11institutetext: Knowledge Technology, Department of Informatics, University
of Hamburg 22institutetext: ZAL Center of Applied Aeronautical Research
33institutetext: Hamburger Informatik Technologie-Center e.V. (HITeC)

# Neural Field Conditioning Strategies for 2D Semantic Segmentation††thanks:
The authors gratefully acknowledge support from the DFG (CML, MoReSpace,
LeCAREbot), BMWK (SIDIMO, VERIKAS), and the European Commission (TRAIL,
TERAIS).

Martin Gromniak 1122 Sven Magg 33 Stefan Wermter 11

###### Abstract

Neural fields are neural networks which map coordinates to a desired signal.
When a neural field should jointly model multiple signals, and not memorize
only one, it needs to be conditioned on a latent code which describes the
signal at hand. Despite being an important aspect, there has been little
research on conditioning strategies for neural fields. In this work, we
explore the use of neural fields as decoders for 2D semantic segmentation. For
this task, we compare three conditioning methods, simple concatenation of the
latent code, Feature Wise Linear Modulation (FiLM), and Cross-Attention, in
conjunction with latent codes which either describe the full image or only a
local region of the image. Our results show a considerable difference in
performance between the examined conditioning strategies. Furthermore, we show
that conditioning via Cross-Attention achieves the best results and is
competitive with a CNN-based decoder for semantic segmentation.

###### Keywords:

neural fields conditioning semantic segmentation

##  1 Introduction

Lately, neural networks for semantic segmentation have been mostly based on
the fully convolutional network (FCN) [11] paradigm. FCN models typically
consist of an encoder and a decoder which are both build with stacked
convolution layers. The purpose of the encoder is to extract features from the
image. With increasing depth of the encoder, the features get more abstract
and the resolution of the feature maps is progressively reduced. The decoder
on the other hand takes the low-resolution feature map from the encoder as an
input and upscales them to the resolution of the original image so that pixel-
level classification can be performed.

While encoders in the form of convolutional neural networks (CNN) have been
rigorously studied, considerably less research has been conducted on the
decoder side of semantic segmentation networks. The main challenge for the
decoder is to upscale the feature map to the image’s original resolution and
simultaneously produce accurate region borders. In CNN-based decoders,
typically upsampling or transposed convolution operators are used to
progressively increase the spatial resolution of the feature maps. These
operations introduce a particular kind of inductive bias. For example,
transposed convolutions can create spectral artifacts in the upscaled feature
maps [5]. Another apparent disadvantage of CNN decoders is that they struggle
to capture long-range dependencies between different parts of the image, due
to their locally connected structure.

In the last few years, neural fields, aka implicit neural representations or
coordinate-based networks, have received much attention for learning a variety
of different signals, for example, 1D audio signals [22], 2D images [4, 26]
and 3D geometries [12, 24]. A neural fields takes (spatial) coordinates
$x\in\mathbb{R}^{d}$ as input and maps them to a task-dependent signal
$y=\Phi(x)$ through a neural network. For example, a neural field representing
an RGB image takes 2D image coordinates as input and produces three RGB values
at each location. One interesting property of neural fields is that they
represent signals as continuous functions on their respective spatial domain.

Inspired by the recent successes of neural fields, we explore the use of
neural fields as decoders in semantic segmentation networks. In this regard,
we hypothesize that (continuous) neural fields provide an inductive bias which
can be better suited for reconstructing high-resolution semantic maps compared
with (discrete) CNN-based decoders. In our work, we examine multiple
conditioning strategies, which enable the neural field decoder to make use of
the information in the latent feature map produced by the encoder. Through our
comparative study, we aim to provide more insights into conditioning methods
of neural fields, as research have been extremely sparse in this regard.
Furthermore, we believe that 2D semantic segmentation provides a well-defined
task for studying conditioning methods, as it has comprehensive metrics and
the possibility for insightful visualizations of the learned geometries.

##  2 Related Work

Semantic Segmentation Encoder-decoder fully convolutional networks [11] have
become the predominant approach for semantic segmentation. They share the
challenge how to encode high-level features in typically low-resolution
feature maps and subsequently upscale these feature maps to retrieve pixel-
accurate semantic predictions. Multiple approaches [19, 1] have introduced
skip-connections between the encoder and the decoder, which help the decoder
to combine global with local information and therefore recover sharp object
boundaries. One drawback of CNNs is that, because of their locally connected
structure, they struggle to combine information which is spatially distributed
across the feature maps. Research attempting to mitigate this drawback has
proposed attention mechanisms over feature maps to selectively capture and
combine information on a global scale [6]. Extending the concept of attention
further, neural network architectures based fully on transformers have been
proposed recently for semantic segmentation [25]. In our work, we utilize a
CNN, which is more efficient than transformers, for extracting features and
use attention in one of our conditioning methods.

CNN Decoders Research on decoders has been more sparse than research on neural
network encoders, i.e. CNN backbones. Wojna et al. [28] compared different
CNN-based decoders for several pixel-wise prediction tasks and observed
significant variance in results between different types of decoders. Multiple
works [14, 5] have provided evidence that upscaling using transposed
convolution operators introduce artifacts in the feature maps and therefore
the decoder’s output. We aim to avoid any explicit or implicit discrete
artifacts by using a continuous neural field decoder.

Neural Fields Neural fields were introduced in 2019 as a representation for
learning 3D shapes [12, 15]. Following works extended neural fields by
learning colored appearance of scenes and objects [24, 13]. Particularly NeRF
[13] has attracted a lot of attention, as it is able to generate very
realistic novel views of a scene, learning from images and associated poses.
NeRF effectively overfits a neural network for one individual scene. This
limits the usability as the neural field needs to be re-trained for every new
scene. Some works have explored the use of neural fields for semantic
segmentation. Vora et al. [27] built a 3D segmentation on top of the NeRF
approach. Hu et al. [9] used neural fields in conjunction with CNNs for
upsampling and aligning feature maps in the decoder of a semantic segmentation
network.

Neural Field Conditioning When a neural field should share knowledge between
different signals, it needs to be conditioned on a latent code which describes
the signal at hand. Several conditioning approaches have been explored in the
literature. Methods based on global conditional codes use one code to describe
the whole signal [12, 23]. Methods based on local conditional codes [29, 4]
use a different code for each spatial area in the signal. On top of these,
there exist multiple methods how a neural field can actually consume a
conditional code, which we describe in detail in Section 3.3. Rebain et al.
[18] compared different methods for conditioning neural fields for 2D and 3D
tasks, but did not consider global and local conditional codes. In the neural
field community, there is a lack of comparative research on what conditioning
strategies work well for which task. We attempt to shed more light on this by
comparing different conditioning strategies for the well-defined task of 2D
semantic segmentation.

##  3 Method

Figure 1: Our high level neural network architecture. A CNN encoder encodes an
image into a feature map. During training, $S$ points per image are sampled
within the image (red) and fed into the decoder. The decoder is a conditional
neural field for which we use different conditioning strategies. For every
point the decoder outputs a prediction of the semantic class at this position
(purple).

###  3.1 Neural Network Architecture and Training Procedure

Our high-level architecture involves a CNN decoder and a neural field
decoder(see Figure 1). We use a CNN to efficiently encode an image into a
feature map with size $c\times h\times w$, where $c$ is the number of
channels, $w$ is the spatial width and $h$ is the spatial height. From this
feature map, we calculate the conditional code for the neural field decoder in
different ways, depending on the conditioning strategy. During training, for
every image, we sample $S$ random points within the image. At test time, the
points are densely sampled so that there exists one point for each pixel. The
point coordinates are normalized to the $\interval{0}{1}$ range, stacked, and
fed to the neural field decoder as input. For every point, the decoder
predicts the semantic class at that position in the image. We use a cross-
entropy loss to train the whole setup in an end-to-end fashion. At test time,
the class predictions per point are arranged into an image. Thereby, we can
compare the predicted feature map with the labeled feature map using standard
image segmentation metrics.

###  3.2 Latent Code Source: Global vs. Local

Figure 2: A visualization of our conditioning strategies. We consider three
conditioning methods: Concat conditioning, FiLM conditioning and Cross-
Attention conditioning (right side). For Concat and FiLM conditioning, one
feature vector is used, which can be calculated from global features (top
path) or local features (mid path). The input to the Cross-Attention
Transformer is the whole feature map, which is reshaped and treated as tokens
(bottom path).

First, we differentiate how the conditional code is calculated based on the
feature map from the encoder. We consider a global code and a local code. The
global code represents the content of the complete image. Naturally, it can
capture the global context in the image well. However, due to its limited
capacity, it might not be able to capture fine, local geometries. On the other
hand, the local code represents a spatially limited area in the image. It can
utilize its full capacity for modeling the geometry in one area with high
fidelity, however, it might lack global context. For example, the probability
of detecting a car rises when a street is detected somewhere in the image.

We calculate the global code vector by applying a global average pooling
operation. It averages all the entries in the feature map across the spatial
dimensions (see the top path in Figure 2). This is a standard operation which
is used, for example, in the ResNet classification head [8]. Through this
procedure, we calculate one global code per image.

For calculating the local code, we utilize the point coordinates, in addition
to the feature map. For every point, we "look up" the value of the feature map
at this position. For this purpose, we normalize the feature map’s spatial
dimensions to the [0,1] range, and therefore effectively align it with the
input image. We then perform a bilinear interpolation within the feature maps
based on the point coordinate to calculate the local code vector (see the
middle path in Figure 2). As a result, we have $S$ local codes per image, one
for every point.

In addition to using either a global or a local code, we also consider the
combination of both to jointly exploit their individual advantages. We do this
by concatenating both codes.

###  3.3 Conditioning the Neural Field Decoder

Conditioning a neural field enables it to effectively adapt the knowledge
which is shared across all signals to the signal at hand.

####  3.3.1 Conditioning by Concatenation

In the simplest conditioning method, the conditional code is concatenated to
the point coordinates and is jointly used as input to the neural field. We re-
concatenate the conditional code to other hidden layers using skip
connections. This approach is used by HyperNeRF [16]. It has the advantage of
being conceptually simple, however, it is computationally inefficient [18],
because it requires $O(k(c+k))$ parameters for the fully connected layers in
the neural field, where $k$ is the hidden layer width and $c$ is the size of
the conditioning vector.

####  3.3.2 Feature-Wise Linear Modulation

Another way to condition a neural field is to use the latent code together
with an MLP to regress the parameters of the neural field. When all parameters
of the neural field are calculated in this way, the approach is known as
hyper-networks [7]. Feature-wise Linear Modulation (FiLM) [17] is a more
constraint subtype of hyper-networks where, instead of predicting all the
weights, feature-wise modulations of activations in the neural field are
predicted. This approach is used in Occupancy Networks [12] and piGAN [2].

####  3.3.3 Cross-Attention

Conditioning by Cross-Attention has been introduced by Jiang et al. [10] and
was extended in the Scene Representation Transformer [21]. The core idea is to
selectively attend to features at different spatial positions, based on the
point coordinates. A transformer architecture with Cross-Attention layers is
used where the queries are derived from the point coordinates and the feature
maps serve as a set of tokens. This approach does have an interesting
connection with using local codes, as both approaches calculate a feature
vector by weighting entries in the feature maps based on the current point
coordinate. However, in difference to the spatial "look up" of local codes,
which can be performed for free, the Cross-Attention operation can flexibly
query both local and global information as needed at the cost of more
computation [18].

##  4 Experiments

We evaluate seven conditioning strategies on a publicly available dataset for
semantic segmentation. Concat conditioning and FiLM conditioning are used in
conjunction with global, local and combined conditional codes each. The Cross-
Attention Transformer uses the reshaped feature map as input (see Figure 2).

###  4.1 Dataset

For our experiments, we used the Potsdam
dataset111https://www.isprs.org/education/benchmarks/UrbanSemLab/2d-sem-label-
potsdam.aspx which is part of the ISPRS semantic labeling contest [20]. It
consists of satellite images of the German city Potsdam together with dense
label masks for six classes: Impervious surfaces, Building, Low vegetation,
Tree, Car and Clutter/background. The orthographic images have a sampling
distance of 0.05 m/px. The total dataset consists of 38 tiles with a size of
$6000\times 6000$ px from which we use the same 24 tiles for training as in
the original contest. From the remaining tiles, we use 7 for validation and 7
for testing. From the tiles, we randomly crop patches of $256\times 256$ or
$512\times 512$ pixels.

###  4.2 Encoder and Decoder Implementations

For the Concat and the FiLM decoder, we use a similar neural network
architecture, which is based on Occupancy networks [12] (see Figure 3(a)). We
use either concatenation plus conventional batchnorm or conditional batchnorm
at the designated places in the neural network architecture. For the Cross-
Attention conditioning, we use a transformer architecture based on the Scene
Representation Transformer [21] (see Figure 3(b)). It uses one multi-head
attention module per block. Keys and values are calculated from the feature
tokens while the queries are calculated from the point coordinates. We can
scale both neural network architectures by repeating the yellow blocks $N$
times or increasing the width of the MLP layers. For all experiments we use a
ResNet34 [8] backbone as the encoder, pre-trained on ImageNet. Its output
feature map has a size of $512\times 8\times 8$ for input images with size
$256\times 256$ pixels and $512\times 16\times 16$ for input images with size
$512\times 512$ pixels respectively.

(a) Concat/FiLM Network

(b) Cross-Attention Transformer

Figure 3: Our neural network architectures used for the Concat and FiLM
conditioning (left) and for the Cross-Attention Transformer (right). The
yellow block can be repeated $N$ times. For the Concat approach, the orange
block denoted with an asterisk represents a concatination followed by a
batchnorm layer. For FiLM, the same block denotes a conditional batchnorm
layer. Other batchnorm and layernorm layers have been omitted for clarity.

###  4.3 Points Embedding

It has been shown that when coordinates are directly used as inputs, neural
fields have a bias towards learning low-frequency signals. To counter this, we
embed both image coordinates independently into a higher dimensional space by
using Fourier features as it is commonly done with neural fields [26]:

$\gamma(x)=(sin(2^{0}{\pi}x),sin(2^{1}{\pi}x),...,sin(2^{l}{\pi}x),cos(2^{0}{\pi}x),cos(2^{1}{\pi}x),...,cos(2^{l}{\pi}x)),$
(1)

where $x$ is an image coordinate and $l$ controls the embedding size.

###  4.4 Training Parameters

The influence of the parameters used in our experiments was evaluated in
preliminary runs, based on the validation performance. For all experiments, we
choose a fixed learning rate of $1\times 10^{-4}$ for the Adam Optimizer and a
batch size of 64. We use horizontal and vertical flipping as data augmentation
and perform early stopping based on the IoU metric on the validation set. For
all neural field architectures, 512 points are sampled per image and we choose
$l=4$ as the size of the points embedding. Empirically, we have found that the
results are not sensitive to both these parameters. We have explored scaling
the neural field architectures by increasing the number of blocks and the MLP
layers’ width. With that approach, we use a hidden size of 512 for all MLP
layers. One block is used within the Concat and FiLM conditioning network and
two blocks are used within the Cross-Attention Transformer. For all
architectures, we try to have approximately the same amount of parameters to
make a fair comparison.

Table 1: Results for all examined decoder architectures. |  | Image Size |
---|---|---|---
Decoder conditioning |  Conditional Code Source | 256 | 512 |  Params
|  |  IoU |  F-score |  IoU |  F-score |
Concatenation |  global |  0.689 |  0.816 |  0.659 |  0.794 |  2.1M
|  local |  0.725 |  0.840 |  0.666 |  0.799 |  2.1M
|  global + local |  0.728 |  0.842 |  0.712 |  0.832 |  4.0M
FiLM |  global |  0.695 |  0.820 |  0.660 |  0.795 |  2.1M
|  local |  0.729 |  0.843 |  0.650 |  0.788 |  2.1M
|  global + local |  0.729 |  0.843 |  0.707 |  0.829 |  3.7M
Cross-Attention |  feature tokens |  0.758 |  0.862 |  0.754 |  0.860 |  2.6M
DeepLabV3+ [3] |  - |  0.760 |  0.863 |  0.763 |  0.866 |  5.4M

Input Image | Ground Truth | Concat + g | Concat + l | Concat + g/l
---|---|---|---|---
|  |  |  |
|  |  |  |
|  |  |  |

FiLM + g | FiLM + l | FiLM + g/l | Cross-Attn | DeepLabV3+
---|---|---|---|---
|  |  |  |
|  |  |  |
|  |  |  |

Figure 4: The predictions of all examined decoder architectures on three
example images ($512\times 512$ px) from the test set. For Concat and FiLM
conditioning, $g$ denotes a global code source, $l$ denotes a local code
source and $g/l$ denotes a concatenation of global and local code. The class
color code is: white=Impervious surfaces, blue=Building, cyan=Low vegetation,
green=Tree, yellow=Car, red=Clutter/background. By comparing the predictions
with the ground truth segmentation masks, it can be observed that the ability
to represent details, e.g. distinct objects or angular corners, varies greatly
between the approaches. Only the Cross-Attention and the DeepLabV3+ decoders
are able to faithfully represent the segmentation masks, while the Concat and
FiLm approaches tend to produce overly smooth geometries.

##  5 Results

In Table 1 we show the Intersection over Union (IoU), F-Score and the number
of parameters for all seven conditioning strategies and two different image
sizes on the test set. We also compare our neural field decoder with the
DeepLabV3+ [3] fully convolutional neural network for semantic segmentation
which also uses a ResNet34 backbone. In Figure 4 we show the predictions of
all decoder architectures for three example images. From the results, we can
make multiple key observations.

First, the Concat and FiLM decoders perform very similarly in all aspects,
regardless of the conditional code source and the image size.

Second, conditioning via Cross-Attention works best amongst all neural field
approaches. Furthermore, it performs similarly to the DeepLabV3+ FCN. Notably,
the Cross-Attention decoder has half as much parameters and no access to the
intermediate feature maps of the encoder.

Third, the performance of the Concat and FiLM approaches can be improved by
using a combination of global and local features, particularly for larger
images. In that case, the performance of both approaches is not much lower
compared with the Cross-Attention decoder.

Fourth, the performance of the Concat and FiLM conditioning decreases with
larger input images when using global codes. This can be expected, as it is
harder to model more geometries in larger images with the same code length.

Fifth, when using local codes, the performance is also degraded when dealing
with larger images. This is unexpected, as the sampling distance (meters per
pixel) remains the same and therefore the size of the features should also
remain the same. This could be an indication that the individual vectors in
the feature map produced by the CNN encoder do not model purely local
features, as stated by methods using this approach [4, 29]. This is further
supported by the fact that modern CNN architectures have very large receptive
fields so that one feature vector in the output feature map receives input
from the complete input image. In our case, the ResNet34 encoder has a
receptive field of 899 pixels which fully covers both our image sizes.

##  6 Conclusion

In this work, we performed a comparative study of neural field conditioning
strategies and explored the idea of a neural field-based decoder for 2D
semantic segmentation. Our results show that neural fields can have a
competitive performance when compared with a classic CNN decoder while
requiring even fewer parameters. We also showed that the performance of the
neural field is considerably affected by the conditioning strategy. The best
conditioning strategy likely depends on the task. For the task of 2D semantic
segmentation, a Cross-Attention-based Transformer is superior to Concat and
FiLM conditioning. However, also the combination of local and global
conditional codes is a promising approach, as the performance is not much
lower. Lastly, for local features, we showed an unexpected degradation in
performance when increasing the image size. Further research is required to
explain this observation and deduce consequences for local conditioning
methods.

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# Precision Rydberg State Spectroscopy with Slow Electrons and Proton Radius
Puzzle

Ulrich D. Jentschura  Department of Physics and LAMOR, Missouri University of
Science and Technology, Rolla, Missouri 65409, USA  Dylan C. Yost  Department
of Physics, Colorado State University, Fort Collins, Colorado 80523, USA

###### Abstract

The so-called proton radius puzzle (the current discrepancy of proton radii
determined from spectroscopic measurements in ordinary versus muonic hydrogen)
could be addressed via an accurate measurement of the Rydberg constant,
because the proton radius and the Rydberg constant values are linked through
high-precision optical spectroscopy. We argue that, with manageable additional
experimental effort, it might be possible to improve circular Rydberg state
spectroscopy, potentially leading to an important contribution to the
clarification of the puzzle. Our proposal involves circular and near-circular
Rydberg states of hydrogen with principal quantum number around $n=18$, whose
classical velocity on a Bohr orbit is slower than that of the fastest
macroscopic man-made object, the Parker Solar Probe. We obtain improved
estimates for the quality factor of pertinent transitions, and illustrate a
few recent improvements in instrumentation which facilitate pertinent
experiments.

##  I Introduction

The Rydberg constant is of consummate importance for our understanding of
fundamental physics. Notably, this constant is an important input datum for
the calculation of transition frequencies in hydrogen and deuterium (see Table
II of Ref. Jentschura  _et al._ (2005) and Refs. Mohr  _et al._ (2008);
Tiesinga  _et al._ (2021)). In addition to the Rydberg constant, accurate
values of the proton and deuteron radii are also required in order to
calculate transition frequencies in hydrogen and deuterium. Conversely, one
can infer proton and deuteron radii from precise values of hydrogen and
deuterium frequencies (see Refs. Jentschura  _et al._ (2005); Tiesinga  _et
al._ (2021) and Table 45 of Ref. Mohr  _et al._ (2008)).

With the advent of muonic hydrogen spectroscopic measurements Pohl  _et al._
(2010, 2016), the CODATA value of the proton radius has shifted from a 2006
value of about $R_{p}\approx 0.88\,{\rm fm}$ to a 2018 value of about
$R_{p}\approx 0.84\,{\rm fm}$, entailing a concomitant change in the Rydberg
constant Mohr  _et al._ (2008); Tiesinga  _et al._ (2021). From the 2006 to
the 2018 CODATA adjustments Mohr  _et al._ (2008); Tiesinga  _et al._ (2021),
the Rydberg constant has shifted by much more than the uncertainty associated
with the 2006 value (see Fig. 1).

One of the most attractive experimental pathways to the determination of the
Rydberg constant involves highly excited Rydberg states in atomic hydrogen, as
described in Ref. Lutwak  _et al._ (1997) by a research group working at the
Massachusetts Institute of Technology (MIT). Within the same group, a value
for the Rydberg constant was obtained in an unpublished thesis by de Vries
deVries  (labelled as “Rydberg state” in Fig. 1),

$\left.cR_{\infty}\right|_{\mathrm{deVries}}=3\,289\,841\,960\,306(69)\,{\rm
kHz}\,,$ (1)

which is consistent with the CODATA 2006 value, and barely consistent with the
2018 CODATA value from Ref. Tiesinga  _et al._ (2021):

$\left.cR_{\infty}\right|_{\rm CODATA,2018}=3\,289\,841\,960\,250(7)\,{\rm
kHz}\,.$ (2)

The 2006 CODATA value is discrepant,

$\left.cR_{\infty}\right|_{\rm CODATA,2006}=3\,289\,841\,960\,360(21)\,{\rm
kHz}\,.$ (3)

A comparison of the three values of the Rydberg constant is made in Fig. 1,
where we use as the reference value

$R_{0}=\left.R_{\infty}\right|_{\rm CODATA,2018}\,.$ (4)

Figure 1:  We examine the values for the Rydberg constant, converted to
frequency units, from CODATA adjustments and from the (unpublished, grayed)
result communicated in Ref. deVries . The CODATA (2006) value was reported in
Ref. Mohr  _et al._ (2008), and the CODATA (2018) value is from Ref. Tiesinga
_et al._ (2021). The reference value $R_{0}$ is from the 2018 adjustment.

The situation is interesting because, before the advent of muonic hydrogen
spectroscopy, values of the Rydberg constant and of the proton radius inferred
from hydrogen and deuterium spectroscopy alone (without any additional input
from scattering experiments) were consistent with the 2006 CODATA values for
both the 2006 CODATA value of the Rydberg constant, as well as the 2006 CODATA
values of the proton and deuteron radii. This is discussed in detail in the
discussion surrounding Table 45 of Ref. Mohr  _et al._ (2008), where it is
pointed out that the proton radius $R_{p}$, the deuteron radius $R_{d}$, and
the Rydberg constant can all be deduced using input data exclusively from
hydrogen and deuterium spectroscopy.

Traditionally, the Rydberg constant has been determined on the basis of
Rydberg-state spectroscopy of atomic hydrogen Kibble  _et al._ (1973);
Gallagher (1994); Biraben  _et al._ (1989); Nez  _et al._ (1992); de Beauvoir
_et al._ (1997); Schwob  _et al._ (1999); de Beauvoir  _et al._ (2000). An
improved measurement of the Rydberg constant would thus constitute an
important contribution to a resolution of the proton radius puzzle Jentschura
(2022). In a remarkable investigation dating about 20 years back, circular
Rydberg states around quantum numbers $n\approx 30$ have been investigated
with the ultimate aim of an improved measurement of the Rydberg constant
deVries . Inspired by the importance of Rydberg states, it has been pointed
out in Refs. Jentschura  _et al._ (2008, 2009, 2010) that Rydberg-state
measurements in hydrogenlike ions of medium charge numbers could potentially
offer an alternative route to a determination of the Rydberg constant.

The purpose of this paper is threefold. First, we update the calculation of
the quality factors for transitions among circular Rydberg states, in
comparison to the estimate provided in Eq. (6) of Ref. Jentschura  _et al._
(2008). Second, we discuss the status of quantum electrodynamic theory of
Rydberg states, demonstrate that the theory is very well under control on the
level of accuracy required for a determination of the Rydberg constant on the
level of precision required for a resolution of the proton radius puzzle, and
discuss the relative suppression of a number of notoriously problematic
quantum electrodynamic corrections for circular and near-circular Rydberg
states. Calculated values for relativistic Bethe logarithms for circular and
near-circular Rydberg states with principal quantum numbers $16\leq n\leq 20$
are also provided. Third, we provide an overview of recent advances in laser
technology and other experimental techniques, which facilitate an improvement
of measurements of the Rydberg constant on the basis of Rydberg state
measurements. SI mksA units are employed throughout this paper.

##  II Quality Factors

Of crucial importance for the feasibility of high-precision spectroscopy
experiments are so-called quality factors of transitions. The quality factor
is the dimensionless ratio of the transition energy to the natural line width
of the transition (measured in radians per second), where the latter is
converted to an energy via multiplication by the reduced Planck constant
$\hbar$. Here, we present the general formula for the one-photon decay rate of
a circular Rydberg state, with principal quantum number $n$ and maximum
orbital angular moment $\ell=n-1$. This reference state can decay, due to
dipole transitions to states with principal quantum number $n-1$ and angular
momentum quantum number $\ell=n-2$. For the decay rate of the state with
principal quantum number $n$ and maximum orbital angular moment $\ell=n-1$, as
parameterized by the imaginary part of the self energy, $E={\rm
Re}\,E-\mathrm{i}\Gamma_{n}/2$, we find the result

$\Gamma^{\ell=n-1}_{n}=\frac{4^{2n}(n-1)^{2n-1}\,n^{2n-4}}{(2n-1)^{4n-1}(2n-3)}\,\frac{\alpha(Z\alpha)^{4}m\,c^{2}}{3n^{5}}\,\left(\frac{\mu}{m}\right)^{3}$
(5)

which can be expanded for large $n$ as follows,

$\Gamma^{\ell=n-1}_{n}=\alpha\frac{(Z\alpha)^{4}mc^{2}}{3n^{5}}\,\left(\frac{\mu}{m}\right)^{3}\\\
\times\left[1+\frac{3}{2n}+\frac{17}{8n^{2}}+\mathcal{O}\left(\frac{1}{n^{3}}\right)\right]\,,$
(6)

where $m$ is the electron mass, $\mu$ is the reduced mass of the two-body
system, $\alpha$ is the fine-structure constant, $Z$ is the nuclear charge
number, and the expansion for large $n$ illustrates that the lifetimes of
circular Rydberg states scale as $n^{5}$. The energy difference for
transitions among circular Rydberg states is

$E_{n}-E_{n-1}=\frac{(Z\alpha)^{2}\mu}{2}\,\left(\frac{1}{(n-1)^{2}}-\frac{1}{n^{2}}\right)\,,$
(7)

which scales as $1/n^{3}$ for large $n$. Due to the $1/n^{5}$ asymptotics of
the decay rate and the $1/n^{3}$ asymptotics of the transition energy, the
quality factor increases, for large $n$, with the square of the principal
quantum number $n$,

$Q=\frac{E_{n}-E_{n-1}}{\Gamma^{\ell=n-1}_{n}+\Gamma^{\ell=n-2}_{n-1}}\\\
=\frac{3n^{2}}{2\alpha\,(Z\alpha)^{2}}\,\left(\frac{m}{\mu}\right)^{2}\,\left[1-\frac{5}{2n}-\frac{17}{8n^{2}}+\mathcal{O}\left(\frac{1}{n^{3}}\right)\right]\,.$
(8)

This formula constitutes an update of the estimate given in Eq. (6) of Ref.
Jentschura  _et al._ (2008) (the quality factor obtained here is larger by a
factor two as compared to Ref. Jentschura  _et al._ (2008)). The estimate in
Eq. (8) illustrates the enormous advantages of Rydberg states for the
measurement of the Rydberg constant. The dramatic increase of the quality
factor with the square of the principal quantum number makes Rydberg state
transitions very attractive. Also, we observe that the quality factor is
inversely proportional to the second power of the nuclear charge number $Z$.
This means that $Z=1$ (atomic hydrogen) offers the best quality factor, for
given principal quantum number $n$.

Figure 2:  The level diagram for Rydberg states illustrates the dipole-allowed
transitions among circular [panel (a)] and near-circular [panel (b)] states.
Transitions driven for high-precision spectroscopy are indicated with two-
sided arrows. Transitions relevant for the calculation of decay rates (quality
factors) are indicated by dashed lines. Circular Rydberg levels with
$\ell=n-1$ are marked in green color, while near-circular Rydberg levels with
$\ell=n-2$ are marked in red color.

Let us also evaluate the quality factor for the transition among near-circular
Rydberg states, where the upper level has orbital angular momentum $\ell=n-2$
and the lower level has orbital angular $\ell=n-3$ (see also Fig. 2). The
calculation of the quality factor proceeds in a similar way, but one needs to
consider two available dipole decay channels, namely, from the reference state
with principal quantum number $n$ and orbital angular momentum quantum number
$\ell=n-2$, to lower states with $n^{\prime}=n-1$ and $\ell=n-3$, and
$n^{\prime}=n-2$ and $\ell=n-3$. The decay width evaluates to

$\Gamma^{\ell=n-2}_{n}=\alpha\frac{(Z\alpha)^{4}mc^{2}}{3n^{5}}\,\left(\frac{\mu}{m}\right)^{3}\\\
\times\left[1-\frac{1}{2n}-\frac{1}{8n^{2}}+\mathcal{O}\left(\frac{1}{n^{3}}\right)\right]\\\
+\alpha\frac{4(Z\alpha)^{4}mc^{2}}{3n^{6}}\,\left(\frac{\mu}{m}\right)^{3}\\\
\times\left[1+\frac{5}{2n}+\frac{25}{4n^{2}}+\mathcal{O}\left(\frac{1}{n^{3}}\right)\right]\,,$
(9)

where the two terms on the right-hand side correspond to the lower states with
$n^{\prime}=n-1$ and $n^{\prime}=n-2$, respectively. The quality factor
evaluates to

$Q^{\prime}=\frac{E_{n}-E_{n-1}}{\Gamma^{\ell=n-2}_{n}+\Gamma^{\ell=n-3}_{n-1}}\\\
=\frac{3n^{2}}{2\alpha\,(Z\alpha)^{2}}\left(\frac{m}{\mu}\right)^{2}\,\left[1-\frac{9}{2n}+\frac{9}{8n^{2}}+\mathcal{O}\left(\frac{1}{n^{3}}\right)\right]\,,$
(10)

which is commensurate with $Q$ given in Eq. (8) and illustrates that no
significant accuracy loss occurs if one measures near-circular as opposed to
circular Rydberg states.

A quick look at Eqs. (1), (2) and (3), and Fig. 1, illustrates that one needs
to resolve the Rydberg constant to roughly one part in $10^{11}$ or better in
order to meaningfully distinguish between the 2006 and 2018 CODATA values of
the Rydberg constant. One can convert this resolution to a splitting factor
$\mathcal{S}$, which measures the fraction to which one needs to split the
resonance line in order to achieve a resolution of one one part in $10^{11}$.
The splitting factor $\mathcal{S}$ is given by the formula

$\mathcal{S}=10^{11}/Q\,.$ (11)

For $Z=1$, one obtains for $\mathcal{S}$ the perfectly reasonable figure
$\mathcal{S}=93$ for $n=18$; expressed differently, one only needs to split
the resonance lines near $n=18$ to one part in 93 is order to achieve a
resolution which meaningfully contributes to a resolution of the proton radius
puzzle.

Cross-damping terms (non-resonant corrections) can be generated by virtual
levels displaced by a fine-structure interval Jentschura and Mohr (2002). A
rough estimate of the corresponding energy (frequency) shift $\delta E_{\rm
CD}$ (we set $\hbar=1$) is given by the expression Jentschura and Mohr (2002)

$\delta E_{\rm CD}\sim\frac{\Gamma_{n}^{2}}{\delta E}\,,$ (12)

Here, $\delta E$ is the displacement of the virtual state responsible for the
cross-damping energy shift. As pointed out in Ref. Jentschura and Mohr (2002),
the nearest virtual states which can contribute to differential cross sections
are states displaced from the upper state of the Rydberg transition by a fine-
structure interval. The maximum angular momentum is $\ell_{\rm max}=n-1$. The
total angular momenta for the circular Rydberg states are $\ell_{\rm max}\pm
1/2$. The two possible values for the total angular momentum quantum numbers
of the upper level are thus $j_{+}=n-1/2$ and $j_{-}=n-3/2$, one of these
being the reference level, the other being the virtual level which contributes
to the cross damping. So, we have potential nonresonant contributions from
virtual levels with an energy displacement

$\delta
E=E_{n,j_{+}}-E_{n,j_{-}}=\frac{(Z\alpha)^{4}m}{2n^{4}(n-1)}\approx\frac{(Z\alpha)^{4}m}{2n^{5}}\,.$
(13)

The ratio of the cross-damping energy shifts relative to transition frequency
is thus estimated by the expression

$\chi\equiv\frac{\delta E_{\rm
CD}}{E_{n}-E_{n-1}}\sim\frac{2}{9}\frac{\alpha^{2}(Z\alpha)^{2}}{n^{2}}\,.$
(14)

For $Z=1$ and $n=18$, this evaluates to $1.9\times 10^{-12}$, which is less
than the accuracy required in order to distinguish between the 2006 and 2018
CODATA values of the Rydberg constant. This estimate suggests that cross-
damping effects are suppressed for Rydberg states and do not represent an
obstacle for the determination of the Rydberg constant from highly excited,
circular Rydberg states.

The above estimates given in Eqs. (12)—(14) are valid for the differential
cross section Jentschura and Mohr (2002). For the total cross section, these
estimates improve even further, consistent with pertinent considerations
reported in Refs. Low (1952); Jentschura and Mohr (2002); Udem  _et al._
(2019).

##  III Quantum Electrodynamic Effects

One might ask if the theory of Rydberg-state transition is well enough under
control in order to facilitate the interpretation of a measurement of
transitions among Rydberg states. As outlined in Ref. Mohr  _et al._ (2008),
the theoretical contributions to the Lamb shift of Rydberg states, on the
level necessary for a determination of the Rydberg constant, can be summarized
into just four terms: (i) the Dirac energy (in the nonrecoil limit) which is
summarized in Eq. (1) of Ref. Jentschura  _et al._ (2008), (ii) the recoil
corrections from the Breit Hamiltonian, which are summarized in Eq. (2) of
Ref. Jentschura  _et al._ (2008), (iii) the relativistic-recoil corrections
summarized in Eq. (3) of Ref. Jentschura  _et al._ (2008), and (iv) the self-
energy effect summarized in Eq. (4) of Ref. Jentschura  _et al._ (2008).
Calculated values of nonrelativistic Bethe logarithms, which enter the
expression for the relativistic recoil correction, have been tabulated for all
states with principal quantum numbers $n\leq 200$ in Ref. Jentschura and Mohr
(2005). This favorable situation illustrates the tremendous simplifications
possible for Rydberg states. Notably, vacuum-polarization, nuclear-size, and
nuclear-structure corrections can be completely ignored for circular Rydberg
states whose probability density at the nucleus vanishes.

Among the four effects listed above, the most interesting contribution
concerns the bound-state self-energy $E_{\rm SE}$, which is described by the
formula

$E_{\rm
SE}=\frac{\alpha}{\pi}\frac{(Z\alpha)^{4}\,m}{n^{3}}\,\biggl{(}A_{40}+(Z\alpha)^{2}\\\
\times\left\\{A_{61}\,\ln\left[\frac{m}{\mu}(Z\alpha)^{-2}\right]+A_{60}\right\\}\biggr{)}\,.$
(15)

The first subscript of the $A$ coefficients counts the number of $Z\alpha$,
while the second counts the number of logarithms
$\ln[\frac{m}{\mu}(Z\alpha)^{-2}]$.

The general result for the $A_{40}$ coefficient for circular Rydberg states
with orbital angular momentum $\ell\neq 0$ and principal quantum number $n\geq
2$ is well known,

$A_{40}=-\left(\frac{\mu}{m}\right)^{2}\frac{1}{2\kappa(2\ell+1)}\,-\frac{4}{3}\,\left(\frac{\mu}{m}\right)^{3}\ln
k_{0}(n,\ell)\,,$ (16)

where $\kappa=(-1)^{j+\ell+1/2}$ is the Dirac angular quantum number and $\ln
k_{0}(n,\ell)$ is the Bethe logarithm. (For values of $\ln k_{0}(n,\ell)$, one
consults Ref. Jentschura and Mohr (2005).) The functional dependence on the
reduced mass is a consequence of the proton’s convection current; an
explanation is given in Chap. 12 of Ref. Jentschura and Adkins (2022). Here,
we will place special emphasis on circular, and near-circular Rydberg states
with $\ell=n-1$ and $\ell=n-2$, with $n\geq 13$, and refer to them as the
following series of states,

  * •

series $\mathcal{A}$: $\ell=n-1$, $j=\ell+1/2$, $\kappa=-(j+1/2)$,

  * •

series $\mathcal{B}$: $\ell=n-1$, $j=\ell-1/2$, $\kappa=(j+1/2)$,

  * •

series $\mathcal{C}$: $\ell=n-2$, $j=\ell+1/2$, $\kappa=-(j+1/2)$,

  * •

series $\mathcal{D}$: $\ell=n-2$, $j=\ell-1/2$, $\kappa=(j+1/2)$.

The $\mathcal{A}$ series has the highest $\ell$ and $j$ for given $n$. The
$A_{40}$ coefficients evaluate to the following expressions for the four
series of states,

$\displaystyle\frac{A_{40}(\mathcal{A},n)}{(\mu/m)^{2}}=$
$\displaystyle\;\frac{1}{2n(2n-1)}-\frac{4}{3}\,\frac{\mu}{m}\,\ln
k_{0}(n,n-1)$ (17) $\displaystyle\frac{A_{40}(\mathcal{B},n)}{(\mu/m)^{2}}=$
$\displaystyle\;-\frac{1}{2(n-1)(2n-1)}-\frac{4}{3}\,\frac{\mu}{m}\,\ln
k_{0}(n,n-1)\,,$ (18)
$\displaystyle\frac{A_{40}(\mathcal{C},n)}{(\mu/m)^{2}}=$
$\displaystyle\;\frac{1}{2(n-1)(2n-3)}-\frac{4}{3}\,\frac{\mu}{m}\,\ln
k_{0}(n,n-2)\,,$ (19)
$\displaystyle\frac{A_{40}(\mathcal{D},n)}{(\mu/m)^{2}}=$
$\displaystyle\;-\frac{1}{2(n-2)(2n-3)}-\frac{4}{3}\frac{\mu}{m}\ln
k_{0}(n,n-2)\,.$ (20)

As a function of the principal quantum number, the Bethe logarithms $\ln
k_{0}(n,n-1)$ and $\ln k_{0}(n,n-2)$ decrease with $n$ for large $n$ as
$n^{-3}$. In the nonrecoil limit $\mu\to m$, and the limit of large $n$, one
has

$A_{40}(\mathcal{A},n)\approx-A_{40}(\mathcal{B},n)\approx
A_{40}(\mathcal{C},n)\\\ \approx-
A_{40}(\mathcal{D},n)\approx\frac{1}{4n^{2}}\,,\qquad n\to\infty\,.$ (21)

The leading quantum electrodynamic corrections for circular and near-circular
Rydberg states are parameterized by the $A_{40}$ coefficient. The quantum
electrodynamic effects are seen to be suppressed, for large $n$, by a factor
$n^{-2}$ which appears in addition to the overall scaling factor $n^{-3}$ in
Eq. (15).

Higher-loop contributions to the anomalous magnetic moment can be taken into
account by the replacement

$-\left(\frac{\mu}{m}\right)^{2}\frac{1}{2\kappa(2\ell+1)}\to-\left(\frac{\mu}{m}\right)^{2}\frac{1}{2\kappa(2\ell+1)}\frac{a_{e}}{\alpha/(2\pi)}$
(22)

where $a_{e}$ contains the higher-loop contributions to the electron anomalous
magnetic moment, which determines the $g$ factor of the electron according to
$g=2(1+a_{e})$. The term $\alpha/(2\pi)$ is the one-loop Schwinger value
Schwinger (1948). The quantity $a_{e}$ can either be taken as the most recent
experimental value of the electron anomalous magnetic moment Fan  _et al._
(2023), which results in $a_{e}=1.159\,652\,180\,59(13)\times 10^{-3}$, or as
a purely theoretical prediction including higher-order effects Laporta (2020).

The suppression of the quantum electrodynamic effects for circular and near-
circular Rydberg states has a physical reason: Namely, the velocity of a
classical electron orbiting the nucleus in a Bohr orbit corresponding to the
principal quantum number $n$ is

$v_{\rm cl}=\frac{Z\alpha c}{n}\,,$ (23)

which evaluates, for $Z=1$ and $n=18$ (this choice of $n$ is motivated in Sec.
IV), to a velocity of $1.21\times 10^{5}{\rm m}/{\rm s}$. This is slower than
the velocity of the fastest macroscopic man-made object, namely, the Parker
Solar Probe which recently reached a velocity of $1.48\times 10^{5}{\rm
m}/{\rm s}$ on its orbit around the Sun PAR ; Raouafi  _et al._ (2023).
Effects originating from relativity and quantum electrodynamics are thus
highly suppressed for circular Rydberg states.

The general result for the $A_{61}$ coefficient, valid for Rydberg states with
$n\geq 13$ and $\ell=n-1$ and $\ell=n-2$, has been given in Eq. (6) of Ref.
Wundt and Jentschura (2008) and Eq. (4) of Ref. Jentschura  _et al._ (2008)
and reads

$A_{61}=\left(\frac{\mu}{m}\right)^{3}\,\frac{3n^{2}-\ell(\ell+1)}{3n^{2}(\ell+3/2)(\ell+1)(\ell+1/2)\ell(\ell-1/2)}\,,$
(24)

a result which is independent of the spin orientation. This expression
evaluates to

$\displaystyle\frac{A_{61}(\mathcal{A},n)}{(\mu/m)^{3}}=$
$\displaystyle\;\frac{A_{61}(\mathcal{B},n)}{(\mu/m)^{3}}=\frac{8}{3n^{2}(n-1)(2n-1)(2n-3)}\,,$
(25) $\displaystyle\frac{A_{61}(\mathcal{C},n)}{(\mu/m)^{3}}=$
$\displaystyle\;\frac{A_{61}(\mathcal{D},n)}{(\mu/m)^{3}}=\frac{32(n+2)}{3n^{2}\prod_{i=2}^{5}(2n-i)}\,.$
(26)

In the large-$n$ limit, one has

$A_{61}(\mathcal{A},n)\approx A_{61}(\mathcal{B},n)\\\ \approx
A_{61}(\mathcal{C},n)\approx
A_{61}(\mathcal{D},n)\approx\frac{2}{3n^{5}}\,,\qquad n\to\infty\,.$ (27)

The suppression with $n^{-5}$, in addition to the overall scaling factor
$n^{-3}$ from Eq. (15), again illustrates the smallness of relativistic and
quantum electrodynamic effects for circular Rydberg states.

Table 1:  Calculated values for the $A_{60}$ coefficients for highly excited
Rydberg states are given for the $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$
and $\mathcal{D}$ series of states, for principal quantum numbers $16\leq
n\leq 20$.

|  |  $\mathcal{A}$ Series |  $\mathcal{B}$ Series
---|---|---|---
$n$ | $\ell$ | $j$ | $A_{60}(n\ell_{j})$ | $j$ | $A_{60}(n\ell_{j})$
16 | 15 | $\frac{31}{2}$ | $1.059\,675(5)\times 10^{-5}$ | $\frac{29}{2}$ | $0.121\,748(5)\times 10^{-5}$
17 | 16 | $\frac{33}{2}$ | $0.805\,212(5)\times 10^{-5}$ | $\frac{31}{2}$ | $0.078\,287(5)\times 10^{-5}$
18 | 17 | $\frac{35}{2}$ | $0.621\,952(5)\times 10^{-5}$ | $\frac{33}{2}$ | $0.049\,885(5)\times 10^{-5}$
19 | 18 | $\frac{37}{2}$ | $0.487\,434(5)\times 10^{-5}$ | $\frac{35}{2}$ | $0.031\,113(5)\times 10^{-5}$
20 | 19 | $\frac{39}{2}$ | $0.387\,025(5)\times 10^{-5}$ | $\frac{37}{2}$ | $0.018\,584(5)\times 10^{-5}$
|  |  $\mathcal{C}$ Series |  $\mathcal{D}$ Series
$n$ | $\ell$ | $2j$ | $A_{60}(n\ell_{j})$ | $2j$ | $A_{60}(n\ell_{j})$
16 | 14 | $\frac{29}{2}$ | $1.540\,182(5)\times 10^{-5}$ | $\frac{27}{2}$ | $0.155\,784(5)\times 10^{-5}$
17 | 15 | $\frac{31}{2}$ | $1.145\,325(5)\times 10^{-5}$ | $\frac{29}{2}$ | $0.096\,026(5)\times 10^{-5}$
18 | 16 | $\frac{33}{2}$ | $0.867\,820(5)\times 10^{-5}$ | $\frac{31}{2}$ | $0.058\,328(5)\times 10^{-5}$
19 | 17 | $\frac{35}{2}$ | $0.668\,553(5)\times 10^{-5}$ | $\frac{33}{2}$ | $0.034\,217(5)\times 10^{-5}$
20 | 18 | $\frac{37}{2}$ | $0.522\,676(5)\times 10^{-5}$ | $\frac{35}{2}$ | $0.018\,690(5)\times 10^{-5}$

The next-higher coefficient is $A_{60}$, which is called the relativistic
Bethe logarithm Jentschura  _et al._ (2003); Le Bigot  _et al._ (2003). Its
absolute magnitude is highly suppressed for circular Rydberg states.
Specifically, according to Refs. Jentschura  _et al._ (2008, 2009, 2010) and
Table 7.2 of Ref. Wundt , one has

$\max\\{|A_{60}(\mathcal{A},n)|,|A_{60}(\mathcal{B},n)|,\\\
|A_{60}(\mathcal{C},n)|,|A_{60}(\mathcal{D},n)|\\}<10^{-4}\,,\qquad n>13\,.$
(28)

Furthermore, according to the calculations reported in Ref. Wundt and
Jentschura (2010); Jentschura  _et al._ (2010), the approximation $G_{\rm
SE}\approx A_{60}$ for the nonperturbative self-energy remainder function
remains valid to excellent approximation for circular Rydberg states, for low
and medium nuclear charge numbers (see Table 1 of Ref. Wundt and Jentschura
(2010) and Tables 1 and 2 of Ref. Jentschura  _et al._ (2010)). The relation
(28) implies that the correction to the transition frequency among circular
Rydberg states induced by the relativistic Bethe logarithm $A_{60}$, for
$Z=1$, is smaller than one part in $10^{-15}$ for $n\geq 13$. Nevertheless, it
is useful to calculate numerical values of relativistic Bethe for the states
under investigation here (see Table 1). We follow the calculational procedure
outlined in Ref. Wundt and Jentschura (2008). For calculated values of
$A_{60}$ for circular and near-circular Rydberg states with $13\leq n\leq 16$,
we refer to Table 1 of Ref. Jentschura  _et al._ (2008) and Table 1 Ref.
Jentschura  _et al._ (2010).

##  IV Experimental Considerations

Let us also include a few considerations relevant to the experimental
realization of a high-precision measurement of the Rydberg constant based on
circular Rydberg states. One might assume that the ultimate experimental
success could be bolstered by choosing transitions with as high a quality
factor $Q$ as possible. As discussed around Eq. (8), since $Q\propto n^{2}$,
high $n$ is desirable.

However, it is also important to consider the sensitivity of a given
measurement to systematic effects. Many systematic effects increase with
powers of $n$. For instance, shifts and distortions of resonances due to the
Stark effect scale as $n^{5}$ Lutwak ; Holley ; deVries , which produces
challenges to measuring transitions between circular Rydberg states with very
high $n$. However, the previous measurement between circular Rydberg states of
hydrogen (Ref. deVries ) between $n=27$ and $n=28$, and $n=29$ and $n=30$, had
negligible contributions from uncertainties in the Stark shifts deVries ). The
experimental accuracy was instead limited by dipole-dipole interactions. Since
the dipole moment for an atom in a superposition of adjacent circular Rydberg
states scales as $n^{2}$, and the systematic effect is related to the
interaction energy of two dipoles, this effect scaled as $n^{4}$.

Therefore, in order to mitigate the dipole-dipole interactions, it may be
interesting to consider transitions between circular Rydberg states with
somewhat lower $n$. For instance, with all other experimental parameters being
similar, a transition between $n=18$ and $n=19$ would reduce the effects of
the dipole-dipole interactions by a factor of $\sim 6$ as compared to the
previous measurement deVries , Another experimental benefit to reducing $n$
below that demonstrated in deVries , is that blackbody-radiation-induced
transitions would be mitigated because the thermal radiation spectral density
for temperatures $\leq 300$ K is reduced for the more energetic transitions
occurring between lower-lying states. This may allow the experiment to be
performed at liquid nitrogen as opposed to liquid helium temperatures.

The MIT measurement used pulsed lasers at a repetition rate of
$61\,\mathrm{Hz}$ to produce circular Rydberg states. Therefore, another
option to mitigate dipole-dipole interactions could be to produce a near-
continuous source of circular Rydberg states using continuous-wave (cw)
lasers. Since the dipole-dipole interaction is related to the peak density of
circular Rydberg states, a near-continuous source of circular Rydberg states
could allow for a large reduction in the peak density while maintaining
sufficient statistics. This could be accomplished by first using the $1S$–$2S$
two-photon transition to populate the $2S$ metastable state as in Refs. Beyer
_et al._ (2017); Brandt  _et al._ (2022), followed by excitation to Rydberg
levels using a 365 nm cw laser. Then circularization would be performed using
the methods outlined in Ref. Lutwak  _et al._ (1997).

To perform spectroscopy of the $n=18$ to $n=19$ circular Rydberg states, a
millimeter-wave Ramsey apparatus akin to the one employed in Ref. deVries
could be used. To excite the transition, a radiation source at 1.04 THz is
needed. While the millimeter wave source in deVries  operated at 256 or 316
GHz, a similar source operating at frequencies above 1 THz is possible using a
planar GaAs Schottky diode frequency multiplier Mehdi  _et al._ (2017). The
output power of such THz sources is relatively low. However, due to the large
transition matrix element between circular Rydberg states, the transition can
be saturated with $<1$ nW and a 3 mm beam waist. Therefore, commercially
available THz sources would likely be sufficient vad .

##  V Conclusions

The main conclusions of this paper are as follows. In Sec. II, we have shown
that the quality factors of transitions among circular Rydberg are sufficient
to comfortably allow for a distinction between the 2006 and 2018 CODATA values
of the Rydberg constant [see Eqs. (2) and (3), and Refs. Mohr  _et al._
(2008); Tiesinga  _et al._ (2021)]. Furthermore, according to the
considerations reported in Sec. II, cross-damping terms do not to present an
obstacle to such a measurement. In Sec. III, we showed that the theory of
bound states is sufficiently under control to allow for a determination of the
Rydberg constant from transitions among circular Rydberg states in atomic
hydrogen. Experimental considerations (Sec. IV) corroborate the advances in
technology which make such a measurement more feasible than reported in Ref.
deVries , in part by reducing several systematic effects through a less dense
atomic beam which can be realized in a continuous-wave excitation scheme into
the circular states.

A few concluding remarks on the proton radius puzzle are in order. We recall
that the proton radius puzzle refers to the difference between the “smaller”
proton radius of $R_{p}\approx 0.84\,{\rm fm}$ obtained in Ref. Pohl  _et al._
(2010) and the larger value of $R_{p}\approx 0.88\,{\rm fm}$ from the 2006
CODATA adjustment (see Refs. de Beauvoir  _et al._ (1997); Schwob  _et al._
(1999); Jentschura  _et al._ (2005); Mohr  _et al._ (2008) and references
therein). Various recent scattering experiments Bernauer  _et al._ (2010);
Xiong  _et al._ (2019) and spectroscopic experiments Beyer  _et al._ (2017);
Fleurbaey  _et al._ (2018); Bezginov  _et al._ (2019); Brandt  _et al._
(2022); Yzombard  _et al._ (2023) come to conflicting conclusions on the
proton radius. A recent measurement described in Refs. Brandt  _et al._ (2022)
has led to a value of $R_{p}\approx 0.86\,{\rm fm}$. It has very recently been
pointed out in Ref. Jentschura (2022) that two older scattering experiments,
carried out in 1969 at Brookhaven (see Refs. Camilleri  _et al._ (1969a, b))
are consistent with an 8% discrepancy in the cross sections between muon-
proton and electron-proton scattering, which translates into 4% for the form
factor slope, which in turn amounts to 2% for the radius. This is precisely
the difference between the “smaller” proton radius of $R_{p}\approx 0.84\,{\rm
fm}$ and the recently obtained (Ref. Brandt  _et al._ (2022)) value of
$R_{p}\approx 0.86\,{\rm fm}$. The MUSE experiment MUS ; M. Kohl et al. [MUSE
Collaboration] (2014); Kohl  at the Paul–Scherrer Institute aims to remeasure
the muon-proton cross sections in the near future.

In conclusion, we have shown that the idea formulated in Refs. Lutwak ; Holley
; deVries ; Lutwak  _et al._ (1997) and Refs. Jentschura  _et al._ (2008,
2009, 2010) could lead to a feasible pathway toward a determination of the
Rydberg constant. This could be interesting because most recent spectroscopic
experiments Beyer  _et al._ (2017); Fleurbaey  _et al._ (2018); Bezginov  _et
al._ (2019); Brandt  _et al._ (2022); Yzombard  _et al._ (2023) focus on
transitions in atomic hydrogen which depend on both constants in question,
namely, the proton radius and the Rydberg constant. Focusing on Rydberg
states, as proposed here, means that one isolates one of these constants,
thereby potentially obtaining a clear and distinct picture of the proton
radius puzzle. The current situation provides not only motivation to carry out
the MUSE experiment at PSI MUS ; M. Kohl et al. [MUSE Collaboration] (2014);
Kohl , but also, to re-double efforts to measure the Rydberg constant.

## Acknowledgements

The authors acknowledge extensive insightful conversations with B. J. Wundt,
and helpful conversations with S. M. Brewer. Support from the National Science
Foundation (grant PHY–2110294) is gratefully acknowledged. Furthermore, U.D.J.
and D.C.Y. acknowledge support from the Templeton Foundation (Fundamental
Physics Black Grant, Subawards 60049570 MST and 60049570 CSU of Grant ID
#61039), is also gratefully acknowledged.

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