# EPIC-Survival: End-to-end Part Inferred Clustering for Survival Analysis,
Featuring Prognostic Stratification Boosting

Hassan Muhammad  Department of Physiology, Biophysics, and Systems Biology -
Weill Cornell Medicine  Chensu Xie  Carlie S. Sigel  Department of Surgery -
Memorial Sloan Kettering Cancer Center  Amber Simpson  Dept. of Biomedical and
Molecular Sciences - Queen’s University  Michael Doukas  Department of
Pathology - Erasmus Medical Center Rotterdam  Lindsay Alpert  Department of
Anatomic Pathology - University of Chicago  William R. Jarnagin  Department of
Surgery - Memorial Sloan Kettering Cancer Center  Thomas J. Fuchs  Department
of Physiology, Biophysics, and Systems Biology - Weill Cornell Medicine
Department of Pathology - Hasso Plattner Institute for Digital Health at Mount
Sinai

###### Abstract

Histopathology-based survival modelling has two major hurdles. Firstly, a
well-performing survival model has minimal clinical application if it does not
contribute to the stratification of a cancer patient cohort into different
risk groups, preferably driven by histologic morphologies. In the clinical
setting, individuals are not given specific prognostic predictions, but are
rather predicted to lie within a risk group which has a general survival
trend. Thus, It is imperative that a survival model produces well-stratified
risk groups. Secondly, until now, survival modelling was done in a two-stage
approach (encoding and aggregation). The massive amount of pixels in digitized
whole slide images were never utilized to their fullest extent due to
technological constraints on data processing, forcing decoupled learning.
EPIC-Survival bridges encoding and aggregation into an end-to-end survival
modelling approach, while introducing stratification boosting to encourage the
model to not only optimize ranking, but also to discriminate between risk
groups. In this study we show that EPIC-Survival performs better than other
approaches in modelling intrahepatic cholangiocarcinoma, a historically
difficult cancer to model. Further, we show that stratification boosting
improves further improves model performance, resulting in a concordance-index
of 0.880 on a held-out test set. Finally, we were able to identify specific
histologic differences, not commonly sought out in ICC, between low and high
risk groups.

_K_ eywords computational pathology $\cdot$ deep learning $\cdot$ clustering
$\cdot$ disease staging $\cdot$ survival analysis

##  1 Introduction

One of the primary purposes of survival analysis in medicine is cancer
subtyping, an important tool used to help predict disease prognosis and direct
therapy—it is the functional application of a survival model in the clinical
setting. Though traditional methods used for discovering cancer subtypes are
extremely labor intensive and subjective, successful stratification of common
cancers, such as prostate, into effective subtypes has only been possible due
to the existence of large datasets. However, working with rare cancers poses
it’s own set of challenges. Under the current largely privatized global
medical infrastructure, access to large datasets is nearly impossible, wide
collaboration is limited, and the traditional human evaluation of recognizing
repeatable tissue morphologies is rendered difficult. Further, histologic
features are limited to the discretion of the manual observer’s past
experiences and subjectivity. EPIC-Survival offers a way forward for subtyping
rare cancers as a unique deep learning-based survival model which overcomes
two key barriers.

Firstly, it is difficult to computationally predict the specific outcome of a
patient. It is more reasonable to predict the subgroup of a cancer population
in which an individual patient falls into. Further, without a robust
prognostic model which learns the relationships between histology and patient
outcome, survival models have minimal use. Thus, it is important that a
survival model produces stratified groups, preferably driven by histology,
rather than simply performing well at ranking patients by risk. Regardless,
survival modelling based on whole slide image (WSI) histopathology is a
difficult task which requires overcoming a second problem.

Because a single digitized WSI can span billions of pixels, it is impossible
to directly use WSIs in full to train survival models, given current
technological constraints. Thus, it is a common technique to sample tiles from
WSIs, often in creative ways, and then aggregating them to represent their
respective WSIs in the final step of training. We can simplify these stages as
the tile _encoding_ stage and the _aggregation_ stages. While the aggregation
stage of survival modelling has historically defaulted to the Cox-proportional
Hazard regression model, recent advancements have made survival modelling more
robust to complex data. We highlight some examples in the next section.
Nevertheless, creative ways to extract features from WSIs and more advanced
techniques to aggregate them still face the limits of operating in detached
two-stage frameworks, in which the information at slide level, e.g. the given
patient prognosis, is never taken into consideration while learning tile
encoding by proxy tasks (cf. Figure 1). This creates a difficulty in being
able to confidently identify specific and direct relationships between tissue
morphology and patient prognosis, even though prognostic performance may be
strong.

In this paper, we introduce a deep convolutional neural network which utilizes
end-to-end training to directly produce survival risk scores for a given WSI
without limitations on image size. Further, we contribute a new loss function
called _stratification boosting_ which further strengthens risk group
separation and overall prognostic performance. Our introduction of
stratification boosting not only improves overall performance, but also forces
the model to identify risk groups. In contrast, other works attempt to find
groups in the distribution of ranking after modelling a dataset. We claim that
this model takes us one step closer to systematically mapping out the
relationships between tissue morphology and patient death or cancer recurrence
times. To challenge our method, we consider the difficult case of small
dataset rare cancers.

###  1.1 Intrahepatic Cholangiocarcinoma

Intrahepatic cholangiocarcinoma (ICC), a cancer of the bile duct, has an
incidence of approximately 1 in 160,000 in the United States [12]. In general,
the clinical standard for prognostic prediction and risk-based population
stratification relies on simple metrics which are not based on histopathology.
These methods have unreliable prognostic performances [3], even when studied
in relatively large cohorts (1000+ samples). Studies which have attempted to
stratify ICC into different risk groups based on histopathology have been
inconsistent and unsuccessful [2, 11, 13].

Figure 1: While other deep learning-based survival modelling approaches employ
a traditional "two-stage" approach, EPIC-Survival introduces end-to-end
learning for prognostic prediction, allowing for a more robust loss function
which encourages the model to learn subgroups within the patient population.

###  1.2 Related Works

Because survival analysis continues to operate in a two-stage approach as
outlined above, advancements in survival analysis largely lie in the feature
extraction front. Muhammad et. al. introduced a deep unsupervised clustering
autoencoder which stratified a limited set of tiles randomly sampled from WSIs
into groups based on visual features at high resolution. These clusters were
then visualized and used as covariates to train simple univariate and
multivariate CPH models [10]. Similarly, in another study by Abbet et. al.,
self-supervised clustering was used to produce subtypes based on histologic
features [1]. These were then visualized and used as covariates in survival
models to measure significance of the clustered morphologies. Zhu et al. takes
the clustering approach one step further by modeling local clusters for a
tile-level prediction before aggregating the results into slide-level survival
predictions [17]. These methods work to build visual dictionaries through
clustering without having direct association to survival data. Slightly
differently, Yao et el. developed a method [16] to build a visual dictionary
through multiple instance learning. Though not completely unsupervised, even
weak supervision can only operate with a decoupled survival regression. Other
studies such as [14, 4, 7] have used even simpler approaches, producing models
which learn to predict prognosis on tiles based on slide-level outcomes and
then aggregate them into a slide-level predictions. These models, however, do
utilize the DeepSurv [8] function, a neural-network based survival learning
loss robust to complex and non-linear data (discussed further in section 2.2).
Unfortunately, the simplified feature extraction methods of the works listed
do not allow the DeepSurv model to operate in its fullest potential—our method
overcomes this barrier.

Recently, Xie et. al. bridged the gap of the two-stage problem in WSI
classification tasks with the introduction of End-to-end Part Learning (EPL)
[15]. EPL maps tiles of each WSI to $k$ feature groups defined as parts. The
tile encoding and aggregation are learned together against slide label in an
end-to-end manner. Refer to [15] for more details. Although the authors
suggested that EPL is theoretically applicable to survival regression,
treatment recommendation, or other learnable WSI label predictions, the effort
has been limited to testing the EPL framework with experiments benchmarking
against classification datasets. In this study, we introduce EPIC-Survival to
extend the EPL method to survival analysis by integrating the DeepSurv
survival function, unencumbered by the limitations of two-stage training.
Moreover, we contribute a new concept called stratification boosting, which
acts as a critical loss term to the learning of distinct risk groups among the
patient cohort. Most importantly, by applying EPIC-Survival, we show that it
is capable of discovering new relationships between histology and prognosis.

##  2 Methods

###  2.1 Survival Modelling

To review, survival modelling is used to predict ranking of censored time-
duration data. A sample is defined as censored when the end-point of its given
time duration, or time-to-event, is not directly associated to the study. For
example, in a dataset of time-to-death by cause of cancer, not all samples
will have end-points associated with a cancer-related death. In some cases, an
end-point may indicate a patient dropping out of the study or dying of other
causes. Rather than filtering out censored samples and regressing only on
uncensored time-to-events, Cox-propotional hazard (CPH) models are used to
regress on a complete dataset and predict hazard, the instantaneous risk that
the event of interest occurs. CPH as defined as:

$H(t)=h_{o}e^{b_{i}x_{i}},$ (1)

where $H(t)$ is the hazard function dependent on time $t$, $h_{o}$ is a
baseline hazard, and covariate(s) $x_{i}$ are weighted by coefficient(s)
$b_{i}$.

In 2016, DeepSurv [8] made an advancement in survival modelling by using a
neural network to regress survival data based on theoretical work proposed in
1995 [5]. Their results showed better performance than the typical CPH model,
especially on more complex data. In the case of a neural network-based
survival function, $b_{i}$ is substituted for model parameters, $\theta_{i}$.
Traditionally, a negative log partial likelihood (NLPL) is used to optimize
the survival function. It is defined as:

$NLPL(\theta)=-\sum_{i:E_{i}=1}(h_{\theta}(x_{i})-log\sum_{j\in\Re(T_{i})}e^{h_{\theta}(x_{j})}),$
(2)

where $h_{\theta}(x_{i})$ is the output risk score for sample $i$,
$h_{\theta}(x_{j})$ is a risk score from ordered set $\Re(T_{i})={i:T_{i}\geq
t}$ of patients still at risk of failure at time $t$, and $i:E_{i}=1$ is the
set of samples with an observed event (uncensored). The performance of a CPH
or CPH-based model can be tested using a concordance index (CI) which compares
the ranking of predicted risks to associated time-to-events. A CI of 0.5
indicates randomness and a CI of 1.0 indicates perfect prognostic predictions.

Further, the Kaplan-Meier (KM) method can be used to estimate a survival
function, the probability of survival past time $t$, allowing for an
illustrative way to see prognostic stratification between two or more groups.
The survival function is defined as:

$S(t)=\prod_{t_{i}<t}\frac{n_{i}-d_{i}}{n_{n}},$ (3)

where $d_{i}$ are the number of observed events at time $t$ and $n_{i}$ are
the number of subjects at risk of death or recurrence prior to time $t$. The
Log-Rank Test (LRT) is used to measure significance of separation between two
survival functions modelled using KM. LRT is a special case of the chi-squared
test used to test the null hypothesis that there is no difference between the
$S(t)$ of two populations.

###  2.2 EPIC Survival

EPIC-Survival bridges the DeepSurv loss with the comprehensive framework of
EPL. In short, EPL assigns tiles to inferred histology parts and
backpropagates the loss against slide labels (time-to-event data) through the
integrated aggregation and encoding graph. For EPIC-Survival, the last fully
connected layer of the original EPL was replaced by a a series of fully
connected layers and a single output node which functions as a risk score for
a given input WSI. Similar to the traditional EPL, NLPL is combined with a
clustering function based on minimizing distances between a sample embedding
and its assigned centroid:

$Loss=NLPL(\theta)+\lambda_{c}\sum_{i=1}^{N}||z_{i}-c_{i}||^{2},$ (4)

where $z_{i}$ is the embedding of randomly sampled tiles, $c_{i}$ is the
centroid assigned during previous training epoch to the WSI from which $z_{i}$
is sampled , and $\lambda$ is a weighting parameter. Figure 2 helps visualize
this combined loss function and the process of slide-level and global
clustering of visual morphology.

Figure 2: Diagram of the proposed EPIC-Survival approach for prognosis
prediction. Top: Whole slide images are tiled into small patches which pass
through an ImageNet pretrained ResNet-34 backbone, outputting a tile feature
vector. Each vector is assigned to a histology feature group defined by global
centroids. Next, local slide-level centroids are calculated and the nearest
tiles to $k$ local centroids are used as part representations of the slide.
This process is repeated for all slides. Bottom: Still within the same
training epoch, parts of all slides are concatenated and trained with survival
data, in conjunction with optimizing local clustering and overall risk group
separation. Note: Global centroids are randomnly initialized before training
and updated between epochs, based on the optimization of the ResNet-34
backbone.

###  2.3 Stratification Boosting

While CPH and DeepSurv regressions serve to optimize the ranking of samples in
relation to time-to-event data, they do not actively form risk groups within a
dataset. In Mayr and Schmid’s work on CI-based learning, they conclude that
"specifically, prediction rules that are well calibrated do not necessarily
have a high discriminatory power (and vice versa)" [9]. One of the most
important applications of survival analysis is cancer subtyping, an important
tool used to help predict disease prognosis and direct therapy. Moreover,
subtyping based on survival analysis creates a functional use for the survival
model, especially if specific morphologies can be identified within each
prognostic group. The DeepSurv loss, which only optimizes ranking, does not
explicitly put a lower bound to the separation between the predicted risks. To
further improve prognostic separation between high and low risk groups in the
patient population, we extend the DeepSurv-EPL function with a stratification
loss term. During training, predicted risks are numerically ordered and
divided into two groups based on the median predicted risk. The mean is
calculated for each group of predicted risks ($R_{high}$ and $R_{low}$) and
the model is optimized to diverge the two values using Huber loss:

$smoothL_{1}(\frac{1}{1+\lvert R_{high}-R_{low}\lvert},0)$ (5)

###  2.4 Dataset

WSIs of ICC cases were obtained from Memorial SLoan Kettering Cancer Center
(MSKCC), Erasmus Medical Center-Rotterdam (EMC), and University of Chicago
(UC) with approval from each respective Institutional Review Boards. In total,
265 patients with resected ICC without neoadjuvant chemotherapy were included
in the analysis. Up-to-date retrospective data for recurrence free survival
after resection was also obtained. A subset of samples (n=157) from MSKCC]
were classified into their respective AJCC [6] TNM and P-Stage groups. 246
slides from MSKCC and EMC were used as training data, split into five folds
for cross validation. 19 slides from UC were set aside as an external held-out
test set. Using a web-based whole slide viewer developed by our group, areas
of tumor were manually annotated in each WSI. Using a touchscreen tablet and
desktop (Surface Pro 3, Surface Studio; Microsoft Inc.), a pathologist painted
over regions of tumor to identify where tiles should be extracted for
training. Tiles used in training were extracted from tumor-regions of tissue
and sampled at 224x224px, 10x resolution.

###  2.5 Architecture and Experiments

An ImageNet-pretrained ResNet-34 was used as the base feature extractor
($\theta_{e}$). A series of three wide fully connected layers (4096, 4096,
256) with dropout were implemented before the single risk output node. Model
hyperparameters $n$, $w$, $b$, $lr$, $d$, and $p$ (number of clusters, waist
size, part-batch size, learning rate, dropout rate, and top-k tiles
respectively) were optimized using random grid search and CI as a performance
metric at the end of each epoch. 16 clusters and a waist size of 16 produced
the best performance. The same 5-fold cross validation was implemented and
held throughout all experiments and models. Predicted risks of the validation
sets from each fold were concatenated for a complete performance analysis
using CI and LRT. Each model was subsequently trained using all training data,
tested on the held-out test set, and evaluated using CI and LRT.

As a baseline, Deep Clustering Convolutional Autoencoder [10] was implemented.
This model was chosen because, like EPIC-Survival, it uses clustering to
define morphological features. However, these features are learned based on
image reconstruction and then used as covariates in traditional CPH modelling,
as a representation for the classic two-stage approach. Further, the subset of
training data with AJCC staging, a clinical standard, was analyzed using a
4-fold cross validation and CPH.

##  3 Results

EPIC-Survival with and without stratification boosting performed similarly on
the 5-fold cross validation producing CI of 0.671 and 0.674, respectively. On
the held out test set, EPIC-survival with stratification boosting performed
significantly better with a CI of 0.880, compared to a CI of 0.652 without
stratification boosting. Unsupervised clustering with a traditional CPH
regression yielded a CI of 0.583 on 5-fold cross validation and 0.614 on the
test set. Table 1 summarizes these results. AJCC staging using the TRN and
P-stage protocols on the subset of ICC produced CIs of 0.576 and 0.638,
respectively. While we recognize that a CI produced on a subset of data may
produce biases from batch effects, these results are not different from the
results of a study which tested multiple prognostic scores on a very large ICC
cohort (n=1054) [3].

Figure 3: Top Left: EPIC-Survival without stratification successfully stratifies (LRT: p $<$ 0.05) the patient population into high and low risks on 5-Fold Cross Validation but fails on the held out test set. Bottom Left: EPIC-Survival with stratification boosting produces strong patient population separation on both 5-Fold Cross Validation and the External Test Set. On the right, we visualize the distribution of time-to-events relative to predicted risk scores ordered from low to high. We find that EPIC-Survival, in general, does well at predicting early recurrence. In general, the inclusion of stratification boosting improves the correlation between predicted risk values and patient outcome. Top Right: EPIC-Survival without stratification boosting Bottom Right: EPIC-Survival with stratification boosting. | Cross Validation | Test
---|---|---
AJCC TNM | 0.576 (n=157) | -
AJCC P-Stage | 0.638 (n=157) | -
Muhammad et. al. | 0.583 (n=244) | 0.614 (n=19)
EPIC (DeepSurv) | 0.671 (n=244) | 0.652 (n=19)
EPIC (Stratfication Boosting) |  0.674 (n=244) |  0.880 (n=19)
AJCC TNM | - | 0.582 (n=1054)
Wang Nomogram | - | 0.607 (n=1054)
LCSGj | - | 0.562 (n=1054)
Okabayashi | - | 0.557 (n=1054)
Nathan Staging | - | 0.581 (n=1054)
Hyder Nomogram | - | 0.521 (n=1054)
Table 1: EPIC-Survival with stratification boosting showed the best
concordance index-based performance. For reference, performance of various
clinical metrics on a very large ICC dataset (n=1054) are provided [3].

In a KM analysis (Figure 3), EPIC-Survival with stratification boosting showed
significant separation between high and low risk populations (p $<$ 0.05).
Epic-Survival without stratification boosting failed on the held out test set.
Although stratification on the 5-fold cross validation is assumed significant,
there remains a risk of crossing survival curves, breaking the assumption of
proportional hazard rates.

To further analyze results, we visualize the distribution of predicted risks
relative to the distribution of time-to-events (Figure 3). We found that EPIC-
Survival with and without stratification boosting performs well at predicting
early recurrence ($<$50 months). Correlation between predicted risks and time
durations of the external test set using EPIC-Survival with stratification
boosting is very strong, as further indicated by the strong CI of 0.880.

Figure 4: Rows: Slide parts, Columns: Patients with their predicted risk
scores highlighted above. Black tiles indicate that there was no assigned tile
to that part of a slide.

###  3.1 Prognosis Predictive Histology Discovery

In Figure 4, we visualize the part representation (rows) in each slide
(columns) from the test set. The slides are ordered by predicted risk scores.
A pathologist with a specialty in gastrointestinal pathology reviewed these
and discovered some general trends indicating that tiles with a low predicted
risk (earlier rate of recurrence) tended to have loose, desmoplastic stroma
with haphazard, delicate collagen fibers, whereas high risk tiles (later
recurrence) tended to have dense intratumoral stroma with thickened collagen
fibers. The quality of nuclear chromatin was vesicular more commonly in the
low risk tiles. The quality of the intratumoral stroma has never been a part
of tumor grading or observed as a prognostic marker. Further, there is no
grading scheme that involves assessment of nuclear features for ICC.

##  4 Discussion

###  4.1 On the Concordance Index

Our test results show a significantly higher CI than the cross validation
experiments. We found that CI on smaller sets are often larger because
correctly ranking a smaller set of data is easier. During hyperparameter
optimization, this was also observed in the case of batch sizes. Smaller batch
sizes produces better CIs—in other words, optimizing the ranking of smaller
batches was easier than optimizing the ranking in larger batches.

###  4.2 Reflections on Stratification Boosting

Our work shows that EPIC-Survival has the capacity to identify specific risk
factors in histology, though these morphologies would need further testing on
a larger study. We hypothesise that altering the stratification boosting
component of the loss function to push separation between $>$2 groups would
further improve performance and has the potential to function as a general
subtyping model.

###  4.3 Conclusion

Our contributions are threefold: (1) we introduce the first end-to-end
survival model, allowing computational pathology to overcome the memory limits
introduced by two-stage approaches; (2) we contribute a new loss term to
strengthen the traditional hazard regression and encourage the learning of
stratified risk groups; (3) we show the power of EPIC-Survival by applying it
to the difficult test case of ICC, surpassing other metrics and providing
insight into new histologic features which may unlock new discoveries in ICC
subtyping.

##  5 Acknowledgements

This work was supported by Cycle for Survival, the generous computational
support given by the Warren Alpert foundation, and spectacular project
management from Christina Virgo.

This study was supported in part by National Institutes of Health/National
Cancer Institute (NIH/NCI) Cancer Center Support Grant P30 CA008748 and U01
CA238444-02.

Thomas J. Fuchs is a founder, equity owner, and Chief Scientific Officer of
Paige.AI.

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# Laser Spectroscopy of the y7P${}_{J}^{\circ}$ states of Cr I

E. B. Norrgard  Sensor Science Division, National Institute of Standards and
Technology, Gaithersburg, Maryland 20899, USA  D. S. Barker  Sensor Science
Division, National Institute of Standards and Technology, Gaithersburg,
Maryland 20899, USA  S. P. Eckel  Sensor Science Division, National Institute
of Standards and Technology, Gaithersburg, Maryland 20899, USA  S. G. Porsev
University of Delaware, Newark, Delaware, USA  Petersburg Nuclear Physics
Institute of NRC “Kurchatov Institute”, Gatchina, Leningrad district 188300,
Russia  C. Cheung  University of Delaware, Newark, Delaware, USA  M. G. Kozlov
Petersburg Nuclear Physics Institute of NRC “Kurchatov Institute”, Gatchina,
Leningrad district 188300, Russia  St. Petersburg Electrotechnical University
“LETI”, Prof. Popov Str. 5, St. Petersburg, 197376, Russia  I. I. Tupitsyn
Department of Physics, St. Petersburg State University, Ulianovskaya 1,
Petrodvorets, St.Petersburg, 198504, Russia  M. S. Safronova  University of
Delaware, Newark, Delaware, USA

###### Abstract

Here we report measured and calculated values of decay rates of the
3d4(5D)4s4p(3Po) y7P${}^{\rm{o}}_{2,3,4}$ states of Cr I. The decay rates are
measured using time-correlated single photon counting with roughly 1 % total
uncertainty. In addition, the isotope shifts for these transitions are
measured by laser induced fluorescence to roughly 0.5 % uncertainty. The decay
rate calculations are carried out by a hybrid approach that combines
configuration interaction and the linearized coupled cluster method (CI+all-
order method). The measurements provide a much needed precision benchmark for
testing the accuracy of the CI+all-order approach for such complicated systems
with six valence electrons, allowing to significantly expand its
applicability. These measurements also demonstrate operation of a cryogenic
buffer gas beam source for future experiments with MgF molecules toward
quantum blackbody thermometry.

##  I Introduction

The decay rates of excited quantum states are important parameters for
characterizing the electronic structure of atomic and molecular systems.
Precise decay rate measurements are necessary for atomic studies of
electroweak symmetries [1, 2, 3], calibration of astrophysical measurements
[4, 5], prediction of optical trap depths and magic wavelengths [6, 7],
fluorescence measurements of the number of magneto-optically trapped atoms or
molecules [8, 9], calibrating atom-based sensors [10], and more.

Here we report measured and calculated values of the $1/e$ decay rates of the
3d4(5D)4s4p(3P∘) y7P${}^{\circ}_{2,3,4}$ states (hereafter,
y7P${}^{\circ}_{J}$ states) of Cr I. We also report the isotope shifts of
these transitions from the ground state. The isotope shifts are measured by
laser induced fluorescence to roughly 0.5 % uncertainty, while the decay rates
are measured using time-correlated single photon counting (TCSPC) with roughly
1 % total uncertainty. TCSPC techniques have previously been used to measured
the decay rates of several atoms [11, 2, 12, 3] and molecules [13] to 1 %
uncertainty or better.

We had two primary motivations for carrying out these measurements. First,
these measurements demonstrate operation of a cryogenic buffer gas beam source
constructed at NIST to create MgF molecules for laser cooling and trapping
experiments toward quantum blackbody thermometry [10]. In particular, the
transition from the ground 3d5(6S)4s a7S3 state of Cr to the
y7P${}^{\rm{o}}_{3}$ state has nearly the same frequency ($\nu=834.0286$ THz
[14]) as the $X^{2}\Sigma^{+},v=0,N=1\rightarrow
A^{2}\Pi_{1/2},v=0,J^{P}=1/2^{+}$ main cycling transition of MgF
($\nu=834.2903$ THz [15]). The decay rates of the Cr y7P${}^{\rm{o}}_{J}$
states [16, 5, 17, 14] are also within 10 % of the expected decay rate for the
MgF cycling transition [18]. Cr thus serves are a suitable proxy for
commissioning the cryogenic buffer gas beam source, lasers, and diagnostic
systems for future work with MgF.

Second, these measurements provide a high-precision benchmark for the first
application of the configuration interaction (CI) and the linearized coupled
cluster method (CI+all-order method) for a system with 6 valence electrons.
The CI+all-order method was demonstrated to produce accurate results for a
wide range of atomic properties for atoms and ions with 2 to 4 valence
electrons [19, 20, 21, 22]. This method has provided precision data for many
applications, including development of ultra-precise clocks [19, 21, 23],
studies of fundamental symmetries [24, 25], searches for the variation of
fundamental constants [26], quantum simulation and computation [27, 28],
plasma physics [29], nuclear physics [22, 30], and many others. It was
previously thought to be unfeasible to apply this method to systems such as Cr
due to extremely large number of configurations that have to be considered as
well as limitation of the initial closed-shell approximation. In this work, we
demonstrated a 1 % to 2.5 % agreement with experiment for Cr electric-dipole
transition matrix elements and a few percent accuracy for a wide range of
energy levels. We also tested the intrinsic accuracy of the approach by
demonstrating the ability to saturate a CI space in such computation to a good
numerical precision. The methodology used here is not specific to Cr and we
expect to be able to apply CI+all-order method for a wide range of other
systems with 5 to 6 valence electrons, which opens many future applications.

To our knowledge, these are the highest precision decay rates reported to date
for any Cr I levels. The decay rates of the y7P${}^{\rm{o}}_{J}$ were
previously measured by Becker et al. [5] with roughly 3 % uncertainty using
laser excitation; by Cooper et al. [17] with 4.5 % uncertainty using laser
excitation; and by Marek and Richter [16] to roughly 10 % uncertainty by the
phase shift method. Tian et al. [31] have measured decay rates of 43 Cr I
levels with an uncertainty of a few percent using pulsed laser excitation in
the wavelength range 216 nm to 649 nm. Despite this extensive survey, the
y7P${}^{\rm{o}}_{J}$ states were omitted, apparently due to a gap in the
coverage of their dye laser. Another compilation of Cr I decay rates is found
in Tozzi et al. [32], which covers lines in the 290 nm to 900 nm range to
roughly 7 % accuracy by Fourier transform spectroscopy on a chromium hollow
cathode lamp. Pulsed laser excitation methods have been used to measure the
decay rates of the z7P${}^{\rm{o}}_{J}$ levels to roughly 3 % precision by
Measures et al. [33], and to roughly 2 % precision for z7P${}^{\rm{o}}_{4}$
only by Hannaford and Lowe [34]. Marek [4] also measured the decay rates of
nine Cr I levels in the 425 nm to 429 nm range including z7P${}^{\rm{o}}_{J}$
to several percent precision. Isotope shifts of several other low-lying Cr I
transitions were reported in Frumann et al. [35].

##  II Theory

Table 1:  Comparison of theoretical energy levels (in cm-1) with experiment [14].  Level | Term | Expt. | Theory | Diff. | Diff. (%)
---|---|---|---|---|---
$J=3$ even |  |  |  |  |
$3d^{5}4s$ | $\rm{a}^{7}\rm{S}$ | 0 | 0 |  |
$3d^{4}4s^{2}$ | $\rm{a}^{5}\rm{D}$ | 8095 | 7128 | 967 | 12%
$3d^{5}4s$ | $\rm{a}^{5}\rm{G}$ | 20521 | 21283 | -762 | -3.7%
$3d^{5}4s$ | $\rm{a}^{5}\rm{P}$ | 21841 | 22702 | -861 | -3.9%
$J=2$ even |  |  |  |  |
$3d^{5}4s$ | $\rm{a}^{5}\rm{S}$ | 7593 | 8040 | -447 | -5.9%
$3d^{4}4s^{2}$ | $\rm{a}^{5}\rm{D}$ | 7927 | 7171 | 756 | 9.5%
$3d^{5}4s$ | $\rm{a}^{5}\rm{G}$ | 20517 | 21386 | -869 | -4.2%
$3d^{5}4s$ | $\rm{a}^{5}\rm{P}$ | 21848 | 22857 | -1010 | -4.6%
$3d^{4}4s^{2}$ | $\rm{a}^{3}\rm{P}$ | 24093 | 24109 | -15 | -0.1%
$3d^{5}4s$ | $\rm{b}^{5}\rm{D}$ | 24300 | 25087 | -787 | -3.2%
$J=4$ odd |  |  |  |  |
$3d^{5}4p$ | $\rm{z}^{7}\rm{P}^{\circ}$ | 23499 | 23596 | -97 | -0.4%
$3d^{4}4s4p$ | $\rm{z}^{7}\rm{F}^{\circ}$ | 25360 | 24545 | 815 | 3.2%
$3d^{4}4s4p$ | $\rm{z}^{7}\rm{D}^{\circ}$ | 27650 | 27045 | 605 | 2.2%
$\bm{3d^{4}4s4p}$ | $\bf{y^{7}P}^{\circ}$ | 27935 | 28629 | -694 | -2.5%
$3d^{4}4s4p$ | $\rm{z}^{5}\rm{F}^{\circ}$ | 31106 | 30588 | 518 | 1.7%
$J=3$ odd |  |  |  |  |
$3d^{5}4p$ | $\rm{z}^{7}\rm{P}^{\circ}$ | 23386 | 23482 | -96 | -0.4%
$3d^{4}4s4p$ | $\rm{z}^{7}\rm{F}^{\circ}$ | 25206 | 24395 | 811 | 3.2%
$3d^{5}4p$ | $\rm{z}^{5}\rm{P}^{\circ}$ | 26787 | 28099 | -1312 | -4.9%
$3d^{4}4s4p$ | $\rm{z}^{7}\rm{D}^{\circ}$ | 27500 | 26900 | 600 | 2.2%
$\bm{3d^{4}4s4p}$ | $\bf{y^{7}P}^{\circ}$ | 27820 | 28528 | -707 | -2.5%
$3d^{4}4s4p$ | $\rm{y}^{5}\rm{P}^{\circ}$ | 29825 | 29114 | 711 | 2.4%
$3d^{4}4s4p$ | $\rm{y}^{5}\rm{F}^{\circ}$ | 30965 | 30448 | 517 | 1.7%
$3d^{4}4s4p$ | $\rm{y}^{5}\rm{D}^{\circ}$ | 33672 | 33412 | 259 | 0.8%
$3d^{4}4s4p$ | $\rm{z}^{3}\rm{F}^{\circ}$ | 36034 | 35877 | 157 | 0.4%
$3d^{4}4s4p$ | $\rm{z}^{3}\rm{D}^{\circ}$ | 38911 | 38883 | 29 | 0.1%
$J=2$ odd |  |  |  |  |
$3d^{5}4p$ | $\rm{z}^{7}\rm{P}^{\circ}$ | 23305 | 23392 | -87 | -0.4%
$3d^{4}4s4p$ | $\rm{z}^{7}\rm{F}^{\circ}$ | 25089 | 24282 | 808 | 3.2%
$3d^{5}4p$ | $\rm{z}^{5}\rm{P}^{\circ}$ | 26796 | 28079 | -1282 | -4.8%
$3d^{4}4s4p$ | $\rm{z}^{7}\rm{D}^{\circ}$ | 27382 | 26785 | 597 | 2.2%
$\bm{3d^{4}4s4p}$ | $\bf{y^{7}P}^{\circ}$ | 27729 | 28441 | -712 | -2.6%
$3d^{4}4s4p$ | $\rm{y}^{5}\rm{P}^{\circ}$ | 29585 | 28886 | 698 | 2.4%
$3d^{4}4s4p$ | $\rm{z}^{5}\rm{F}^{\circ}$ | 30859 | 30343 | 515 | 1.7%
$3d^{4}4s4p$ | $\rm{z}^{5}\rm{D}^{\circ}$ | 33542 | 33282 | 260 | 0.8%
$3d^{4}4s4p$ | $\rm{z}^{3}\rm{P}^{\circ}$ | 34190 | 33757 | 433 | 1.3%
$3d^{4}4s4p$ | $\rm{z}^{3}\rm{F}^{\circ}$ | 35898 | 35735 | 163 | 0.5%
$3d^{4}4s4p$ | $\rm{z}^{3}\rm{D}^{\circ}$ | 38731 | 38704 | 26 | 0.1%

We calculated Cr energies and transition rates using the method that combines
linearized coupled-cluster and configuration interaction approaches; the
CI+all-order method [36]. In this method, the coupled-cluster approach is used
to construct an effective Hamiltonian $H_{\rm{eff}}$ that includes core and
core-valence correlations. The many-electron wave function is obtained using
the CI method as a linear combination of all distinct many-electron states of
a given angular momentum $J$ and parity:

$\Psi_{J}=\sum_{i}c_{i}\Phi_{i}.$

The energies and wave functions of the low-lying states are determined by
diagonalizing this effective rather than bare Hamiltonian.

The CI+all-order method was demonstrated to produce accurate results for a
wide range of atomic properties for atoms and ions with 2 to 4 valence
electrons [19, 20, 21, 22]. The main problem in extending this approach to
more complicated systems such as Cr is an exponential increase in the number
of configurations in the CI expansion with the number of valence electrons.
Another problem is the degrading quality of one-electron functions, which are
built in a closed-shell Dirac-Hartree-Fock potential, i.e., $V^{N-M}$, where
$N$ is the total number of electrons and $M$ is the number of valence
electrons.

In some cases a different potential can be used to construct one-electron
orbitals for the CI+all-order calculations. For example, in Ref. [25] Pb I was
considered as a system with four valence electrons but the $V^{N-2}$ potential
was used for the construction of the basis-set orbitals. Such an approach is
expected to provide better initial approximation but it is more complicated
and the area of its applicability requires further research. While one can
construct one-electron orbitals for partially open shells, an all-order
effective Hamiltonian $H_{\rm{eff}}$ cannot be constructed for such a basis.

The problems mentioned above hampered application of the effective Hamiltonian
approach to the systems with more than 4 valence electrons. However, in 2005
Dzuba [37] used this method to calculate properties of the Kr atom. An
effective Hamiltonian was formed using second order many-body perturbation
theory in the $V^{N-8}$ potential and Kr was considered as an 8-electron
system. Only the ground state was calculated and 2.4 % accuracy for the
ionization potential was obtained.

In this work, we have successfully applied the CI+all-order approach for the
case of six valence electrons for the first time. We start from Dirac-Hartree-
Fock one-electron wave functions for the low-lying valence electrons: $3d$,
$4s$, $4p$, $5s$, $4d$, $5p$, and $4f$ with the
$1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}$ core, i.e., constructed in a $V^{N-6}$
potential. All other orbitals (up to $35spdfghi$) are constructed in a
spherical cavity using B-splines. Such a finite basis set method discretizes
the continuum spectrum: a sum over the finite basis is equivalent (to a
numerical precision) to the sum over all bound states and integration over the
continuum. This large basis is used for all coupled-cluster computations but
can be reduced for CI computations. Taking into account that we are starting
from a potential of the Cr6+ core, which is significantly more compact than
neutral Cr, we reduced the cavity size from 60 $a_{0}$ used for 2 to 3 valence
electrons to 30 $a_{0}$ ($a_{0}\,\approx\,52.9$ pm is the Bohr radius).

Table 2:  Electric-dipole reduced matrix elements (in $ea_{0}$, where $e$ is the electron charge and $a_{0}$ is the Bohr radius) and transition rates (in s-1) in Cr. The CI+all-order values include RPA correction; the contributions from the core-Brueckner ($\sigma$), two-particle (2P), structural radiation (SR), and normalization (Norm) corrections are listed separately. Final matrix elements are given in column“Total”. The transition rates are given in the last column.  Transition | CI+all-order+RPA | $\sigma$ | SR | 2P | Norm | Total | Tr. rate
---|---|---|---|---|---|---|---
$3d^{4}4s4p$ $\rm{y}^{7}\rm{P}_{4}^{\circ}-3d^{5}4s$ $\rm{a}^{7}\rm{S}_{3}$ | 6.089 | 0.004 | 0.010 | 0.038 | -0.704 | 5.437 | 145$\times 10^{6}$ s-1
$3d^{4}4s4p$ $\rm{y}^{7}\rm{P}_{3}^{\circ}-3d^{5}4s$ $\rm{a}^{7}\rm{S}_{3}$ | 5.295 | 0.003 | 0.008 | 0.027 | -0.611 | 4.722 | 139$\times 10^{6}$ s-1
$3d^{4}4s4p$ $\rm{y}^{7}\rm{P}_{2}^{\circ}-3d^{5}4s$ $\rm{a}^{7}\rm{S}_{3}$ | 4.476 | 0.002 | 0.007 | 0.022 | -0.511 | 3.996 | 138$\times 10^{6}$ s-1

Usually, we construct the set of CI configurations by starting from the
configurations of interest, here $3d^{5}4s$, $3d^{4}4s^{2}$, $3d^{5}4p$, and
$3d^{4}4s4p$, and make all possible single and double excitations to a large
basis, such as $20spd18f16g$. Then, one accounts for some triple and quadruple
excitations, which are generally much smaller. However, because the one-
electron orbitals were built in the $V^{N-6}$ potential, $4s$ orbital for
neutral Cr is a mixture of our $4s$, $5s$, and higher $ns$ basis set orbitals.
This leads to increased weights of the $3d^{5}5s$, $3d^{4}4d4s$, $3d^{4}4s5p$,
etc. configurations. A solution is to use all such configurations as basis
configurations and allow all single and double excitations for such
configurations into as many basis set states as numerically feasible. In
addition, we allow all single excitations from a large number of dominant
configurations ($\sim$130 non-relativistic configurations) to the
$20spd18f16g$ basis when constructing the CI space for odd states.

An obvious problem is a very large number of resulting configurations and
corresponding Slater determinants (207,000 and 223,000 relativistic
configurations and 45 million and 51 million Slater determinants for odd and
even states, respectively). We note that even and odd states are computed
separately in CI. Our recently developed Message Passing Interface (MPI)
version of the CI code [38, 19] allows us to carry our computations with such
a large number of configurations in a reasonable time, using 500 to 1000
central processing units (CPUs). Using very large number of configurations
also significantly affects convergence of the Davidson procedure that extracts
low-lying eigenvalues and eigenstates. We carried out additional improvements
to the code, allowing for fast diagonalization of a much larger matrix used to
initialize a Davidson procedure [39], improving convergence. To further
alleviate the convergence issues, we carry out computations for different $J$
separately, reordering configurations by weight and omitting those with
negligible weights, reducing the size of the computations.

The results for the energy levels in cm-1 are listed in Table 1, grouped by
$J$ and parity. Comparing with experimental energy values [14] required
resolving another issue, i.e. correct identification of the
$\rm{y}^{7}\rm{P}^{\circ}$ levels, as $\rm{z}^{7}\rm{D}^{\circ}$ levels have
similar energy and both ${}^{5}\rm{P}^{\circ}$ levels mix with the
$\rm{y}^{7}\rm{P}^{\circ}$ levels for $J=2$ and $J=3$ states (there is no
${}^{5}\rm{P}^{\circ}$ level with $J=4$). We wrote a code that computes the
expectation values of $\langle\bm{L}^{2}\rangle\ $ and
$\langle\bm{S}^{2}\rangle\ $with the final wave functions, where $\bm{L}$ and
$\bm{S}$ are total electron orbital angular momentum and spin operators,
respectively. Computing these quantities for all states allowed us to find
approximate quantum numbers $L$ and $S$, where $\langle\bm{L}^{2}\rangle\
=L(L+1)$ and $\langle\bm{S}^{2}\rangle\ =S(S+1)$, and unambiguously identify
all terms in Table 1. The energies are generally in good agreement with the
experiment. The differences between the theory and experiment for
$\rm{y}^{7}\rm{P}^{\circ}$ levels are $-2.5$ % (shown in bold in Table 1).

We use the resulting CI + all-order wave functions to calculate the electric-
dipole reduced matrix elements, with the effective electric-dipole operator
$D_{\rm{eff}}$ in the random-phase approximation (RPA). Such effective
operator accounts for the core-valence correlations in analogy with the
effective Hamiltonian discussed above [40, 41, 42]. We also include other
correction to the E1 operator beyond RPA; the core-Brueckner ($\sigma$), two-
particle (2P) corrections, structural radiation (SR), and normalization (Norm)
corrections [40, 41, 42]. In most computations of E1 matrix elements, they are
omitted, as SR and normalization corrections tend to partially cancel for
divalent atoms [43]. Reference [23] discussed the size of these corrections in
Sr, concluding that they cannot be omitted at the 1 % level of accuracy. For
Cr, we find that a normalization correction is very large, 11 %, which is
another consequence of starting from the $V^{N-6}$ potential. All other
corrections are small ($<1$ %). The results are listed in Table 2. The CI+all-
order values include RPA correction, the contributions from the core-Brueckner
($\sigma$), two-particle (2P), structural radiation (SR), and normalization
(Norm) corrections are listed separately. Final matrix elements are given in
column “Total”. The transition rates are given in the last column in s-1.

We also studied the effects of saturating the configuration space for all
three transitions. In these tests, we compared results of two computations
obtained when the odd configurations with weights above $10^{-6}$ and
$10^{-7}$ were truncated, corresponding to 44878 relativistic configurations,
13.2 million Slater determinants and 115359 relativistic configurations, 38.6
million Slater determinants, respectively. Truncating configurations with
small weights is frequently done in CI computations to improve convergence,
compute more states, or calculate small corrections. However, our tests show
that truncating configurations at the 10-6 level may introduce significant
differences in cases where mixing is important. For $J=4$, truncation of odd
configurations with weight smaller than $10^{-6}$ only changed the value of
the $\rm{y}^{7}P_{4}^{\circ}-a^{7}S_{3}$ matrix element by 0.1 %, while the
$\rm{y}^{7}\rm{P}_{3}^{\circ}-a^{7}\rm{S}_{3}$ and
$\rm{y}^{7}\rm{P}_{2}^{\circ}-a^{7}\rm{S}_{3}$ matrix elements changed by 2.7
% and 4.9 %, respectively. This sensitivity is due to mixing with the
${}^{5}\rm{P}^{\circ}$ levels for $J=2$ and $J=3$ cases mentioned above.

We confirm the sensitivity of matrix elements to mixing of
${}^{5}\rm{P}^{\circ}$ levels by studying changes in approximate quantum
numbers $L$ and $S$ computed as described above. For $J=4$ level, $S=2.9994$,
even if we omit configurations with weights above $10^{-5}$. For
$\rm{z}^{5}\rm{P}_{2}^{\circ}$, $\rm{y}^{7}\rm{P}_{2}^{\circ}$, and
$\rm{y}^{5}\rm{P}_{2}^{\circ}$ levels, $S$ is equal to 2.033, 2.965, 2.023,
respectively, when configurations with weights above $10^{-7}$ are included.
These values change to 2.094, 2.859, 2.118, respectively, when only
configurations with weights above $10^{-6}$ are included. We note that the
$\rm{y}^{7}\rm{P}_{J}^{\circ}-\rm{a}^{7}\rm{S}_{3}$ matrix elements are nearly
purely nonrelativistic – i.e. very nearly proportional to $\sqrt{2J+1}$ – so
we expect $L$ and $S$ to be good quantum numbers.

We also calculate transitions from the $\rm{y}^{7}\rm{P}_{J}^{\circ}$ levels
to other states besides the ground state. This contribution is negligible for
$J=4$. The largest contribution to total transition rates for $J=2$ and $J=3$
comes from transitions to the $3d^{5}4s$ $\rm{a}^{5}\rm{S}_{2}$ state, with
the branching on the order of 1 %. We note that mixing strongly affects these
small matrix elements, and large CI spaces are needed for branching ratio
computations. The final total decay rates $\Gamma$ for the
$\rm{y}^{7}\rm{P}_{J}^{\circ}$ for $J=2,3,4$ are $140\,\times\,10^{6}$ s-1,
$141\,\times\,10^{6}$ s-1, and $145\,\times\,10^{6}$ s-1, respectively. We
expect slightly worse accuracy for $J=2$ and $J=3$ cases as these are more
sensitive to the saturation of the CI space as described above.

##  III Experimental Details

Our experimental apparatus is based on a pulsed beam of Cr atoms produced
using a cryogenic buffer gas beam (CBGB) source [44, 45]. The CBGB source cell
dimensions are based on the design outlined in Truppe et al. [46]. Atoms are
laser ablated from a solid chromium metal sample with a nominally 9 ns long,
10 mJ pulse of 532 nm light. In our design, the chromium sample is stationary.
The ablation laser focus is repositioned when the yield of an ablation spot
has dropped by roughly half, typically after several thousand ablation pulses.
The source is run with a He buffer gas flow rate of 30 mL/min at standard
temperature and pressure. We orient our experiment by taking $\hat{z}$ to be
the direction of travel of the atomic beam (roughly parallel to the ground),
$\hat{y}$ vertically upward, and $\hat{x}$ parallel to the ground and forming
a right-handed coordinate system.

Excitation light is produced using a titanium-doped sapphire laser. The light
is frequency doubled by a lithium triborate crystal in a bow-tie resonator
cavity to produce the ultraviolet light resonant with the transitions from the
ground state to the y7P${}^{\rm{o}}_{J}$ states. An acousto-optic modulator
(AOM) allows the laser light to be turned on and off rapidly; we observe a
typical turn-off times of 4.5 ns. In order to minimize the presence of stray
light in the experiment, the first-order diffracted beam is directed to the
atomic beam vacuum chamber via a 1 m-long polarization maintaining optical
fiber. The laser beam has a nominal 1/e2 diameter of 15 mm. It is
retroreflected to intersect the atomic beam twice, propagating in the
$\pm\hat{y}$-directions, and polarized linearly in the $\hat{z}$-direction.

A hybrid photomultiplier and avalanche photodiode (Picoquant PMA Hybrid-06
[47]) images the atomic beam along the $-\hat{x}$ direction. This detector
combines the single photon sensitivity of a photomultiplier (roughly 20 %
quantum efficiency for the wavelengths of interest here), but has negligible
afterpulsing [48], a common systematic effect inherent to standard PMT
detectors [13]. Because afterpulsing is not present, the data do not need to
be cut to exclude events where multiple fluorescence photons are detected from
a single excitation. However, care must still be taken to ensure the photon
count rate is sufficiently low that multiple photons are not incident on the
detector within the the detector dead time. The detector is cooled and
typically produces 10 dark counts per second. A complementary metal-oxide-
semiconductor (CMOS) camera viewing along $+\hat{x}$ images fluorescence from
the atomic beam on the a7S${}_{3}\rightarrow$ y7P${}^{\rm{o}}_{J}$ transition.
The camera images were used to measure isotope shifts from the observed
fluorescence, and to monitor the ablation yield during decay rate
measurements.

##  IV Isotope shift measurement

The laser is scanned over a roughly 2 GHz range centered on each transition.
Typical fluorescence spectra for each transition measured by total counts on
the CMOS camera are shown in Fig. 1. As is typical with CBGB sources, the
yield tends to decrease with a halflife of several thousand ablation pulses,
and the observed peak heights in Fig. 1 do not identically match the natural
isotopic abundance of Cr [49]. Systematic asymmetry in the observed lineshapes
is mitigated by summing multiple scans of the laser up and down in frequency.
The 52Cr and 53Cr isotopes are readily identified. By collecting fluorescence
for multiple sets of 10 ablation pulses with the laser alternately resonant
with the 50Cr (2.4 % natural isotopic abundance) and 54Cr (4.3 % natural
isotopic abundance) isotopes, we achieve an average signal ratio in good
agreement with their natural isotopic abundance ratio.

The titanium-doped sapphire laser is referenced to a confocal Fabry-Pèrot
cavity locked to a frequency stabilized helium-neon laser. The cavity has a
nominal 1.5 GHz free spectral range. The cavity nonlinearity is measured to be
less than 2 MHz per free spectral range by scanning the laser frequency while
applying 115 MHz sidebands using an electro-optic modulator.

The line centers for the bosonic isotopes are determined by fitting to a Voigt
profile with Lorentzian width equal to the measured decay rate $\Gamma$ given
in Table 4 and detailed in Sec. V. Due to the partially resolved nature and
low signal to noise ratio of the 53Cr features, we have not attempted to fit
this isotope. Our measured isotope shifts
$\nu_{A-A^{\prime}}=\nu_{A}-\nu_{A^{\prime}}$ are listed in Table 3.

Figure 1: Laser induced fluorescence signal as a function of laser frequency for the a7S${}_{3}\rightarrow$ y7P${}^{\rm{o}}_{J}$ transitions. Laser frequency is relative to the fitted 52Cr center frequency. Black horizontal bands depict the frequency extent of the 53Cr hyperfine structure.  Isotope shift | y7P${}^{\circ}_{2}-\rm{a}^{7}\rm{S}_{3}$ | y7P${}^{\circ}_{3}-\rm{a}^{7}\rm{S}_{3}$ | y7P${}^{\circ}_{4}-\rm{a}^{7}\rm{S}_{3}$
---|---|---|---
$\nu_{50-52}$ | 807.9(1.4) | 811.6(1.9) | 836.8(1.2)
$\nu_{52-54}$ | 738.7(2.8) | 744.2(2.8) | 763.3(1.8)
Table 3: Measured isotope shifts in MHz for the Cr I
y7P${}^{\circ}_{J}-\rm{a}^{7}\rm{S}_{3}$ transitions. Values in parentheses
are the combined $1\sigma$ statistical and systematic uncertainty.

##  V Decay Rate measurement

In this section we detail the measurement and analysis procedures used to
determine the decay rates $\Gamma$. Our measured values of $\Gamma$ for the
y7P${}^{\rm{o}}_{J}$ states are presented in Table 4 with a list of
uncertainty estimates.

The output of an arbitrary waveform generator in pulse mode is split and sent
to both the AOM switching electronics and the sync channel of a multichannel
event timer device (Picoquant Multiharp 150 [47]). The output of the
photodetector is sent to an input channel of the event timer, and the time of
arrival is measured against the sync channel. A histogram of events is
collected with 80 ps time bins by the event timer. Timing, computer control of
equipment, and data collection are performed using the Labscript Suite [50].

Each pulse of the cryogenic buffer gas beam of Cr lasts roughly 8 ms. At
helium flow rates of 30 mL/min, the temporal distribution of the pulse is
observed to be bimodal, with roughly half of the atoms passing through the
detection region between 0.8 ms and 2.8 ms after ablation. We therefore
typically only collect decay rate data for this fraction of the atom pulse. In
the standard measurement configuration, the AOM produces light pulses with 16
ns full width at half maximum at a pulse rate of 5 MHz for a duration of 2 ms
(i.e. $10^{4}$ light pulses per Cr ablation pulse). The Cr target is ablated
at a repetition rate of 15 Hz. Data is read out from the multichannel event
timer after 300 ablation pulses. We then perform the same procedure, absent
the ablation pulse, to obtain a background scattered light signal. We define a
run of the experiment as 50 iterations of this signal plus background
measurement procedure. A single run lasts roughly 40 min and achieves about 1
% to 2 % statistical uncertainty on the decay rate for the state of interest.
We performed 6, 12, and 7 runs measuring the decay rate of the $J=2,3,4$
states, respectively. In addition, several runs were performed on each state
with one experimental parameter varied in order to quantify potential
systematic errors.

For the $J=2,3$ states, typical laser power is 10 mW, and narrow bandpass
interference filters are used to monitor the fluorescence decay of the
$\lambda\,=\,$ 495 nm, 494 nm transitions to a5S2, respectively. For $J=4$,
the transition to a5S2 is electric dipole forbidden, and we initially
attempted to monitor the combined fluorescence decay on the 504 nm and 509 nm
transitions to the a5D3,4 states with another bandpass filter. However, the
y7P${}^{\rm{o}}_{4}\rightarrow$a5D3,4 signal was observed to be roughly 100
times lower than the y7P${}^{\rm{o}}_{2,3}\rightarrow$a5S2 decays, in
agreement with our calculated transition dipole matrix elements as well as the
relative line strengths reported by Wallace and Hinkle [51]. We therefore
proceeded to measure the y7P${}^{\rm{o}}_{4}$ decay by monitoring the 358 nm
y7P${}^{\rm{o}}_{4}\rightarrow$a7S3 transition using a third interference
filter set with the laser power reduced to 10 $\mu$W. This measurement
configuration produces similar signal and background count rates as the one
used for y7P${}^{\rm{o}}_{2,3}$, but the 358 nm fluorescence is too weak for
useful ablation monitoring with the CMOS camera.

Parameter ($\times 10^{-6}\,\rm{s}^{-1}$) | y7P${}^{\circ}_{2}$ | y7P${}^{\circ}_{3}$ | y7P${}^{\circ}_{4}$
---|---|---|---
Decay Rate $\Gamma$ | $147.5$ | $148.4$ | $148.4$
Statistical Uncertainty | $1.1$ | $0.6$ | $0.5$
Truncation Error | $1.0$ | $0.5$ | $0.4$
Hyperfine Quantum Beats | $0.15$ | $0.07$ | $0.07$
Zeeman Quantum Beats |  |  |
$\mathcal{B}_{x}$ | $0.05$ | $0.03$ | $0.02$
$\mathcal{B}_{y}$ | $0.3$ | $0.26$ | $0.5$
$\mathcal{B}_{z}$ | $0.09$ | $0.08$ | $0.05$
Laser Polarization | $0.26$ | $0.19$ | $0.07$
Pulse Pileup | $0.017$ | $0.014$ | $0.05$
Differential Nonlinearity | $0.017$ | $0.018$ | $0.018$
Time Calibration | $0.003$ | $0.003$ | $0.003$
Systematic Uncertainty, Total | $1.1$ | $0.6$ | $0.6$
Total Uncertainty | $1.6$ | $1.0$ | $0.8$
Table 4: Measured decay rates $\Gamma$ and $1\sigma$ error budget for the Cr
y7P${}^{\circ}_{J}$ levels.

###  V.0.1 Pulse Pileup

In order to minimize missed counts due to coincident events, or “pulse
pileup”, we typically collect data with roughly 1 count per 1000 excitation
cycles. Fluorescence photons are counted using a multihit-capable event timer
with a dead time of 650 ps. The photodetector has a maximum average count rate
of 80 MHz. Therefore, we correct the raw signal counts $N_{i}$ in each time
bin $i$ to obtain $N_{i}^{\prime}$ corrected counts per time bin using

$N_{i}^{\prime}=\frac{N_{i}}{1-\frac{1}{N_{\rm{cycle}}}\sum_{j=i-k_{d}}^{i}N_{j}}$
(1)

where $N_{\rm{cylcle}}$ is the total number of excitation cycles, and
$k_{d}=t_{\rm{bin}}\times 80\,$MHz is the number of time bins of width
$t_{\rm{bin}}$ in the detector-limited deadtime [52]. We estimate the error
$\delta\Gamma$ due to this correction by analyzing the data assuming the
detector has no response after the first detection of an excitation cycle. The
difference is less than $4\times 10^{-4}$ fractional uncertainty
$\delta\Gamma/\Gamma$ for our measurements.

###  V.0.2 Fit Procedure and Truncation Error

After correcting for pulse pileup, background counts $N_{i}^{\rm{bg}\prime}$
are subtracted from the signal counts $N_{i}^{\rm{sig}\prime}$, and the result
is fit to a single decaying exponential plus offset

$N_{i}^{\rm{sig}\prime}-N_{i}^{\rm{bg}\prime}=Ae^{-\Gamma t_{i}}+c.$ (2)

As an example, the pulse-pileup-corrected data measuring the
y7P${}^{\rm{o}}_{4}$ decay are shown in upper portion of Fig. 2.

The single largest systematic uncertainty on the measured decay rate is the
choice of the start time $t_{\rm{start}}$ of the fit (sometimes called
truncation error). We vary $t_{\rm{start}}$, setting this parameter to the
start of each $t_{\rm{bin}}=80$ ps time bin over a 12 ns range which
corresponds to between about 30 % to 5 % of the peak observed counts (Fig. 2
lower portion). For these late times, the influence of the excitation laser on
the decay is negligible. In an effort to reduce possible systematic error due
to an arbitrary choice of $t_{\rm{start}}$, we assign a value for $\Gamma$ by
taking the average fitted $\Gamma$ over this 12 ns range of start times,
weighted by the nonlinear least-squares $1\sigma$ confidence interval when
fitting to Eq. (2). We assign a statistical uncertainty equal to the median
$1\sigma$ nonlinear least-squares fit uncertainty in the range, and assign a
truncation uncertainty equal to the standard deviation of the fitted $\Gamma$
values.

The truncation error is observed to typically be of comparable magnitude to
the nonlinear least-squares fit confidence interval for $\Gamma$, and
therefore decreases with improved counting statistics. We minimize the effects
of truncation error by applying the fit to the combined data set of all runs
taken under standard conditions. We also analyzed each data run individually
and took a weighted average of the fitted $\Gamma$. That analysis obtained the
same results for each state within the fit confidence intervals.

Figure 2: (Top) Histogram of counts as a function of time used to determine
decay rate $\Gamma$ for the y7P${}^{\rm{o}}_{4}$ state (358 nm center
wavelength). The histogram bin width is 80 ps. The line labeled “fit” is a
representative fit to Eq. (2) with $t_{\rm{start}}\,=\,40$ ns. (Bottom) Fitted
$\Gamma$ as a function of $t_{\rm{start}}$, with error bars indicating the fit
uncertainty (blue line). The range of $t_{\rm{start}}$ chosen for our analysis
and the standard deviation of fitted $\Gamma$ over this range is indicated by
the red line. The orange band indicates the current consensus range for
$\Gamma$ in the NIST atomic spectral database [14].

###  V.0.3 Hyperfine Quantum Beats

The laser pulse has a full width at half maximum of 16 ns, corresponding to a
Fourier limited linewidth of about 100 MHz. Decay rate measurements are taken
with the laser tuned to the peak in fluorescence which nominally corresponds
to resonance with the 52Cr isotope (84 % natural isotopic abundance).
Hyperfine quantum beats are nonexistent for the nuclear spin $I=0$ 52Cr
isotope. However, hyperfine splitting is present in 53Cr ($I=3/2$, 9.5 %
natural isotopic abundance), and the 52Cr–53Cr isotope shift is sufficiently
small that several 53Cr lines are not resolved from the 52Cr line within the
Doppler width of the beam (roughly 20 MHz) for all three y7P${}^{\rm{o}}_{J}$
states considered here (Fig. 1).

The hyperfine structure of the y7P${}^{\rm{o}}_{2,3}$ states were measured by
Becker et al. [53]. The smallest hyperfine interval is 101 MHz
($J=2,F=1/2-3/2$), comparable to the Fourier limited laser linewidth. While
unlikely, we cannot a prior rule out the possibility of a small contamination
of the signal due to hyperfine quantum beats.

We test for systematic errors due to hyperfine quantum beats by varying the
laser detuning by $+20\,$MHz and $-20\,$MHz from the 52Cr fluorescence peak.
These detunings correspond to roughly 10 times the root-mean-square error of
the laser’s cavity transfer lock. This tests the possibility of different
laser frequencies exciting superpositions of 53Cr hyperfine levels with
greater or less efficiency. In these tests, we detect no statistical
difference in the fitted $\Gamma$. The uncertainties obtained by linear
regression are all $\delta\Gamma/\Gamma<0.15$ %.

Attempts were also made to measure decay rates by exciting 54Cr ($I=0$, 4.3 %
natural isotopic abundance), which is shifted sufficiently from 53Cr to rule
out potential hyperfine quantum beats. However, a single run could only
provide a statistical uncertainty of roughly $10$ %, which was deemed
insufficient.

###  V.0.4 Zeeman Quantum Beats

After truncation error, the largest uncertainty in our measurement is the
presence of Zeeman quantum beats. There is no polarizer in the imaging optics,
and we did not make an effort to cancel the ambient magnetic field in the
detection region, which is measured to be
$(\mathcal{B}_{x},\mathcal{B}_{y},\mathcal{B}_{z}$) $\approx$ (0.0, 0.2, 0.4)
G by a three axis Hall probe (1 G $\equiv\,0.1$ mT). The interaction region is
surrounded by three pairs of Helmholtz coils which produce a field uniform to
3 % over the detection volume. We test for systematic errors due to Zeeman
quantum beats by applying a variable magnetic field along each axis between
$\pm 5$ G.

As expected for our geometry (excitation light polarized along $\hat{z}$ and
imaging along $\hat{x}$), when varying $\mathcal{B}_{x}$ and
$\mathcal{B}_{z}$, we observe no statistical difference in the fitted
$\Gamma$. We estimate error due to these magnetic field components by linear
regression assuming an uncertainty of 0.2 G. Zeeman quantum beats are
definitively observed when $\mathcal{B}_{y}$ is varied, which produce an
inflated truncation error due to the fit amplitude $A$ of Eq. (2) oscillating
with fit start time $t_{\rm{start}}$. For each state, the fitted $\Gamma$ is
observed to reach a local extremum around $\mathcal{B}_{y}\approx 3$ G; we
therefore empirically fit the fitted $\Gamma$ to a quartic function of
$\mathcal{B}_{y}$ and assume an uncertainty in $\mathcal{B}_{y}$ equal to its
measured value of 0.2 G to estimate the error in $\Gamma$.

Figure 3: Fitted decay rate $\Gamma$ as a function of applied magnetic field
$\mathcal{B}_{y}$. The line shows a quartic polynomial determined by a least-
squares fit to the data. Error bars indicate one standard error.

We attempted to model the quantum beats for all three decays. Because the
initial Zeeman phase of the excited state is unknown, it is difficult to
predict the quartic fit curves in Fig. 3 a priori. Nevertheless, given that
the decay rates and transition wavelengths are roughly equal for all three
states, the reduced matrix elements are all roughly the same. The relative
scale of the curves in the panels in Fig. 3 is then set by the relative
Clebsch-Gordan coefficients for the decays that can produce quantum beats. Our
model shows that the amplitude of the beats from y7P${}^{\rm{o}}_{2}$ should
be roughly half that of the y7P${}^{\rm{o}}_{3}$ and y7P${}^{\rm{o}}_{4}$,
which are roughly expected to be identical. These expectations in line with
the fit curves of Fig. 3.

As the ambient magnetic field is not exactly collinear with the applied laser
polarization, we further test for Zeeman quantum beats by intentionally
rotating the laser polarization to lie along the $\hat{x}$ axis. In this
configuration, we again detect no statistical difference in the fitted
$\Gamma$. The fractional uncertainty obtained by linear regression are all
$\delta\Gamma/\Gamma<0.1$ %

###  V.0.5 Time Calibration

The multievent timer has have a differential nonlinearity (that is, non-
cumulative uncertainty in the bin width of the time digitizer) of less than 1
% [48]. This amounts to an uncertainty of 800 fs, or about $8\times 10^{-5}$
fractional uncertainty in the decay rate. Of secondary concern is the
nonlinearity of the time base of the electronics. The timing for the
experiment is referenced to the internal clock of the TCSPC electronics, which
has a specification of $10^{-5}$ stability; we assign a systematic error of
twice this value.

##  VI Comparison of results and conclusion

Method | Decay Rate $\Gamma(\times 10^{6}$ s-1) | Reference
---|---|---
| y7P${}^{\rm{o}}_{2}$ | y7P${}^{\rm{o}}_{3}$ | y7P${}^{\rm{o}}_{4}$ |
Experiment | 147.5(1.6) | 148.4(1.0) | 148.4(0.8) | this work
Theory | 140 | 141 | 145 | this work
Experiment | 174(17) | 165(17) | 164(16) | [16]
Experiment | 152(7) | 152(7) | 152(7) | [17]
Experiment | 158(5) | 140(4) | 144(4) | [5]
Compilation | 162(16) | 150(15) | 148(15) | [14]
Table 5: Comparison of our measured and calculated decay rates to values from
previous works. Values in parentheses are the combined $1\sigma$ statistical
and systematic uncertainty.

Here we have presented laser spectroscopy of the y7P${}^{\rm{o}}_{J}$ states
of Cr I with a comparison to theory. We measure the isotope shifts to roughly
0.5 % uncertainty. This appears to be the first reported measurement of these
values. Table 5 compares our measured and calculated decay rates as well as
prior reported values. Our measured decay rates uncertainties are roughly a
factor of three better than the next smallest reported uncertainty. We find
excellent agreement, 2 % to 5 %, between theory and experiment decay rates,
therefore, the differences for the matrix elements are only 1 % to 2.5 %. We
use experimental energies in decay rate calculations. As noted above, slightly
worse accuracy for $J=3$ and $J=2$ is explained by the sensitivity of the
mixing with the ${}^{5}\rm{P}^{\circ}$ states to the saturation of the CI
space as described above.

We verified that the $J=4$ matrix element value is numerically stable with
addition of more configurations. We also tested the accuracy of the effective
Hamiltonian by carrying out an additional computation where $H_{\rm{eff}}$ is
constructed using second-order perturbation theory, resulting in only 1.7 %
difference for the $J=4$ matrix element. Therefore, such excellent agreement
of the $J=4$ value with experiment confirms our calculations are robust to
large (11 %) normalization correction, which is important for further
calculations of various properties of atoms and ions with 5 to 6 valence
electrons.

These experiments demonstrate the operation of a cryogenic buffer gas beam
source, laser system, and detection system suitable for spectroscopic studies
of a variety of atomic and molecular species. We intend to use this apparatus
to next perform spectroscopic characterization of the MgF molecule toward
laser cooling and trapping. With minimal modifications, this system could also
perform similar studies of more complex molecules such as MgNC, which has been
proposed as a candidate for precision measurements of nuclear spin-dependent
parity violating physics [54, 55].

###### Acknowledgements.

The authors thank NIST colleagues Zeeshan Ahmed, Steven Buntin, Tanya Dax,
Patrick Egan, James Fedchak, Gerald Fraser, Nikolai Klimov, Jabez McClelland,
William Phillips, Joe Rice, Julia Scherschligt, Ian Spielman, Stephan
Stranick, and Wes Tew for providing equipment used in these experiments. We
thank Chris Billington for software assistance, and thank Mahder Abate and
Alessandro Restelli for help with AOM switching electronics. We thank Charles
Clark for helpful discussions. We acknowledge support from the National
Institute of Standards and Technology, ONR Grant No. N00014-20-1-2513, US NSF
Grant No. PHY-2012068, and by the Russian Science Foundation under Grant No.
19-12-00157. This research was supported in part through the use of University
of Delaware HPC Caviness and DARWIN computing systems: DARWIN - A Resource for
Computational and Data-intensive Research at the University of Delaware and in
the Delaware Region, Rudolf Eigenmann, Benjamin E. Bagozzi, Arthi Jayaraman,
William Totten, and Cathy H. Wu, University of Delaware, 2021, URL:
https://udspace.udel.edu/handle/19716/29071.

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