz, which agrees with our
calculated value of $339.6$ MHz within uncertainties. The value $f_{2}=340$
MHz is chosen for the $P_{1}$\- and $P_{2}$-scans in Figs. 4 (c) and (e).

Investigating the power dependence, we observe that as $P_{2}$ is increased in
Fig. 4 (c), while holding $P_{1}$ at $-5$ dBm, the rightmost peak blue-shifts
and disappears at $P_{2}\gtrsim$ 8 dBm. In the corresponding calculations in
Fig. 4 (d), the rightmost peak $m_{J}$-splits into three lines, while their
approximate signal strengths (proportional to symbol area in the theory plots)
significantly diminish, in agreement with the experimental data in Fig. 4 (c).
At the same time, the leftmost and the central lines remain relatively strong
throughout the tested $P_{2}$-range and experience small red shifts. As
$P_{2}$ is increased, in experiment [Fig. 4 (c)] and calculation [Fig. 4 (d)]
the oscillator strength redistributes into the central line. Comparing Figs. 4
(c) and (d), we calibrate $P_{2}$ and find
$P_{2}(\text{dBI})=P_{2}(\text{dBm})-$ 13.5 dB $\pm$ 1 dB. The main source of
uncertainty is the step size in $P_{2}$ of 1 dBm. Again, in the present
context an approximate atomic calibration is sufficient. At a power of
$P_{2}=0$ dBm the electric-field amplitude $E_{RF2}=5.8\pm 0.7$ V/m. For this
field, a distance of 3 cm between the legs of the rabbit-ears antenna, and for
impedance-matched coupling, the RF power feeding the antenna would be about
-10 dBm. Factoring in transmission-line losses and some (likely) impedance
mismatch, this is reasonably close to the actual signal generator power of
$P_{2}=0$ dBm.

Finally, we explore the effect of varying $P_{1}$ on the three-peak EIT
spectra in Fig. 4 (e). Here $f_{2}=$ 340 MHz and $P_{2}=$ 0 dBm. As can be
seen in Fig. 4 (e), the observed shifts as well as the signal-strength
distribution change quite dramatically; this is because the $F$ to $H$
transition is a 2-photon transition. First, the rightmost peak exhibits
positive shifts, similar to Fig. 4 (c). However, opposite to Fig. 4 (c), it
becomes the strongest peak. This peak would be expected to $m_{J}$-split [see
Fig. 4 (f)]. Due to the intrinsic EIT linewidth and likely inhomogeneities of
the RF intensities within the cell, in the experiment we observe line
broadening instead of splitting [see Fig. 4 (e)]. Second, the central peak,
while slightly red-shifting, almost disappears at $P_{1}>$ 3 dBm, in
accordance with the calculations in Fig. 4 (f). Finally, we observe that the
leftmost line massively red-shifts (by $\approx$ 150 MHz at $P_{1}\sim$ 0
dBm). Here, the $m_{J}$-splitting, seen in the calculation in Fig. 4 (f), is
borderline resolved in the experiment.

##  IV DC-field sensing

FIG. 5: Panels (a), (c), (e), and (g): Calculated EIT spectra vs coupler-laser
detuning, $\Delta_{c}$, and DC electric field, $E_{DC}$, with the indicted RF
transitions applied. Corresponding experimental EIT spectra in selected light-
induced DC electric fields are shown in panels (b), (d), (f), and (h) (bold
red curves), together with corresponding spectra at fixed $E_{DC}$ from the
plots above that best match the experimental data (gray curves). The best-fit
$E_{DC}$-values extracted from the match of each measured spectrum to the
calculated spectra are indicated. Best-fit $E_{DC}$-values versus 453-nm
photo-illumination laser power, $P_{453}$, are shown in panel (i).

In this Section we utilize Rydberg states with $\ell\leq$ 6 prepared as
described in the previous Section for DC-electric-field sensing. The method of
creating DC fields by photo-illumination of the vapor-cell wall is first
reviewed, followed by experimental measurements of the DC fields. The atomic-
calibration-based estimates for $E_{RF1}$ and $E_{RF2}$ are employed to model
the experimental data.

###  IV.1 Creating DC fields in vapor cell by wall photo-illumination

Applying DC electric fields to atomic vapors confined in glass cells has
turned out to be a considerable challenge Mohapatra  _et al._ (2007); Sedlacek
_et al._ (2012); Holloway  _et al._ (2014), as these devices exhibit a robust
isolation against external static fields. Random, weak electric fields
(sometimes referred to as microfields) Hooper (1966); Potekhin  _et al._
(2002); Demura (2010) may still be present and require an experimental
characterization Holloway  _et al._ (2017). Charges generating the microfields
may arise from effects including photo-ionization of atoms, and collisions
involving atoms in Rydberg Gallagher (2005); Beterov  _et al._ (2009) or low-
lying intermediate states (such as $5P_{1/2}$ and $5D_{3/2}$) Cheret  _et al._
(1982); Barbier  _et al._ (1986, 1987); Barbier and Cheret (1987).

To demonstrate DC electric field measurement using high-$\ell$ Rydberg states,
in the present work we require a simple method to generate DC test fields in
the cell. While electrodes inside the cell or the cell walls Barredo  _et al._
(2013); Grimmel  _et al._ (2015); Anderson  _et al._ (2018); Ma  _et al._
(2022) can be used for that, no such electrodes are present in our
experiments. Therefore, we use an alternative method in which free charges are
generated on the cell walls by laser illumination and photo-electric effect
Rousseau  _et al._ (1968). In Hankin  _et al._ (2014); Jau and Carter (2020),
this approach was utilized to create patch regions with local electric fields
on glass surfaces. It has also been shown that the light-induced DC fields can
be quite homogeneous and reach up to $\sim$ 60 V/m within the vapor cell Ma
_et al._ (2020).

In our work, a 453-nm laser beam of $\approx$ 1 cm diameter, that forms a
small ($\approx$ 20∘) angle with respect to the cell axis, is aligned in a way
that it illuminates a cell-wall section close to one of the legs of the
rabbit-ears antenna along the entire length of the cell. This generates an
approximately homogeneous DC electric field, $E_{DC}$, within the EIT laser
beams passing through the cell. The field $E_{DC}$ is approximately parallel
to all laser and RF electric fields in our setup, which all point along $x$ in
Fig. 2 (a). The dependence of the $E_{DC}$-magnitude on 453-nm laser power is
highly nonlinear Ma  _et al._ (2020). Hence, in addition to showing that
high-$\ell$ Rydberg-EIT is suitable for measurement of DC electric fields in
vapor cells, in effect we also determine what 453-nm laser power results in
what level of $E_{DC}$ in our experiment.

###  IV.2 Overview of results

In Figs. 5 (a), (c), (e) and (g), we plot calculated EIT maps versus coupler-
laser detuning and $E_{DC}$. The calculation for the RF-dressed Rydberg levels
is similar to those in Secs. III.2 and III.3, with the addition of the DC
Stark shifts using the DC polarizabilities from Table 1. This yields RF-
dressed Rydberg-level energies, $\Delta W(k,m_{J})$, and corresponding
$|25F,m_{J}\rangle$-amplitudes, $c(k,m_{J})$, where $k$ is a dressed-state
label counting from 1 to up to 3 for each $m_{J}$ (which is conserved for all-
parallel electric fields). Only levels with $|m_{J}|=1/2$, $3/2$ or $5/2$ are
coupled. We then use an empirical finding according to which the EIT line
strengths scale with $|\Omega_{C}\Omega_{P}|^{2}$, where $\Omega_{P}$ is the
probe Rabi frequency on the transition
$|5S_{1/2},F=3,m_{F}\rangle\leftrightarrow|5P_{1/2},F^{\prime}=3,m_{F^{\prime}}=m_{F}\rangle$,
and $\Omega_{C}$ is the effective two-photon coupler Rabi frequency on the
transition
$|5P_{1/2},F^{\prime}=3,m_{F}\rangle\leftrightarrow|25F_{5/2},F^{\prime\prime\prime},m_{F}\rangle$
transition. $\Omega_{C}$ is coherently summed over all intermediate, detuned
states $|5D_{3/2},F^{\prime\prime},m_{F}\rangle$, where the summing index is
$F^{\prime\prime}$. Performing all sums over $m_{F}$ and the unresolved
Rydberg hyperfine structure, $F^{\prime\prime\prime}$, one finds EIT line
strengths proportional to $S(k,|m_{J}|)=|c(k,|m_{J}|)|^{2}w(|m_{J}|)$, where
the Rydberg physics is contained in the determination of the
$|25F,|m_{J}|\rangle$-amplitudes in the RF-dressed states, $c(k,|m_{J}|)$, and
the EIT readout physics in the $|m_{J}|$-weights, $w(|m_{J}|)$. The magnitude-
squares of $c(k,|m_{J}|)$ correspond with the areas of the symbols in Figs. 3
and 4, whereas in Fig. 5 we use the $S(k,|m_{J}|)$. The weights $w(|m_{J}|)$
depend somewhat on the intermediate-state detuning $\Delta_{d}$ in Fig. 2 (b)
and are, in our case, about $w(|m_{J}|)=0.43$, 0.29, and 0.28 for
$|m_{J}|=1/2$, $3/2$, and $5/2$, respectively. EIT is observed when the
coupler detuning $\Delta_{c}$ equals to one of the dressed-state energies,
$\Delta W(k,|m_{J}|)$. To obtain the model spectra displayed in the top row in
Fig. 5, we convolute the EIT lines with identical Gaussian line profiles with
FWHM of 18 MHz, the approximate width of the $25F$ EIT line without RF
dressing fields, and sum over all $(k,|m_{J}|)$.

In Figs. 5 (b), (d), (f) and (h), exemplary experimental EIT spectra for the
corresponding cases are shown, together with the best-matching spectra from
the calculated maps above. For clarity, the calculated spectra are slightly
offset from the experimental data. Measured and computed spectra are matched
empirically, which is sufficient at the present level. In Fig. 5 (i) we plot
the matched values of $E_{DC}$ from all other panels of the figure versus
453-nm photo-illumination power, $P_{453}$. There, we approximately calibrate
the DC electric field as a function of illumination power using high-$\ell$
Rydberg EIT spectroscopy for our experimental setup. In the remainder of the
paper, we discuss these results in detail.

###  IV.3 Detailed analysis of results

In Figs. 5 (a) and (b) it is apparent that the $25F_{5/2}$-state, which has
the lowest polarizabilities among the states tested here, only exhibits
significant Stark shifts, exceeding about 10 MHz, for $E_{DC}\gtrsim 40$ V/m,
which already is near the largest fields that we can generate with the photo-
illumination method used. As $E_{DC}$ is increased further to 50 V/m (by
increasing $P_{453}$), the experimental spectra merely shift by an additional
3-5 MHz and exhibit a slight broadening. Increasing $n$ aids in achieving a
stronger response of $nF_{5/2}$ states. For example, the $57F_{5/2}$ EIT peak
was seen to disappear entirely at our maximum $P_{453}$ (not shown). In the
following, we employ RF-dressing at $n=25$ to access higher-$\ell$ Rydberg
states at the same $n$ to elicit a stronger DC electric-field response,
afforded by the higher polarizabilities of those states (see Table. 1).

Increasing $\ell$ by driving the $25F_{5/2}\leftrightarrow 25G_{7/2}$
transition with the log-periodic antenna ($f_{1}=$ 5.234 GHz, $P_{1}$ = -20
dBm), moderately enhances the response, as seen in Figs. 5 (c) and (d). The
two AT peaks start to notably change at $E_{DC}\gtrsim$ 20 V/m. The left AT
peak shifts and broadens considerably more than the right one. At fields
$E_{DC}\gtrsim$ 40 V/m, the left AT peak diminishes greatly in signal strength
and eventually becomes indiscernible while becoming more responsive to changes
in $E_{DC}$. This behavior reflects the fact that the dressed state associated
with the left AT peak turns into the stronger-shifting $25G_{7/2}$ state,
which has a larger polarizability than the $25F_{5/2}$ state. As the
$25G_{7/2}$-character of the lower AT peak increases, its EIT signal strength
diminishes due to the diminishing optical coupling to the $5D_{3/2}$ state
[see Fig. 2 (b)]. Conversely, with increasing $E_{DC}$ the right AT peak
acquires an increasing amount of $25F_{5/2}$ character, which reduces its
response to changes in $E_{DC}$ while its EIT signal strength increases.

Increasing $\ell$ yet more by driving the $25F_{5/2}\leftrightarrow 25H_{9/2}$
two-photon microwave transition (see Sec. III.2) allows for further
$E_{DC}$-response enhancement, as seen in Figs. 5 (e) and (f). Here $f_{1}$
and $P_{1}$ for the log-periodic antenna are 3.158 GHz and -5 dBm,
respectively. Already at $E_{DC}\gtrsim$ 10 V/m the two AT peaks significantly
change. The left AT peak in Figs. 5 (e) and (f) shifts and broadens
considerably more than in Figs. 5 (c) and (d), and at $E_{DC}\gtrsim$ 30 V/m
its strength drops to hardly noticeable. These features reflect the even
larger polarizability of the $25H_{9/2}$ state, which lowers the detectable
$E_{DC}$. The excess broadening [Fig. 5 (e)] and splitting [Fig. 5 (f)] of the
$25F_{5/2}$-like upper AT peak at fields $E_{DC}\gtrsim$ 30 V/m results from
an increase of the $m_{J}$-splitting caused by the AC Stark effect of the
$25F_{5/2}$-state in the rather strong two-photon RF drive field,
$E_{RF1}\approx 80$ V/m.

Finally, we utilize both the PCB and the rabbit-ears antennas to assess the
response of the three-peak EIT on the $25F_{5/2}\leftrightarrow
25H_{9/2}\leftrightarrow 25I_{11/2}$ ladder to the DC fields in Fig. 5 (h).
Here, $f_{1}$ and $P_{1}$ are same as in Fig. 5 (f), while $f_{2}$ = 345 MHz
and $P_{2}$ = 4 dBm. In this most $E_{DC}$-field-sensitive case, we have
utilized finer steps in $P_{453}$ (and, hence, in $E_{DC}$) to resolve the
spectral changes in fields $E_{DC}\lesssim$ 10 V/m. First, the rightmost peak
appears to red-shift and to merge into the central peak, and the signal
strength redistributes between the peaks. At $E_{DC}\gtrsim$ 20 V/m first the
most red-shifting line acquires increasing $25I_{11/2}$-character and greatly
diminishes in signal strength, followed by an increasingly $25H_{9/2}$-like
state departing to the red-shifted side. At larger fields, the
$25F_{5/2}$-like peak dominates and eventually becomes the only visible peak
left, and the spectrum looks similar to Figs. 5 (e) and (f).

In all four cases in Fig. 5, the theoretical predictions qualitatively agree
with the experimental data. Fig. 5 demonstrates the step-wise increase in DC
electric-field sensitivity with increasing $\ell$, probed via three-photon
optical Rydberg EIT. The minor discrepancies between measurements and
calculations are attributed to imperfections in the optical and RF
polarizations, inhomogeneities in both the RF and DC fields, and environmental
magnetic fields (which were not shielded). Especially at small $E_{DC}$, the
latter may cause $m_{J}$-mixing and Zeeman shifts of the high-$\ell$ states on
the order of $\pm$5 MHz. In view of Table 1, the $m_{J}$-mixing may alter the
spectra due to the large tensor polarizabilities.

At $E_{DC}\sim 0$ and for RF parameters that lead to efficient mixing of the
coupled levels, the polarizabilities of the mixed levels are approximately
given by the averages over the polarizabilities of the coupled states. In Fig.
5, these averages are to be taken over the levels printed over the tops of the
four data sets. Assuming that the $|m_{J}|$-range is restricted to
$|m_{J}|\leq 5/2$, i.e. the states that are optically coupled for all
$x$-polarized fields, the averages are 0.010, 0.027, 0.068 and 0.145
MHz/(V/m)2 for Figs. 5 (a), (c), (e) and (g), respectively. At small $E_{DC}$,
all dressed levels in a given plot should Stark-shift with roughly equal
curvatures, and the curvature ratios between the four sets of plots in Figs. 5
(a), (c), (e) and (g) should roughly accord with the stated numbers. This
simplified analysis explains the trends that are observed in Fig. 5 for
$E_{DC}\lesssim 15$ V/m, i.e. in regions where the RF coupling amplitudes
exceed the Stark shifts for all four cases.

In Fig. 5 (i) we plot the $E_{DC}$-values obtained by matching observed and
calculated spectra from Figs. 5 (a) through (h) versus the 453-nm photo-
illumination power, $P_{453}$, used to generate the DC field. The plot
reflects fairly large, conservative estimates for the $E_{DC}$-uncertainty.
The latter are obtained from matching the experimental spectra to different
calculated spectra empirically and are usually between 10-20% of respective
$E_{DC}$. For $P_{453}\lesssim 10$ mW the field increases approximately
linearly in $\log(P_{453})$ and saturates at higher powers at values near 50
V/m, in general agreement with earlier findings Ma  _et al._ (2020). The
observed highly nonlinear behavior reflects a complex charging or discharging
equilibrium within the vapor cell. This topic is beyond the scope of our
present work and may be explored further in future studies.

##  V Discussion and Outlook

We have prepared Rydberg states with $\ell\leq$ 6 in a room-temperature vapor
cell using a combination of three-photon optical EIT and up to three-photon RF
dressing. The RF dressing fields were calibrated by Rydberg-EIT spectroscopy.
The resultant well-characterized dressed Rydberg states were then used to
measure and calibrate DC electric fields generated by photo-electric effect on
the cell walls. The utility of the increased electric polarizabilities of the
high-$\ell$ Rydberg states in measuring weak electric fields was demonstrated.
The underlying physics was modeled by accounting for all AC and DC shifts and
resonant couplings of the dressed Rydberg-atom system, as well as the
specifics of the utilized three-optical-photon Rydberg-EIT probe.

Possible improvements in the setup to extend the electric field sensitivity
could include stray-magnetic-field shielding, reduction of the EIT laser
linewidths, and reduction of RF field inhomogeneities and unwanted
polarization variations Shaffer and Kübler (2018); Jing  _et al._ (2020);
Prajapati  _et al._ (2021); Holloway  _et al._ (2022b); Dixon  _et al._
(2023); Legaie  _et al._ (2023). Solving the Lindblad equation for our case of
collinear three-optical-photon Rydberg EIT in a Rb vapor cell, we find that
EIT linewidths $<5$ MHz should be possible. At the low $n$-value used in our
work, optical electric-dipole transition matrix elements are relatively large
(scaling $\sim n^{-3/2}$) and RF-transition ones relatively small (scaling
$\sim n^{2}$). These scalings favor economical setups, as laser power is more
expensive than RF power at sub-5-GHz frequencies. We therefore anticipate
applications in high-precision measurements on relatively low-$n$ and
high-$\ell$ Rydberg states Lee  _et al._ (2016); Berl  _et al._ (2020); Moore
_et al._ (2020) in room-temperature vapor cells for tests of ab initio atomic-
physics calculations Marinescu  _et al._ (1994); Safronova  _et al._ (2004);
Safronova and Safronova (2011); Duspayev and Raithel (2022), the exploration
of pathways towards excitation of circular Rydberg atoms Anderson  _et al._
(2013); Zhelyazkova and Hogan (2016); Larrouy  _et al._ (2020); Cardman and
Raithel (2020); Wu  _et al._ (2023), and other applications in fundamental-
physics research Safronova  _et al._ (2018).

Along the lines of applications mentioned in the introduction, the
demonstrated methods could be of interest for sensing electric fields in low-
pressure discharge plasmas Harry (2013); Eden  _et al._ (2013); Nijdam  _et
al._ (2023). Combining Rydberg-EIT in low-pressure buffer-gas vapor cells,
which was recently observed Thaicharoen  _et al._ (2023), with plasma
generators, one may anticipate future plasma-physics research in discharge
plasmas with Rydberg-EIT as an electric-field probe. This may include dusty
plasmas Shukla (2001); Shukla and Eliasson (2009), which are not only relevant
to applications Boufendi  _et al._ (2011); Ratynskaia  _et al._ (2022) but
also in astrophysics Mendis and Rosenberg (1994), nonlinear dynamics Morfill
and Ivlev (2009) and condensed matter Thomas  _et al._ (1994). Rydberg-EIT as
a non-invasive electric-field probe with high spatial resolution Anderson  _et
al._ (2017) employing high-$\ell$ states could become useful in future dusty-
plasma research.

## Acknowledgments

We would like to thank Bineet Dash, Nithiwadee Thaicharoen, Michael Viray and
Jamie MacLennan for useful discussions. This work was supported by the U.S.
Department of Energy, Office of Science, Office of Fusion Energy Sciences
under award number DE-SC0023090, NSF Grant No. PHY-2110049, and Rydberg
Technologies, Inc.. A.D. and R.C. acknowledge support from the respective
Rackham Predoctoral Fellowships at the University of Michigan. D.A.A and G. R.
have an interest in Rydberg Technologies, Inc.

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