# Large Language Model Inference with Lexical Shortlisting

Nikolay Bogoychev, Pinzhen Chen, Barry Haddow, Alexandra Birch

###### Abstract

Large language model (LLM) inference is computation and memory intensive, so
we adapt lexical shortlisting to it hoping to improve both. While lexical
shortlisting is well-explored in tasks like machine translation, it requires
modifications before being suitable for LLMs as the intended applications vary
significantly. Our work studies two heuristics to shortlist sub-vocabulary at
LLM inference time: Unicode-based script filtering and corpus-based selection.
We explore different LLM families and sizes, and we find that lexical
shortlisting can reduce the memory usage of some models by nearly 50% and has
an upper bound of 25% improvement in generation speed. In this pilot study, we
also identify the drawbacks of such vocabulary selection methods and propose
avenues for future research.

##  1 Introduction

Pre-trained large language models are gaining increasing attention given their
strong performance in various natural language tasks (Radford et al. 2019;
Brown et al. 2020; Kaplan et al. 2020; Ouyang et al. 2022). Most LLMs are
Transformer-based (Vaswani et al. 2017; Scao et al. 2022; Touvron et al.
2023), where the output layer with hidden size $D$ is associated with the most
expensive matrix multiplication over space $D\times|V|$, where $D\ll|V|$ with
a vocabulary $V$. This expensive operation leads to increased inference cost
of both memory and speed given the autoregressive nature of LLM decoding. As
LLMs are already massive in size, this inference latency makes LLM deployment
costly.

In practical scenarios, shortlisting a sub-vocabulary $V^{\prime}$ with
$|V^{\prime}|\ll|V|$, and only loading the corresponding sub-embedding matrix
for inference seems favourable, since the majority of the logits from the
output layer do not affect the hypothesis token(s) at each time step. This has
been actively explored in machine translation, it is feasible to infer the
probable output tokens given the input (Schwenk, R. Costa-jussà, and R.
Fonollosa 2007; Le, Allauzen, and Yvon 2012; Devlin et al. 2014; Bogoychev et
al. 2022). This can result in a 10-to-30-fold reduction in vocabulary size,
therefore massively speeding up the decoding process.

For translation, sub-vocabulary selection is usually performed by pre-
computing likely word-level alignments. While anticipating certain limitations
such as domain mismatch (Bogoychev and Chen 2021; Domhan et al. 2022), lexical
shortlisting in LLMs has a fundamental difficulty: often the outputs are
variable and open-ended, making it less straightforward to determine the
vocabulary needed.

In this pilot study, we adapt lexical shortlisting to LLMs by relaxing the
selection criterion to the potential language of candidate generations
(Abdaoui, Pradel, and Sigel 2020; Ushio, Zhou, and Camacho-Collados 2023). In
specific, we propose and experiment with two strategies: script-based token
filtering where vocabulary items are removed if they do not belong to the
output language, and corpus-based pre-selection where we keep items based on
vocabulary hits from a large representative corpus. Our contributions can be
summarized as follows:

  * •

We propose two vocabulary selection methods: one based on writing script
filtering, and one based on corpus selection.

  * •

We experiment with models of different families and sizes (LLaMA and BLOOM),
and report varying output layer behaviours in terms of performance.

  * •

We perform an extensive controlled study, measuring potential speed-ups for
different LLMs and hardware.

  * •

We discuss practical limitations for applying these methods in the wild.

##  2 Vocabulary Shortlisting

Large Language models, especially multilingual ones, hold vocabulary items for
many languages and scripts, which are rarely used simultaneously in a single
generation pass. This work filters unnecessary vocabulary tokens according to
the language of the query. We describe two ways to prepare sub-vocabulary for
LLMs, both focusing on retaining tokens relevant only to the language being
generated. They are tested on-the-fly in a batched setting: we determine a
sub-vocabulary for an entire batch, because running shortlisting separately
for each instance in a batch diminishes the return. Furthermore, we always
include all tokens appearing in the inputs in our vocabulary selection.

##### Script-based selection

We first propose to only keep the tokens which match the writing script of a
desired language. This can be done by filtering token strings based on the
Unicode range. It should be especially effective for languages operating on
unique scripts, such as Armenian, Chinese, Korean, etc, since it allows for
concise vocabulary space restriction to the necessary items. This method might
be less effective for languages that share a writing system with others, e.g.
languages using Cyrillic or extended Latin alphabets. It would be infeasible
to limit the vocabulary to the set of subwords that belong to a specific
language solely, resulting in a relatively large vocabulary. Moreover, this
method would strictly rule out code-mixed tokens, emojis, etc.

##### Corpus-based selection

A more comprehensive way is to tokenize a representative corpus in the desired
language in advance and use the vocabulary entries that have been recorded to
build a sub-vocabulary. Nonetheless, this method is non-exhaustive because we
could miss rare but valid tokens, or suffer from domain mismatch between the
vocabulary selection corpus and the inference prompts. This approach is
similar to the work of Abdaoui, Pradel, and Sigel (2020) and Ushio, Zhou, and
Camacho-Collados (2023).

##  3 Experimental Setup

###  3.1 Languages

We experiment on four languages: Bulgarian, Chinese, English, and Spanish.
This offers a combination of distinct conditions that cover most use cases.
English and Spanish use the same script and would have a high overlap in
vocabulary after tokenization. Since we test on English-centric models, we
examine how good of a sub-vocabulary we can find when it is not possible to
shortlist merely based on the script. Bulgarian is a low-resource language
written in the Cyrillic script. Most multilingual language models have lower
amounts of Cyrillic tokens, so we expect that script-based filtering will
leave a small sub-vocabulary; however, since Cyrillic is used by a number of
languages, we will inevitably end up with vocabulary items that do not belong
to the Bulgarian. Finally, Chinese is a high-resource language with a unique
script.111Chinese characters also appear in Japanese and Korean, but those
languages also use Kana and Hangul, which will be omitted from the vocabulary
automatically. Unicode vocabulary selection would be the most effective in
this case.

###  3.2 LLM selection

We experiment with instruction-tuned LLMs based on BLOOM at various sizes
(Scao et al. 2022) as well as LLaMA-7B (Touvron et al. 2023). In specific, we
adopt Chen et al. (2023)’s models fine-tuned on machine translations of the
Alpaca dataset (Taori et al. 2023) to test for open domain question answering,
which maximizes the difficulty for shortlisting as explained earlier.

BLOOM is a multilingual LLM that explicitly supports English, Spanish and
Chinese, but not Bulgarian. Consequently, it has sizeable vocabulary of 250K
and is therefore a prime candidate to reduce vocabulary for a specific
language during inference. We experiment with the 560M, 1B7, and 7B1
checkpoints, with diminishing computational burden on the embedding and output
layers.

LLaMA is an English-centric LLM with a small 32K vocabulary. In this case, we
might have reduced benefit from vocabulary shortlisting, because a
proportionally lower amount of computation occurs in the output layer. On the
other hand, since the LLM is English-focus, we expect drastic vocabulary
reductions compared to BLOOM for Bulgarian and Chinese.

###  3.3 Evaluation

Our test set is 50 prompt questions from the OpenAssistant dataset (Köpf et
al. 2023) that are human translated into all testing languages. The potential
answers are of various degree of open-endedness and lengths. We decode them
with a beam size of 1. Our evaluation measures the time taken to decode the
test set (in mm:ss). We measure end-to-end time including model loading and
tokenization with Hugging Face’s implementation throughout. As a generation
quality indicator, we count the number of times a model fails to produce the
exact same output with a full vocabulary or a shortlisted vocabulary, given
the same input question. Ideally, the mismatch should be zero, which implies
no impact from shortlisting.

###  3.4 Shortlisting details

For vocabulary selection, we tokenize the test inputs and always include the
tokens to the sub-vocabulary. We then apply either of the proposed selection
method. Script-based selection checks whether a token falls in a specific
Unicode subset: Cyrillic for Bulgarian, ASCII for English, Latin Extended-A
for Spanish, and Chinese characters for Chinese. For corpus-based selection,
we utilize a subset of the WikiMatrix corpus (Schwenk et al. 2021) which
contains Wikipedia texts for each language. In addition for both selection
methods, we keep the first 300 vocabulary entries too, as those usually
correspond to special tokens, Unicode bytes (for byte-level BPE), numbers, and
punctuation.

We pre-compute the vocabulary subset offline and we do not record the time
spent on pre-tokenizing a large corpus or extracting a Unicode subset in the
measurements. Although being a time-consuming stage of vocabulary selection,
these can be reused for every batch during inference once done. In detail,
script-based filtering takes under 60 seconds and corpus-based selection takes
up to 10 minutes. Adding the inputs’ tokens to the existing pre-selected sub-
vocabulary is extremely fast in comparison; we do however count model slicing
in our runtime, as it needs to be done for every processed batch.

Language | $|V|$ | BLOOM-560M | BLOOM-1B7 | BLOOM-7B1
---|---|---|---|---
0time | mis | time | mis | time | mis
bg full | 250680 | 05:26 | – | 15:18 | – | 65:01 | –
bg Unicode | 22912 | 04:39 | 1/50 | 13:44 | 4/50 | 51:46 | 10/500
bg corpus | 58642 | 04:49 | 0/50 | 09:34 | 1/50 | 60:28 | 3/50
bg oracle | 1408 | 04:22 | 0/50 | 12:31 | 0/50 | 61:06 | 0/50
en full | 250680 | 07:37 | – | 16:35 | – | 55:08 | –
en Unicode | 186752 | 07:40 | 1/50 | 16:05 | 0/50 | 58:18 | 0/50
en corpus | 113024 | 07:00 | 1/50 | 15:08 | 3/50 | 54:20 | 2/50
en oracle | 4736 | 06:14 | 0/50 | 13:06 | 0/50 | 48:46 | 0/50
es full | 250680 | 05:58 | – | 12:26 | – | 63.15 | –
es Unicode | 187008 | 05:48 | 0/50 | 12:01 | 0/50 | 59:15 | 0/50
es corpus | 112128 | 05:37 | 0/50 | 11:34 | 4/50 | 57:41 | 4/50
es oracle | 4736 | 04:53 | 0/50 | 09:26 | 0/50 | 51:43 | 0/50
zh full | 250680 | 06:29 | – | 15:27 | – | 55:09 | –
zh Unicode | 51584 | 05:54 | 16/500 | 13:09 | 21/500 | 50:50 | 22/500
zh corpus | 104320 | 06:08 | 11/500 | 14:08 | 16/500 | 46:39 | 17/500
zh oracle | 4096 | 05:16 | 0/50 | 12:07 | 0/50 | 50:50 | 0/50
Table 1: CPU shortlisting results for BLOOM. The time format is mm:ss. The mis(match) column refers to the number of sentences that do not exactly match the full vocabulary output. Language | $|V|$ | LLaMA-7B
---|---|---
time | mis
bg full | 32000 | 117:15 | –
bg Unicode | 4736 | 125:55 | 19/500
bg corpus | 26496 | 132:24 | 5/50
bg oracle | 2048 | 123:06 | 0/50
en full | 32000 | 113:52 | –
en Unicode | 27520 | 125:57 | 6/50
en corpus | 30720 | 111:30 | 19/500
en oracle | 4480 | 119:32 | 0/50
es full | 32000 | 131:03 | –
es Unicode | 27648 | 128:00 | 8/50
es corpus | 30336 | 129:26 | 2/50
es oracle | 3456 | 123:25 | 0/50
zh full | 32000 | 130:42 | –
zh Unicode | 2688 | 114:39 | 13/500
zh corpus | 28160 | 119:58 | 2/50
zh oracle | 1536 | 126:16 | 0/50
Table 2: CPU shortlisting results for LLaMA-7B. The time format is mm:ss. The
mis(match) column refers to the number of sentences that do not exactly match
the full vocabulary output.

###  3.5 Upper bound performance

We conduct an oracle vocabulary selection experiment to find the theoretical
upper bound for speed and memory improvements. For oracle shortlisting, we run
inference using full vocabulary and we select the used vocabulary items (and
the first 300 vocabulary entries) the oracle sub-vocabulary. We perform this
in a batched setting, because per-input slicing is prohibitively expensive in
practice and will offset any potential performance benefit.

###  3.6 Test hardware

We perform our experiments both on CPU and GPU devices. For the CPU tests we
use Intel(R) Xeon(R) Gold 6248 (40 Cores, 80 Threads), and for GPU tests we
use an Nvidia GeForce RTX 3090. Inference on CPU is performed in float32
precision, whereas on GPU it is in int8 with bitsandbytes quantization
(Dettmers et al. 2022).

Language | $|V|$ | BLOOM-560M | BLOOM-1B7 | BLOOM-7B1
---|---|---|---|---
0time | mis | time | mis | time | mis
bg full | 250680 | 05:22 | – | 08:29 | – | 14:43 | –
bg Unicode | 22912 | 05:23 | 0/50 | 08:45 | 6/50 | 14:35 | 17/500
bg corpus | 58642 | 05:22 | 0/50 | 08:38 | 1/50 | 14:33 | 10/50
bg oracle | 1408 | 05:21 | 0/50 | 09:06 | 0/50 | 14:33 | 0/50
en full | 250680 | 06:50 | – | 09:02 | – | 11:54 | –
en Unicode | 186752 | 06:54 | 0/50 | 08:52 | 0/50 | 11:46 | 0/50
en corpus | 113024 | 06:38 | 2/50 | 08:56 | 3/50 | 11:59 | 3/50
en oracle | 4736 | 06:43 | 0/50 | 09:00 | 0/50 | 11:52 | 0/50
es full | 250680 | 06:17 | – | 07:05 | – | 12:35 | –
es Unicode | 187008 | 06:13 | 0/50 | 07:03 | 0/50 | 12:17 | 0/50
es corpus | 112128 | 06:15 | 1/50 | 7:10 | 3/50 | 12:30 | 3/50
es oracle | 4736 | 06:26 | 0/50 | 07:23 | 0/50 | 12:17 | 0/50
zh full | 250680 | 05:37 | – | 08:47 | – | 11:58 | –
zh Unicode | 51584 | 06:10 | 15/500 | 08:34 | 20/500 | 11:22 | 29/500
zh corpus | 104320 | 06:03 | 11/500 | 09:01 | 16/500 | 11:35 | 13/500
zh oracle | 4096 | 05:35 | 0/50 | 08:42 | 0/50 | 11:46 | 0/50
Table 3: GPU shortlisting results for all languages with BLOOM. The time
format is mm:ss. The mis(match) column refers to the number of sentences that
do not exactly match the full vocabulary output.

Language | $|V|$ | BLOOM’s memory (GB)
---|---|---
560M | 1B7 | 7B1
Full model | 250680 |  02.100 |  06.100 | 27.100
Embedding matrix |  |  |
full vocab | 250680 | 0.90 | 1.90 | 3.80
1-5 0bg Unicode | 22912 | 0.09 | 0.18 | 0.36
bg corpus | 58642 | 0.22 | 0.45 | 0.90
en Unicode | 186752 | 0.70 | 1.40 | 2.80
en corpus | 113024 | 0.44 | 0.88 | 1.70
es Unicode | 187008 | 0.70 | 1.40 | 2.80
es corpus | 112128 | 0.43 | 0.86 | 1.70
zh Unicode | 51584 | 0.20 | 0.40 | 0.80
zh corpus | 104320 | 0.40 | 0.80 | 1.60

Table 4: Theoretical memory footprint for the BLOOM family with float32
featuring the embedding matrix.

Language | $|V|$ |  memory (GB) LLaMA-7B
---|---|---
Full model | 32000 | 27.100
Embedding matrix |
full vocab | 32000 | 0.50
1-3 0bg Unicode | 4736 | 0.07
bg corpus | 26496 | 0.41
en Unicode | 27520 | 0.43
en corpus | 30720 | 0.48
es Unicode | 27648 | 0.43
es corpus | 30336 | 0.47
zh Unicode | 2688 | 0.04
zh corpus | 28160 | 0.44

Table 5: Theoretical memory footprint for LLaMA-7B with float32 featuring the
embedding matrix.

##  4 Results and Discussions

###  4.1 CPU results

#### Oracle shortlisting

We present our CPU results on the BLOOM family in Table 1 and the results on
LLaMA-7B in Table 2. The oracle rows indicate theoretical upper bound of speed
improvements using lexical shortlisting. We observe around 20% improvements
with the smaller BLOOM-560M and BLOOM-1B7, but only 5-10% in the 7B models. As
the models grow in size, the upper bound gain decreases. This could be
explained by the fact that the proportion of the embedding and output matrices
becomes smaller in a larger model. By comparing BLOOM-7B1 with LLaMA-7B, we
also find that the larger the base vocabulary, the more speed gain we can get
by reducing the vocabulary. We note that the oracle vocabulary is more than an
order of magnitude smaller than the other shortlisting approaches, but in
practice it would be difficult to reduce the vocabulary size by as much.

Speed numbers of LLaMA-7B on CPU are relatively inconsistent and had wide
variance across test runs. We attribute this to the small vocabulary size and
thus less computational footprint in the output layer affected by
shortlisting. Also, there could be various scheduling issues and non-
deterministic cache accesses as GEMM operations are split across the 40 cores
of the CPU we used.

#### Script-based shortlisting

When applying script-based shortlisting by filtering vocabulary items
according to the Unicode range, we observe different trends in English and
Spanish compared to Bulgarian and Chinese. For BLOOM, the sub-vocabulary size
for Bulgarian and Chinese can be reduced to only 10-20%, whereas for English
and Spanish it has about 60% of the original size. This is potentially because
BLOOM allocated more vocabulary items for European languages which are the
dominant ones when the tokenizer is trained. Generally, the inference time
reduces to between using the full vocabulary and the oracle shortlisting. In
terms of mismatch, the model can maintain almost the same outputs with and
without shortlisting for English and Spanish. However there are 10-20%
mismatches for Bulgarian and 30-40% for Chinese.

LLaMA-7B results are less favourable: script-based shortlisting does not
significantly reduce the vocabulary size for English and Spanish, and all
languages suffer from relative high mismatches between 10-40%. Specifically
for Bulgarian and Chinese, we argue that Unicode filtering could be too harsh
as sometimes English characters are code-mixed in the language and cannot be
avoided, e.g., when generating a website link. Therefore, we conclude that
shortlisting based on the writing script can improve inference efficiency
without degrading performance for a multilingual LLM to generate Latin
languages, but it is less feasible for non-Latin languages or an English-
centric LLMs with a smaller vocabulary.

#### Corpus-based shortlisting

As opposed to shortlisting based on Unicode, corpus-based shortlisting leaves
a much larger vocabulary for Bulgarian and Chinese, but reduces the vocabulary
to half or less for English and Spanish. This method achieves a more balanced
shortlisting effect for each language potentially because the inclusion of
many tokens outside of the output language’s writing system. However, for
LLaMA-7B which has a small vocabulary in the first place, this approach keeps
majority of the entries for all languages, and is thus not useful.

Corpus-based shortlisting also ameliorates the quality problem to some extent,
although the Chinese models still struggle to produce identical output as the
full vocabulary models. Overall, we see a small but consistent reduction in
runtime with BLOOM too for this shortlisting approach, indicating its
practicality at least for English.

###  4.2 Memory

In Table 3.6 we show that lexical shortlisting leads to ample memory footprint
reduction, especially for smaller models like BLOOM-560M, where the model size
is dominated by the vocabulary. In practice, these models are small enough to
fit in modern GPUs and CPUs, so the reduced memory is not game-changing for
latency, but we argue that the saved memory could be used to accommodate a
larger batch size so as to enhance throughput. On the other hand, when looking
at bigger models like BLOOM-7B1 or LLaMA-7B in Table 3.6, the relative
reduction in model size is more modest and will not enable the use of smaller
GPUs.

For small models such as BLOOM-560M, the vocabulary makes up nearly 50% of all
parameters in the model, but as we increase the model size, this percentage
diminishes. This can also be used as a proxy judgement about where the
majority of the computational time is spent during generation: the larger the
model, the lesser the time spent in the output layer. Hence, we expect to have
lower speed improvements with shortlisting bigger models in the wild.

###  4.3 GPU results

In addition to CPU tests, we performed the same BLOOM experiments on GPU and
we observe that all three selection criteria including the oracle do not lead
to improved inference speed as shown in Table 3. Unfortunately, the small
performance differences might amount to little more than noise, when the
overhead of model slicing is considered. We hypothesize that GPUs are designed
for multiplying large matrices, so reducing the matrix size, even to the
extremity of an oracle sub-vocabulary, is not able to offer any speedup. This
is consistent with Bogoychev et al. (2020)’s findings in applying shortlists
to neural machine translation on GPUs.

##  5 Related Work

Lexical shortlisting for neural machine translation has been extensively
adopted to speed up neural machine translation inference (Kim et al. 2019;
Bogoychev et al. 2020; Behnke et al. 2021; Bogoychev et al. 2022). With larger
vocabulary sizes, and more model parameters, large language models could
benefit from the memory efficiency brought by shortlisting too.

Updating the vocabulary of a pre-trained large language model to support to an
unseen language has been a common practice, where in most situations the
vocabulary is augmented with new entries. For instance, Cui, Yang, and Yao
(2023) augmented the LLaMA vocabulary with 20K Chinese characters and
continued training the model to adapt to the Chinese Language. More
aggressively, Gee et al. (2022) switched the vocabulary of a pre-trained
language model to a new one that is more suitable for the target domain and
more compact, with a consideration of efficiency. This method improves
inference speed, but unlike our work, it requires further training the
embedding matrix for the new vocabulary.

The most similar to our work are vocabulary trimming by Abdaoui, Pradel, and
Sigel (2020) and Ushio, Zhou, and Camacho-Collados (2023). They slice the
vocabulary of a model based on language criteria. Their work focuses on
reducing the model size with no speed considerations. Furthermore, they
experimented with multilingual pre-trained models with large vocabulary sized
at hundreds of thousands, whereas we explore large language models including
the multilingual BLOOM with 250K vocabulary and the English-centric LLaMA with
32K vocabulary .

##  6 Conclusion

We presented a study of using lexical shortlisting to speed up inference with
large language models. We show that while we can achieve speed improvements,
it does not guarantee that the output is not altered compared to full
vocabulary generation. This is due to the lack of a heuristic that gives a
compact and at the same time high-quality sub-vocabulary. With the models
tested, we see feasibility of our proposed approaches for English and Spanish,
but there are shortcomings when considering languages written in non-Latin
script.

## Limitations

We used (mis)matches to indicate the quality of generation when applying
shortlisting. However, this is a strict metric that penalizes a different
output regardless of its length or the number of incorrect tokens. Since the
inference time of an autoregressive LLM can be affected by the number of
tokens generated for each input, it is reasonable to include metrics like BLEU
and ROUGE to take the generation length or recall into account. Further more,
the shortlisted vocabulary, inference time, and mismatch are entangled and we
are not aware of an ideal evaluation setup.

## Acknowledgement

This project has received funding from UK Research and Innovation (UKRI) under
the UK government’s Horizon Europe funding guarantee [grant numbers 10052546
and 10039436].

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# Gauge effects in bound-bound Rydberg-transition matrix elements

A. Duspayev<EMAIL_ADDRESS>Department of Physics, University of Michigan,
Ann Arbor, MI 48109, USA  G. Raithel  Department of Physics, University of
Michigan, Ann Arbor, MI 48109, USA

###### Abstract

Accurate data on electric-dipole transition matrix elements (EDTMs) for bound-
bound Rydberg-atom transitions become increasingly important in science and
technology. Here we compute radial EDTMs of rubidium using the length,
velocity and acceleration gauges for electric-dipole-allowed transitions
between states with principal and angular-momentum quantum numbers ranging
from 15 to 100. Wave-functions are computed based upon model potentials from
Marinescu et al., Phys. Rev. A 49, 982 (1994). Length-gauge EDTMs, often used
for low-$\ell$ transitions, are found to deviate from the fundamentally more
accurate velocity-gauge EDTMs by relative amounts of up to $\sim 10^{-3}$. We
discuss the physical reasons for the observed gauge differences, explain the
conditions for applicability of the velocity and length gauges for different
transition series, and present a decision tree of how to choose EDTMs.
Implications for contemporary Rydberg-atom applications are discussed.

##  I Introduction

Rydberg atoms Gallagher (2005) are an active field of modern physics with
applications in precision measurements Hare  _et al._ (1993); Jentschura  _et
al._ (2010); Ramos  _et al._ (2017); Berl  _et al._ (2020), molecular physics
Shaffer  _et al._ (2018); Fey  _et al._ (2020), quantum control Thaicharoen
_et al._ (2019); Cardman and Raithel (2020), field sensing Sedlacek  _et al._
(2012); Holloway  _et al._ (2014); Anderson  _et al._ (2021, 2020), nonlinear
quantum optics Chang  _et al._ (2014); Liang  _et al._ (2018); Cantu  _et al._
(2020), and as a platform for quantum computing and simulations Saffman
(2016); Browaeys and Lahaye (2020); Morgado and Whitlock (2021). Being at the
core of several directions in fundamental research and emerging quantum
technologies Adams  _et al._ (2019), Rydberg transitions from low-lying states
or other Rydberg atomic states require accurate calculations Safronova  _et
al._ (2004); Šibalić  _et al._ (2017). Of particular interest are electric and
magnetic multipole transition matrix elements Safronova and Safronova (2011),
static and dynamic polaizabilities Safronova  _et al._ (2004); Chen  _et al._
(2015), and collisional and photo-ionization Dinneen  _et al._ (1992); Markert
_et al._ (2010); Viray  _et al._ (2021); Cardman  _et al._ (2021) cross-
sections. Among the former, electric-dipole transition matrix elements (EDTMs)
are the most important due to their wide usage Safronova  _et al._ (2004).

A common framework for the computation of EDTMs between different bound
Rydberg levels is based on model potentials Marinescu  _et al._ (1994). In
most cases, the computations are performed in the length gauge (LG) Bethe and
Salpeter (2012); Friedrich (2017), also referred to as Babushkin gauge
Papoulia  _et al._ (2019). Another common form is the velocity gauge (VG). In
addition, there exists an acceleration gauge (AG). Although quantum mechanics
is gauge-invariant Bethe and Salpeter (2012); Sakurai and Napolitano (2017),
different approximations and assumptions must be made to transform the
expressions for EDTMs between different gauges, which can result in notable
discrepancies in the final results Grant (1974); Fischer (2009). These are
revealed in high-precision calculations and naturally raise the question which
gauge should be used. This issue has been discussed in a range of research
fields, including interaction of atoms and molecules with strong fields Leone
_et al._ (1989); Cormier and Lambropoulos (1996); Bauer  _et al._ (2005); Chen
and Hu (2009); Han and Madsen (2010); Bandrauk  _et al._ (2013), solid-state
physics Virk and Sipe (2007); Passos  _et al._ (2018) and astrophysical
spectroscopy A. Pehlivan Rhodin  _et al._ (2017); Papoulia  _et al._ (2019).

FIG. 1: Decision tree to select EDTMs from pre-computed length-gauge and
velocity-gauge databases of EDTMs, $M_{L}(n_{i},\ell_{i},\Delta n,\Delta\ell)$
and $M_{V}(n_{i},\ell_{i},\Delta n,\Delta\ell)$, respectively. Initial- and
final-state principal and angular-momentum quantum numbers are denoted
$(n_{i},\ell_{i})$ and $(n_{f},\ell_{f})$, respectively, the difference
$\Delta n=n_{f}-n_{i}$, the difference $\Delta\ell=\ell_{f}-\ell_{i}=\pm 1$,
and $\ell_{<}$ is the lesser of $\ell_{i}$ and $\ell_{f}$.

To our knowledge, gauge effects in EDTMs have not been broadly discussed yet
in context with emerging applications of Rydberg-atom transitions in the
aforementioned quantum technologies. Here we provide a comparison of EDTMs
computed in the LG, VG and AG gauges for electric-dipole electromagnetic
transitions in rubidium Rydberg atoms. The main emphasis is placed on the LG
and VG forms, the most widely used and the most accurate gauges, respectively,
while the AG is discussed for instructive purposes. We show that the angular-
momentum type of the Rydberg-Rydberg transition plays a decisive role in what
is the best choice for the gauge that should be used. For the Rb model
potentials Marinescu  _et al._ (1994) that we use to compute the Rydberg-
electron wave-functions, we find that the VG should be used if the lesser of
the involved angular momenta, $\ell_{<}$, is 2 or lower, and that the VG or
the LG can be used otherwise, as they produce identical results within the
numerical accuracy level of the calculations. As an exception, only the LG
should be used for transitions between near-degenerate Rydberg levels with
$l_{<}\geq 3$. The decision tree that follows is summarized in Fig. 1 and is
rationalized throughout the paper. While the AG should not be used in
applications, for instructive purposes we show that the AG results converge
with the VG and LG results in the appropriate limits.

The paper is organized as follows: The three gauge forms are reviewed and the
relevant expressions are provided in Sec. II, the results are presented in
Sec. III and discussed in Sec. IV. The paper in concluded in Sec. V.

##  II Matrix elements in different gauge forms

In this section, we review the formalism for calculations of EDTMs in the
three gauges, and justify the validity of the electric-dipole approximation
(EDA). We begin with a spin-less, non-relativistic $N$-electron atom with
nuclear charge $Z$ placed in a plane-wave electromagnetic field. The linearly
polarized field has a vector potential
$\mathbf{A}({\bf{r}},t)={\bf{A}}_{0}\cos({\bf{k}}\cdot{\bf{r}}-\omega t)$,
wavevector $\mathbf{k}$, angular frequency $\omega$, and an amplitude of the
electric field of $E_{0}=\omega A_{0}$. The atom can be described by the
following Hamiltonian:

$\displaystyle\hat{H}=\hat{H}_{0}+\hat{H}_{int},$ (1)

where the field-free part, $\hat{H}_{0}$, is (in SI units)

$\displaystyle\hat{H}_{0}=\sum_{i=1}^{N}\left(\frac{\hat{\mathbf{p}}_{i}^{2}}{2m_{e}}-\frac{Ze^{2}}{4\pi\epsilon_{0}\hat{r}_{i}}\right)+\frac{1}{4\pi\epsilon_{0}}\sum_{\begin{subarray}{c}i=1\\\
i\neq
j\end{subarray}}^{N}\frac{e^{2}}{|\hat{\mathbf{r}}_{i}-\hat{\mathbf{r}}_{j}|},$
(2)

and the atom-field interaction part, $\hat{H}_{int}$, resulting from including
the kinetic momentum,
$\hat{\mathbf{P}}_{i}=\hat{\mathbf{p}}_{i}+e\hat{\mathbf{A}}(\hat{\mathbf{r}}_{i},t)$,
is

$\begin{split}\hat{H}_{int}=\sum_{i=1}^{N}\Bigl{(}\frac{e}{2m_{e}}[\hat{\mathbf{p}}_{i}\cdot\mathbf{A}(\hat{\mathbf{r}}_{i},t)+\mathbf{A}(\hat{\mathbf{r}}_{i},t)\cdot\hat{\mathbf{p}}_{i}]\\\
+\frac{e^{2}\mathbf{A}^{2}(\hat{\mathbf{r}}_{i},t)}{2m_{e}}\Bigr{)}.\end{split}$
(3)

Here, $e$ is the magnitude of the electron charge, $m_{e}$ is the electron
mass, and $\hat{\mathbf{r}}_{i}$ are the position operators of the electrons.
The ${\bf{A}}^{2}$-term in Eq. 3 describes the ponderomotive interaction,
which can be employed to realize ponderomotive optical lattices for Rydberg
atoms Dutta  _et al._ (2000); Anderson  _et al._ (2011). In the present work,
we focus on the first term, which describes, among other phenomena, Rydberg-
atom microwave transitions.

We consider an alkali atom with a single electron excited to Rydberg state.
The sum in Eq. 3 can be dropped, and the interactions of the Rydberg electron
with the core electrons can be compounded into a set of $\ell$-specific model
potentials Marinescu  _et al._ (1994). Adopting the Coulomb gauge,
$\nabla\cdot\mathbf{A}=0$, in which $\hat{\mathbf{p}}$ and
$\mathbf{A}(\hat{\mathbf{r}}_{i})$ commute, and also dropping the
$\mathbf{A}^{2}$-term, which is irrelevant for the low-field transitions
considered here, Eq. 3 can be written as:

$\hat{H}_{int}=\frac{e}{m_{e}}\mathbf{A}(\hat{\mathbf{r}})\cdot\hat{\mathbf{p}}.$
(4)

We then express the electric-field-normalized atom-field interaction matrix
element between initial, $\left|i\right\rangle$, and final,
$\left|f\right\rangle$, states of the Rydberg atom, in the rotating-wave
approximation, as:

$\displaystyle M_{V}=\frac{\left\langle
f\right|\hat{H}_{int}\left|i\right\rangle}{(E_{0}/2)}=-\frac{e\hbar}{{\rm{i}}m_{e}\omega}(\hat{\mathbf{n}}\cdot\int\psi_{f}^{*}e^{{\rm{i}}\mathbf{k}\cdot\mathbf{r}}\nabla\psi_{i}d^{3}r)\quad.$

Making the electric-dipole approximation (EDA), which is valid for microwave
transitions of Rydberg atoms and is addressed further below,

$M_{V}=-\frac{e\hbar}{{\rm{i}}m_{e}\omega}(\hat{\mathbf{n}}\cdot\int\psi_{f}^{*}~{}\nabla\psi_{i}~{}d^{3}r),$
(5)

where $\hat{\mathbf{n}}$ is the electric field’s polarization vector. The
electric-dipole transition matrix element (EDTM) in Eq. 5 is commonly referred
to as the velocity(VG)-form, as it essentially involves the linear-momentum
operator. In order to transform it into the often-used length-gauge(LG) form,
one invokes the commutation relation
$[\hat{\mathbf{r}},\hat{H}_{0}]=\frac{i\hbar}{m_{e}}\hat{\mathbf{p}}$, which
is exact only if $\hat{H}_{0}$ does not contain momentum-dependent potentials
Bethe and Salpeter (2012). For the case of an on-resonant interaction,
$\omega=|E_{f}-E_{i}|/\hbar$ with final- and initial-state energies $E_{f}$
and $E_{i}$, respectively, the EDTM takes the commonly employed LG form,

$M_{L}=e(\hat{\mathbf{n}}\cdot\int\psi_{f}^{*}~{}\mathbf{r}~{}\psi_{i}~{}d^{3}r).$
(6)

For our third, the acceleration-gauge (AG) form, to be valid, the valence-
electron potential in $\hat{H}_{0}$ must be of pure Coulomb form,
$\hat{V}=-Ze^{2}/(4\pi\epsilon_{0}\hat{r})$, such that the commutation
relation $[\hat{\mathbf{p}},\hat{H}_{0}]=-i\hbar\nabla\hat{V}$ is true Bethe
and Salpeter (2012). Then, the EDTM can be expressed in AG form,

$M_{A}=-\frac{Ze^{3}}{m_{e}\omega^{2}}(\hat{\mathbf{n}}\cdot\int\psi_{f}^{*}~{}\frac{\hat{\mathbf{r}}}{4\pi\epsilon_{0}r^{3}}~{}\psi_{i}~{}d^{3}r).$
(7)

Note that the EDTMs in Eqs. 5-7 are in SI units. In order to convert them to
atomic units used in the next Section, one needs to divide by $ea_{0}$, where
$a_{0}$ is the Bohr radius. (Alternately, set
$\hbar=m_{e}=Z=e=4\pi\epsilon_{0}=1$ and convert $\omega$ from SI into atomic
units.)

The VG is the most general form, whereas for the LG and AG to be applicable
specific requirements must be satisfied. These always hold in the non-
relativistic hydrogen atom. However, the alkali model potentials Marinescu
_et al._ (1994), which we use here, are $\ell$-dependent for $\ell\leq 3$,
which makes the LG inaccurate at these low $\ell$-values. Rydberg atoms with a
polarizable ionic core, such as Rb and Cs, exhibit a long-range, non-Coulombic
core polarization potential, the leading term of which scales as
$-\alpha_{d}/(2r^{4})$. There, $\alpha_{d}$ is the dipolar core
polarizability. Due to the non-Coulombic perturbation, the AG is inaccurate at
most $\ell$-values. It is an objective of our work to quantify the deviations
of EDTMs computed in the LG and the marginally applicable AG forms from those
computed in the fundamentally accurate VG form.

It is noteworthy that the EDA must be applied to the VG, the fundamental gauge
that follows directly from the ${\bf{A}}\cdot{\bf{p}}$-part of the minimal-
coupling Hamiltonian in Eq. 4, before the matrix element can be transformed
into the LG and AG forms (under the applicable respective conditions). For the
microwave transitions considered in the present work, the EDA is naturally
satisfied due to the long radiation wavelength (${\bf{k}}\cdot{\bf{r}}\ll 1$).
It has been shown elsewhere that the EDA even applies to electric Rydberg-atom
couplings in the UV wavelength range Cardman  _et al._ (2021), where the
Rydberg-atom size typically exceeds the wavelength
(${\bf{k}}\cdot{\bf{r}}>1$).

In the numerical results presented in Sec. III, we show comparisons between
the EDTMs in the three gauges for several selected angular-momentum channels.
All EDTMs shown are radial parts; matrix elements for transitions with
specific magnetic quantum numbers follow from multiplication with angular
matrix elements provided, for instance, in Bethe and Salpeter (2012).

The fine structure of the Rydberg levels is ignored in our analysis, as it
would primarily only add Clebsch-Gordon factors to the angular matrix
elements. The gauge effects we focus on in our work result from the
$\ell$-dependence and the non-Coulombic long-range terms in the model
potentials Marinescu  _et al._ (1994) used to compute the radial wave-
functions of the $(n,\ell)$-states of the atom. To capture gauge effects, it
is therefore sufficient to consider radial EDTMs with fine-structure-averaged
quantum defects Gallagher (2005) for the Rydberg energy levels. As the model
potentials vary if $\ell\in[0,3]$ and are identical if $\ell\geq 3$, for
$\ell_{<}\leq 2$ we expect to find errors of both the LG and AG results
relative to the VG result. The latter is considered correct within our model.
For $\ell_{<}\geq 3$ we still expect to find errors of the AG result due to
the non-Coulombic, long-range ion-core polarization potential.

For the numerical calculation of the required wave-functions, we use a method
described in Reinhard  _et al._ (2007) that allows a non-uniform spatial grid.
The spatial finesse is essentially characterized by a specifiable number of
grid points per local de-Broglie wavelength of the electron wave-function
(which varies widely within the atomic potential). For the bulk of our
calculations, we chose a lower limit of 5,000 grid points per electron
wavelength. To estimate numerical uncertainty, several computation series were
performed on finer grids with a minimum of 10,000 grid points per electron
wavelength. All computations were performed in Fortran using REAL*16
precision. For reference, less than 500 grid points per electron wavelength,
in a REAL*8 implementation, suffice to compute EDTMs of strong bound-bound
Rydberg transitions with three to four significant digits.

##  III Results

We have computed the EDTMs in all three gauges for all electric-dipole-allowed
Rydberg-Rydberg transitions $n_{i}\rightarrow n_{f}$, with initial, $n_{i}$,
and final, $n_{f}$, principal quantum numbers covering the full range from 15
to 100, and for all combinations of initial and final angular momenta,
$\ell_{i}$ and $\ell_{f}$, that satisfy the electric-dipole selection rule
$\Delta\ell=\ell_{f}-\ell_{i}=\pm 1$. The results are stored in data banks
$M_{V}(n_{i},\ell_{i},\Delta n,\Delta\ell)$, $M_{L}(n_{i},\ell_{i},\Delta
n,\Delta\ell)$, and $M_{A}(n_{i},\ell_{i},\Delta n,\Delta\ell)$, with $\Delta
n$ defined as $\Delta n=n_{f}-n_{i}$.

As a convenient measure for the differences between the radial EDTMs in the
three gauges, we define the relative gauge error

$\displaystyle\delta_{i,V}=\left|1-\frac{M_{i}}{M_{V}}\right|,i=L,A,$

where the subscripts V, L and A stand for VG, LG and AG gauges, respectively.

FIG. 2: (Color online) Comparison of EDTMs calculated in LG vs VG using Eqs. 5
and 6 for the indicated $n_{i}S\leftrightarrow n_{f}P$ series (a),
$n_{i}D\leftrightarrow n_{f}P$ series (b), and $n_{i}D\leftrightarrow n_{f}F$
series (c).The principal quantum numbers $n$ range between 15 (left margin, in
most series), and 100 (right margin, in most series).

In Fig. 2, we show $\delta_{L,V}$ vs $|M_{V}|$ on double-log scales for
several transition series with $\ell_{<}\leq 2$. We consider transitions with
$|\Delta n|\leq 4$, which include the series with the largest EDTMs (that are
relevant in numerous applications). It is first noted that each series falls
on a well-defined line characteristic to the series. All characteristic lines
have similar slopes, with the distributions of data points
$(|M_{V}|,\delta_{L,V})$ along those lines showing some irregularity for weak
transitions (transitions with small $|M_{V}|$).

Rb $n_{i}S_{1/2}$ $\leftrightarrow$ $n_{f}P_{J}$ microwave transitions are of
some importance because the $n_{i}S_{1/2}$ Rydberg atoms can be initialized by
two-photon laser excitation from the Rb $5S_{1/2}$ ground state. As seen in
Fig. 2(a), for the strongest transitions, $nS\leftrightarrow nP$ and
$nS\leftrightarrow(n-1)P$, $\delta_{L,V}$ typically ranges between $10^{-4}$
and $10^{-5}$, while for the weaker transitions with larger $|\Delta n|$ this
number can be as large as $\sim 2\times 10^{-3}$. Further, from the observed
approximate scaling $\delta_{L,V}\propto 1/|M_{V}|$ it follows that
$\Delta_{L,V}:=M_{L}-M_{V}$ tends to be a constant for each series. For the
strongest series, $nS\leftrightarrow nP$ and $nS\leftrightarrow(n-1)P$, it is
$\Delta_{L,V}\approx-0.062$ ea0 and $-0.058$ ea0, respectively. For the weaker
series, the approximately constant values of $\Delta_{L,V}$ generally drop.
For instance, for the $nS\leftrightarrow(n+4)P$ series it is
$\Delta_{L,V}\approx-0.0066$ ea0.

Rb $n_{i}D$ $\leftrightarrow$ $n_{f}P$ microwave transitions are often used
because of the large values of $|M_{V}|$ on the $nD$ to $(n+1)P$ series, and
because the two-photon laser preparation rates of $n_{i}D$ Rydberg atoms from
the Rb $5S_{1/2}$ ground state are greater than those of $n_{i}S$ Rydberg
atoms. It is seen in Fig. 2(b) that these transitions have
$\delta_{L,V}$-values on the order of three times larger than those of
comparable $n_{i}S$ $\leftrightarrow$ $n_{f}P$ transitions, with numerous
instances of $\delta_{L,V}>10^{-2}$. On the $n_{i}D\leftrightarrow n_{f}P$
series, the $\delta_{L,V}$ are much larger and the $M_{V}$ much smaller for
$\Delta n\leq 0$ than for $\Delta n>0$. The strongest series,
$nD\leftrightarrow(n+1)P$, has $M_{L}-M_{V}\approx-0.34$ ea0. Interestingly,
the $nD\leftrightarrow nP$ series is very weak and exhibits a peculiar
behavior at small $|M_{V}|$. We attribute the unusual features of the
$nD\leftrightarrow nP$ series to the exact level energies and wave-functions
of the involved states, which depend on the Rb quantum defects of both initial
and final levels. In fact, further check revealed that this series exhibits
what may be characterized as a bound-bound analog of a Cooper minimum Cooper
(1962); Viray  _et al._ (2021). (A Cooper minimum Cooper (1962) is a class of
EDTM zeros at certain quantum-defect-dependent photo-ionization energies.) In
the present case, the $nD\leftrightarrow nP$ EDTMs pass through a pronounced
minimum of $|M_{V}|\sim 0.16$ ea0 at $n=20$ (off-scale in Fig. 2(b)), which is
about three orders of magnitude lower than the $|M_{V}|$-value of the nearby
$20D\leftrightarrow 21P$ transition (which has $|M_{V}|=472$ ea0). In Fig.
2(b), this results in an irregularity of the distribution of data points on
the $nD\leftrightarrow nP$ series at low $|M_{V}|$. Since the relation
$\delta_{L,V}\propto 1/|M_{V}|$ still holds, $\delta_{L,V}$ reaches as high as
$\sim 0.5$ for the $20D\leftrightarrow 20P$ transition. Further examination
reveals another unusual series, the $nD\leftrightarrow(n-1)P$ series. We found
that in that case the EDTMs pass through a shallow maximum of $|M_{V}|$ of
only $\approx 7$ ea0 at $n\sim 65$. This maximum results in a “bunching” of
data points in Fig. 2(b) at the right margin of the $nD\leftrightarrow(n-1)P$
series.

We further show several $n_{i}D\leftrightarrow n_{f}F$ series in Fig. 2(c).
Figures 2(b) and 2(c) have some resemblance, which one may attribute to
similar differences in the quantum defects of the respective involved states.
In the $n_{i}D\leftrightarrow n_{f}F$ series, the $\delta_{L,V}$ are much
larger and the $M_{V}$ much smaller for $\Delta n\geq 0$ than for $\Delta
n<0$. Investigating further, we find that the $\Delta n\geq 0$ cases all show
a Cooper-minimum-like behavior, similar to the $nD\leftrightarrow nP$ series
in Fig. 2(b). In addition, the EDTMs of the $n_{i}D\leftrightarrow n_{f}F$
series with $\Delta n>0$ pass through maxima at series-dependent
$n_{i}$-values located below the respective “Cooper-minima”. These effects
result in irregular appearances of the distribution of data points on the
$n_{i}D\leftrightarrow n_{f}F$ series with $\Delta n\geq 0$ in Fig. 2(c).

For $\ell\geq 3$, the utilized model potentials become $\ell$-independent, and
numerical readings of $\delta_{L,V}\neq 0$ are only due to numerical error.
For instance, for $n_{i}F\leftrightarrow n_{f}G$ transitions (not shown here)
we find $\delta_{L,V}$-values peaking near $10^{-6}$. This indicates a
relative numerical confidence level of the EDTMs below $\sim 10^{-5}$. This
was verified by comparing several results from Fig. 2 with similar results
obtained with a finer spatial grid in the wave-function calculations. This
estimation of the numerical error is important because it shows that the
deviations between $M_{L}$ and $M_{V}$, discussed in Figs. 2(a)-(c) and in the
corresponding text passages above, are indeed due to the different gauge forms
used.

Finally, we show a comparison between EDTMs computed in AG and VG. This
comparison is of some fundamental interest but not relevant in applications of
EDTMs. For low-$\ell$ states, $\delta_{A,V}$ can reach very large values (up
to $\sim 10^{6}$, not shown here), reflecting the basic unsuitability of the
AG for calculating EDTMs in non-Coulombic potentials. However, if $\ell$
becomes sufficiently large, the AG results tend to gradually approach the
EDTMs calculated in VG and LG. As an example, in Fig. 3 we show results for
several series for transitions between $\ell=10$ and $\ell=11$ states. On the
series shown, the AG’s gauge error is small but it still exceeds the numerical
error by several orders of magnitude. It is noteworthy that there is no
apparent (simple) relationship between $\delta_{A,V}$ and $|M_{V}|$, and that
$\delta_{A,V}$ increases with $|M_{V}|$. Both of these observations
qualitatively differ from Fig. 2. In additional analyses, not presented, it is
seen that $\delta_{A,V}$ drops below our numerical error for $\ell\gtrsim 20$
(with a caveat discussed in Sec. IV). This shows that the always-present error
of the AG that is caused by the long-range ion-core polarization potential,
$-\alpha_{d}/(2r^{4})$, drops below our present numerical error when
$\ell\gtrsim 20$.

FIG. 3: (Color online) Comparison of EDTMs calculated in AG vs VG using Eqs. 5
and 7, respectively, for the indicated transition series with $\ell_{i/f}=10$
or 11, for which gauge errors in the AG calculation are still significant. The
principal quantum number $n$ ranges between 15 (left) and 100 (right margins).

##  IV Discussion

In Sec. III we have established and quantified deviations between LG- and VG-
computations of EDTMs for transitions that involve Rydberg levels with
$\ell_{<}\leq 2$. These deviations are caused by the dependence of the
utilized model potentials Marinescu  _et al._ (1994) on $\ell$ for $\ell\leq
3$. For transitions with $\ell_{<}\geq 3$, the model potentials are identical,
and the EDTMs in LG and VG agree within our numerical error of $\delta_{L,V}$
of $<10^{-5}$. The gauge error of the AG, caused by the long-range, non-
Coulombic core polarization potential $-\alpha_{d}/(2r^{4})$, rapidly drops
with increasing $\ell_{<}$ and becomes undetectable for $\ell_{<}\gtrsim 20$.
Due to the ubiquitous use of low-$\ell$ Rb Rydberg states in science and
technology, the following discussion is focused on aspects of the gauge errors
of the LG pertaining to low-$\ell$ states.

For the strongest transitions within the $\ell_{<}\leq 2$-series displayed in
Fig. 2, the choice of the gauge form does not play an overwhelming role, as
$\delta_{L,V}$ \- the relative EDTM error incurred by adopting the LG instead
of the VG - barely exceeds $10^{-3}$ even in the worst cases. Picking
$n_{i}=40$ as a specific example, we find $\delta_{L,V}=4.0\times 10^{-5}$ for
the strongest $S\leftrightarrow P$ transitions, $\delta_{L,V}=1.7\times
10^{-4}$ for the $40D\leftrightarrow 41P$ transition, and
$\delta_{L,V}=3.6\times 10^{-5}$ for the $40D\leftrightarrow 39F$ transition.
In high-precision spectroscopy, it would be fairly challenging to probe Rabi-
frequency discrepancies at that level of precision (and to even calibrate RF
electric fields with sufficient accuracy to enable such measurements).
However, the weak $n_{i}P\leftrightarrow n_{f}D$ series and several of the
$n_{i}D\leftrightarrow n_{f}F$ series have $\delta_{L,V}$-values in the
1$\%$-and-above range and exhibit several unusual behaviors, including Cooper-
minimum-like features. The properties of these weak series may offer
opportunities to explore gauge effects in Rydberg electric-dipole transitions
for fundamental investigations. The same holds for photo-ionization cross
sections, which generally show larger differences between LG and VG Viray  _et
al._ (2021); Cardman  _et al._ (2021) than bound-bound EDTMs.

It is abundantly evident in Fig. 2 that all data points for any given series
that involves $\ell$-dependent model potentials fall on approximately straight
lines, regardless of any added complexity such as Cooper-like minima in
$|M_{V}|$. Fitted slopes are $\approx-1$ for all three types of transitions in
Fig. 2. This observation implies that the absolute difference in EDTMs,
$\Delta_{L,V}=M_{L}-M_{V}$, remains constant over a wide range of $n$. In all
series presented in Fig. 2, the relative variation of $\Delta_{L,V}$ across
the entire series is less than 2$\%$. Further investigation might be needed to
explain this finding. The value of $\Delta_{L,V}$ is largest for the strongest
series, $nD\leftrightarrow(n+1)P$, at $\Delta_{L,V}=-0.34$ ea0.

The $1/\omega^{2}$-dependence of $M_{A}$ (see Eq. 7) and the
$1/\omega$-dependence of $M_{V}$ (see Eq. 5) cause an issue of numerical
accuracy for near-degenerate transitions. The integral expressions in the
respective equations can drop to near or even below the numerical error. As a
result, for near-degenerate transitions the expressions in Eqs. 5 and 7 can
become numerically unstable and lead to non-physical results. (This is why in
Fig. 3 we have excluded the result for $\Delta n=0$.) Since the LG (Eq. 6) has
no equivalent numerical problem, the frequency-denominator issue is of no
practical concern. For $\ell_{<}\leq 2$, where the VG must be used, there are
no near-degenerate transitions, so the issue does not arise, while for
$\ell_{<}\geq 3$, where near-degenerate transitions occur when $\Delta n=0$,
the LG may be used.

Following the discussion, we are using the method summarized in Fig. 1 to pick
EDMTs for any electric-dipole transition of interest. First, we compute
databases $M_{V}(n_{i},l_{i},\Delta n,\Delta_{\ell})$,
$M_{L}(n_{i},l_{i},\Delta n,\Delta_{\ell})$ and $M_{A}(n_{i},l_{i},\Delta
n,\Delta\ell)$. (The database $M_{A}$ is for added insight only.) After
entering the transition labels $(n_{i},l_{i},\Delta n,\Delta\ell)$, the
following simple rules are applied:

  * •

If $\ell_{<}\leq 2$, pick $M_{V}(n_{i},l_{i},\Delta n,\Delta\ell)$.

  * •

If $\ell_{<}\geq 3$ and $|\Delta n|\geq 1$, use the average of
$M_{V}(n_{i},l_{i},\Delta n,\Delta\ell)$ and $M_{L}(n_{i},l_{i},\Delta
n,\Delta\ell)$.

  * •

If $\ell_{<}\geq 3$ and $|\Delta n|=0$, pick $M_{L}(n_{i},l_{i},\Delta
n=0,\Delta\ell)$.

##  V Conclusion

We have presented results for EDTMs calculated in three different gauge forms:
velocity, length and acceleration. Based on the analysis of the observed
differences between these forms, we outlined a method for choosing the most
appropriate gauge, depending on the quantum numbers of the transition. We have
discussed aspects of the underlying physics. In the computations we have used
model potentials for Rb from Marinescu  _et al._ (1994). Analogous
computations could be performed for cesium or other alkali atoms for which
there is a set of model potentials.

The observed differences between the forms have negligible or only minor
consequences in applications of Rydberg atoms for microwave field sensing
Sedlacek  _et al._ (2012); Holloway  _et al._ (2014); Anderson  _et al._
(2021, 2020), where strong transitions are used. For these, the relative
effect of gauge on the EDTMs, $\delta_{L,V}$, remains below $\sim 10^{-3}$.
Gauge effects with a significance $>1\%$ may be detectable in weak Rydberg-
Rydberg transitions, which one may study to experimentally explore gauge
effects.

It is finally noted that the DC Stark effect occurs at $\omega=0$, while the
present work deals with resonant transitions at frequencies that are typically
in the radio-frequency and microwave ranges. A limiting case of
$\omega\rightarrow 0$, covered in our work, occurs for near-degenerate $\Delta
n=0$ transitions between high-$\ell$ states. The limit $\omega\rightarrow 0$
requires caution because of the rise of Bloch-Siegert shifts caused by the
counter-rotating terms Berman and Malinovsky (2010), which we have dropped
early-on when making the rotating-wave approximation, and because of other
complications. The limit $\omega\rightarrow 0$ may be treated better as a
quasi-static DC Stark effect in a slowly-varying electric field. In that
approach, no resonance condition is assumed and the static atom-field
interaction operator $e{\bf{E}}\cdot\hat{\bf{r}}$ applies, for which the
length-gauge EDTMs are exact. Hence, the (quasi-static) DC quadratic Stark
effect of low-$\ell$ atomic states requires the length-gauge form of the
EDTMs, (even at low $\ell$), whereas interactions with microwave electric
fields involving the same low-$\ell$ states require the velocity-gauge form.
This difference may become relevant in precision measurements of resonant AC
versus DC electric fields as well as applications of Rydberg atoms in defining
voltage standards Holloway  _et al._ (2021).

## Acknowledgments

We thank Ryan Cardman, Bineet Kumar Dash, and Dr. David Anderson for useful
discussions. This work was supported by the NSF Grant No. PHY-2110049.

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