# Mapping the Universe with Gamma-Ray Bursts

Istvan Horvath1, Zsolt Bagoly1,2, Lajos G. Balazs3,4, Jon Hakkila5, Zsuzsa
Horvath1 Andras Peter Joo4, Sandor Pinter1, L. Viktor Tóth4,6, Peter Veres7,8,
and Istvan I. Racz1
1University of Public Service, Budapest, Hungary
2Department of Physics of Complex Systems, Eötvös University, Budapest,
Hungary
3Konkoly Observatory, Research Centre for Astronomy and Earth Sciences,
Budapest, Hungary
4Department of Astronomy, Eötvös Loránd University, Budapest, Hungary
5Department of Physics and Astronomy, The University of Alabama in Huntsville,
Huntsville, AL, USA
6University of Debrecen, Faculty of Science and Technology, Egyetem tér 1,
H-4032 Debrecen, Hungary
7Department of Space Science, University of Alabama in Huntsville, Huntsville,
AL 35899, USA
8Center for Space Plasma and Aeronomic Research, University of Alabama in
Huntsville, Huntsville, AL 35899, USA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYc; in original form ZZZ)

###### Abstract

We explore large-scale cosmic structure using the spatial distribution of 542
gamma-ray bursts (GRBs) having accurately-measured positions and spectroscopic
redshifts. Prominent cosmological clusters are identified in both the northern
and southern galactic hemispheres (avoiding extinction effects in the plane of
the Milky Way) using the Bootstrap Point-Radius method. The Northern Galactic
hemisphere contains a significant group of four GRBs in the redshift range
$0.59\leq z\leq 0.62$ (with a Bootstrap probability of $p=0.012$) along with
the previously-identified Hercules–Corona Borealis Great Wall (in the revised
redshift range $0.9\leq z\leq 2.1$; $p=0.017$). The Southern Galactic
hemisphere contains the previously-identified Giant GRB Ring ($p=0.022$) along
with another possible cluster of $7-9$ GRBs at $1.17\leq z\leq 1.444$
($p=0.031$). Additionally, both the Hercules–Corona Borealis Great Wall and
the Giant GRB Ring have become more prominent as the GRB sample size has
grown. The approach used here underscores the potential value of GRB
clustering as a probe of large-scale cosmic structure, complementary to galaxy
and quasar clustering. Because of the vast scale on which GRB clustering
provides valuable insights, it is important that optical GRB monitoring
continue so that additional spectroscopic redshift measurements should be
obtained.

###### keywords:

(cosmology:) large-scale structure of Universe - methods: data analysis -
methods: statistical - (transients:) gamma-ray bursts - (stars:) gamma-ray
burst: general - cosmology: observations

††pubyear: 2023††pagerange: Mapping the Universe with Gamma-Ray Bursts -9

##  1 Introduction

Since their discovery, gamma-ray bursts (GRBs) have continued to pique the
interest of scientists and captivate them with their enigmatic origins and
remarkable energy outputs (Klebesadel et al., 1973). Despite the passage of
decades, formulating a comprehensive theory that elucidates their genesis and
satisfactorily explains every observed aspect remains an ongoing challenge
(Pe’er, 2015; Zhang, 2018; Bošnjak et al., 2022).

GRBs are widely acknowledged as one of the most intense and brilliantly
luminous phenomena in the cosmos. They are believed to result from either the
implosion of massive stars (Woosley, 1993; Woosley & Bloom, 2006) or from the
merging of binary compact objects (Berger, 2014). Because of their extreme
luminosities, a thorough examination of their celestial coordinates, along
with their intrinsic properties, holds promise to shed light on the Universe’s
largest-scale structures (Meszaros, 2006; Andrade et al., 2019; Horvath et
al., 2020; Tugay & Tarnopolski, 2023).

The GRBs’ angular distribution was also studied over the past few decades.
Generally, GRBs have been found to be uniformly distributed on the sky (Briggs
et al., 1996; Balázs et al., 1998, 1999; Mészáros et al., 2000b; Magliocchetti
et al., 2003; Vavrek et al., 2008; Tarnopolski, 2015, 2016, 2019; Řípa &
Shafieloo, 2019; Andrade et al., 2019; Tarnopolski, 2022), although some
subsamples appear to deviate from isotropy (Balázs et al., 1998; Cline et al.,
1999; Mészáros et al., 2000a, b; Litvin et al., 2001; Magliocchetti et al.,
2003; Vavrek et al., 2008; Řípa & Shafieloo, 2017; Haslbauer et al., 2023;
Kumar Aluri et al., 2023). Apart from the Hercules–Corona Borealis Great Wall
identified by our group (Horváth et al., 2014, 2015), one of the most
significant recent developments in this field has been the detection of a
large-scale structure in the redshift range of $0.78<z<0.86$, dubbed the
"Giant GRB Ring", which appears to be consists of a circular arrangement of
GRB positions at a distance of about 6 billion light-years (Balázs et al.,
2015), although have been some debates about the existence of that structure
(Fujii, 2022b). It appears to be somewhat smaller ($1.72$ Gpc) than the
Hercules–Corona Borealis Great Wall ($2-3$ Gpc), but its presence in the data
has been confirmed by further, elaborate statistical analysis (Balázs et al.,
2018). While the physical and astrophysical nature of both structures are not
conclusively understood (cf. the discussions in Balázs et al., 2015, 2018),
their existence may provide a challenge to standard assumptions about
universal homogeneity and isotropy (i.e. the cosmological principle; see
however Li & Lin 2015). This means that studying structures like these is of
high scientific importance (Mandarakas et al., 2021; Marchã & Browne, 2021;
Migkas et al., 2021; Fujii, 2022a).

Aside from GRBs, other large structures have been discovered during the 21st
century. These have included the Sloan Great Wall (which is a giant filament
of galaxies) with a size of $\sim$ 0.4 Gpc (Gott et al., 2005), the Huge Large
Quasar Group with a size of 1.2 Gpc (Clowes et al., 2013), and the Giant
Quasar Arc with a size of 1 Gpc (Lopez et al., 2022). Both GRB-defined
structures (the Giant GRB Ring and the Hercules–Corona Borealis Great Wall)
exceed these in size.

Overall, these findings suggest that GRBs can serve as powerful probes of the
large-scale structure of the Universe (Tugay & Tarnopolski, 2023), providing
valuable information about the distribution of matter and the evolution of
cosmic structures over time (Racz et al., 2017). Due to their large absolute
brightness they can be observed at very large cosmological distances.
Consequently, the study of their locations in the sky coupled with their
properties can provide insights into the large-scale structure of the
Universe.

##  2 Data and Methods

In this study we use a database of gamma-ray bursts (GRBs) having measured
positions on the celestial sphere, optical afterglows, and spectroscopic
redshifts; a majority of which were detected by NASA’s Swift and Fermi
experiments. Here we use the same dataset of Horvath et al. (2022) and Bagoly
et al. (2022), which is based on the dataset found in Horvath et al. (2020).
The redshifts have been primarily obtained from the Gamma-Ray Burst Online
Index (GRBOX) database; more than five hundred spectroscopic redshifts have
been measured for GRBs as of today. It should be noted that, while the
datasets were derived from different sources in the public domain, the GRBOX
database heavily relies on notes from the Gamma-ray Coordination Network.
GRBOX also relies on a publicly available dataset compiled by Joachim
Greiner111https://www.mpe.mpg.de/ jcg/grbgen.html, which provides extensive
information on nearly all GRBs observed by any instrument. A thorough review
was conducted during the assembly of the GRBOX catalog database222 Published
by the Caltech Astronomy Department:
http://www.astro.caltech.edu/grbox/grbox.php to exclude observations with
significant systematic uncertainties. The redshift data used in Horvath et al.
(2022); Bagoly et al. (2022) and Horvath et al. (2020) exclusively consisted
of spectroscopic redshifts, disregarding photometric redshifts and redshift
estimates based on Ly-alpha limits because these data tend to introduce
significant radial distance uncertainties. These uncertainties generally
exceed several hundred Mpcs.

We have supplemented the data from Horvath et al. (2022) and Bagoly et al.
(2022) with new observations, obtained up until August 31st, 2022, that have
resulted in a total of 542 GRBs with accurate position and spectroscopic
redshifts (up to $z=8.26$). We have used Joachim Greiner’s table and relevant
Gamma-ray Coordination Network messages to extend the dataset so that it
includes the most recent data.

###  2.1 The dataset

The dataset used here is the most extensive one currently available for global
studies, containing as it does precise GRB distances obtained from
spectroscopic redshifts. It may be difficult to improve upon this data set,
since the peak rate of GRBs with spectroscopic redshifts was $\approx$120/year
after 2006, but has decreased rapidly since then, with each year yielding only
$\approx 90\%$ of the previous year’s redshift observations. This trend has
only been disrupted in recent years by a few dedicated optical campaigns, but
still, the current rate of new redshift observations remains embarrassingly
low, preventing the possibility of significant statistical advancements in the
near future (Bagoly et al., 2022).

Figure 1: The cumulative distribution function (CDF) of log10(1+z) for GRBs in
this study. Purple indicates the 262 GRBs found in the northern galactic sky,
while green indicates the 280 GRBs found in the southern galactic sky. The
difference between the two CDFs is insignificant (see text). Figure 2: The
cumulative distribution function (CDF) of the galactic longitudes for GRBs in
this study. Purple (green) indicates the distributions of GRBs found in the
northern (southern) galactic hemispheres. The difference between the two CDFs
is insignificant ($p=0.32$).

Two GRBs have been removed from the dataset based on erroneous measurements.
Bagoly et al. (2015) used the Spatial Two-Point Correlation Function to
identify an interesting neighbouring pair of GRBs (GRB 020819B and GRB
050803). These two GRBs created a prominent peak in the Spatial Two-Point
Correlation Function at a distance of $\approx 56$ Mpc. Interestingly, both
GRBs’ redshift values were later questioned in the literature (more than a
decade after the events!). The Optically Unbiased GRB Host (TOUGH) survey
measured a redshift of $z=3.5$ for the host galaxy of GRB 050803 which is
completely inconsistent with the previous value of $z=0.422$ (Schulze et al.,
2015). Similarly, Perley et al. (2017) analyzed the spiral galaxy at a
redshift of $z=0.41$ that was believed to be the host galaxy of GRB 020819B.
By employing VLT MUSE and X-shooter observations, the authors concluded that
the galaxy was an unrelated foreground galaxy. Consequently, given that both
previous estimated distances are likely incorrect, we have removed these GRBs
from the dataset.

In the first three figures we demonstrate the distributions of spatial
parameters. The cumulative distribution of the redshifts of the 542 GRBs can
be seen in Fig. 1. Of these, 262 are found in the northern galactic
hemisphere, while the remaining 280 are found in the southern galactic
hemisphere. The two-sample Kolmogorov-Smirnov test of these redshift
distributions shows no significant difference between the two hemispheres
($D=0.098$, $p=0.151$). Fig. 2 shows the galactic longitude ($l$) coordinate
distributions for both hemispheres. Because the maximum deviation of these
distributions is $D=0.082$, the two-sample Kolmogorov-Smirnov test gives us a
p-value of $p=0.325$, indicating that we cannot reject the hypothesis that the
two samples are drawn from the same distribution. For the galactic longitude
($b$) coordinates, the KS test result similarly shows the two hemisphere
distributions are not significantly different ($D=0.109$, $p=0.082$), see Fig.
3.

The sky distribution of the 542 GRBs can be seen in Fig. 4. There are only a
very few spectroscopic redshift observations near the galactic plane as the
visible gap for the low galactic $|b|$ coordinates shows. This reduced
observational probability area will generate low bootstrap point-radius counts
hence it will introduce large Poisson noise into the statistics. Avoiding this
we analyze the two galactic hemispheres separately. This separation will
reduce the method’s sensitivity for any clustering around the galactic
equator, where the low observational probability (which showed a low number of
GRBs) would anyway prevent any statistically reliable detection.

Figure 3: The cumulative distribution function (CDF) of the galactic latitude
magnitudes for GRBs in this study. Purple (green) indicates the distributions
of GRBs found in the northern (southern) galactic hemispheres. The difference
between the two CDFs is not significant.

###  2.2 Method of analysis

In order to test for the existence of clustering in the current database while
accounting for known sampling biases, we apply the bootstrap point-radius
method. This method was described in Sec. 5. of Horváth et al. (2014). A
similar analysis is performed here, albeit for a much larger dataset (as
explained in Sect. 3).

The bootstrap point-radius method searches for an overdensity in the point
distribution. We choose a slice of $n$ consecutive GRBs in their radial ($z$)
distribution, starting with position $k$, and determine the number of GRBs
within a volume defined by a spherical surface $A$. We refer to this surface
as a spherical cap. We look for the $K$ maximum number of GRBs within the
spherical cap, for all $k$ radial starting position and for all spherical
cap’s center position on the sky, therefore $K$ will depend on the given $A$
cap area and $n$ slice size only.

During the analysis we assume that the sky exposure is independent of $z$
(e.g. this was shown in Bagoly et al. (2022)). Hence the same sky exposure
factor was assumed within a given spherical cap for the slice and the other
GRBs outside the slice, and differences in the spatial distribution could be
analyzed.

For the spherical cap centers a Healpix partition (Górski et al., 2005) was
chosen with $N=6$, providing 49152 quasi-isotropic positions. The average
distance between them is $\approx 10$ times smaller than the average distance
between the GRBs, providing good coverage.

We do a Monte Carlo simulation by mixing the radial and angular positions 1000
times and repeating the process. Here we also select the largest number of
GRBs found ($K$) within a given spherical cap size $A$ and slice size $n$, and
compare them with the real $K$ value (for more details about this method see
our previous works, Horváth et al. 2014, 2015, where the same method was
applied). We should mention that for similar $A$ and $n$ values the
distribution is expected to be similar (c.f. Figs. 5. and 8.).

The method will be sensitive to a density enhancement in a radial slice within
a given angular area. Clearly, the distribution of $K$ depends on the angular
two-point correlation function filtered by the spherical cap scale. The
numbers inside and outside the slice will depend non-linearly on the
correlation function because of the maximum cutoff. Also, the geometrical
change in the angular diameter distance with the cosmological co-moving
distance makes the real connection with the spatial two-point correlation
function more complicated.

Figure 4: The sky distribution of the 542 analysed GRBs.

##  3 Results

Figure 5: The bootstrap probabilities from the northern hemisphere on the (n,
A) plane. White $p\geq 0.08$, light blue $0.08>p\geq 0.05$, blue $0.05>p\geq
0.03$, red $0.03>p\geq 0.01$. Only blue and red blocks are significant.

###  3.1 Northern galactic hemisphere

Part of the results, the $K(n,A)$ function, for the northern galactic
hemisphere can be seen in Table 6. The complete $K(n,A)$ function can be found
in the supplement materials. To calculate the probability of getting the
number $K$ (for a certain $n$ and $A$), we apply the bootstrap method. We do
this by retaining the sky position of the GRB and replacing its redshift with
that of another GRB selected randomly from the sample. We do this for all 262
GRBs in the northern galactic hemisphere. We repeat the process as discussed
in Sect. 2.2 with this configuration and recalculate $K$ (with the same $n$
and $A$). We repeat this process 1000 times, counting how many times $K$ is
greater than or equal to the measured value of $K$. We denote this number as
$P$, and $P/1000$ estimates the probability of measuring a count of $K$ or
larger.

Table 7 summarizes the probabilities of our analysis. Figure 5 shows also
these results. There are three separated area in the $n$, $A$ plane on which
clustering is found to occur, indicative of significant deviations from
isotropy/homogeneity. Notice that the light blue dots are not significant.

The first of these occurs for $n=5$ and $A=0.0628$ ($0.02\times\pi$
steradian), with an observed $K=4$. Only one case ($5$, $0.0628$) reached the
significant level in our analysis, with the bursts found within a redshift
range of $0.59\leq z\leq 0.62$. Fig. 6 shows the locations of these 5 GRBs on
the sphere of the sky. We ran another 3000 bootstraps and only 47 runs (from
the total 4000 runs) reached $K=4$, indicating that the probability of getting
this fluctuation by chance is only $0.01175$. We also checked the $n=5$ and
$A=0.0628$ parameters on the southern hemisphere. For 4000 bootstrap runs only
37 reached $K=4$, therefore the probability of getting this result by chance
is even smaller ($p=0.00925$). Table 1 shows the properties of these 5 GRBs.
Four out of the five GRBs in this group are positioned close to each other,
raising the question of whether these GRBs or their host galaxies have
something in common. We examined this by looking for similar properties of the
GRBs and their host galaxies in literature data. We found $T90$ durations
ranging from $\sim 35-365s$ classifying all these as long gamma-ray bursts
(LGRBs), with peak energies of similar magnitudes $E_{p,obs}$ from $\sim
95-137keV$ (Xue et al. (2019), Tarnopolski & Marchenko (2021)). Bi et al.
(2018) calculated a star formation rate $SFR=81yr^{-}1$ and a metallicity
$log(Z/Z_{\odot})=-0.4$ for the host galaxy of GRB070612, supporting the high-
mass collapsar model for LGRBs. GRB Coordinates Network reports, however,
suggest differences in the metallicities of the four GRBs, with some metal
lines only appearing in some of the spectra (Mg I, Mg II and Fe II in the case
of GRB150323A, and Ca H and K, and C I in the case of GRB110106B (Chornock et
al. (2011), Cucchiara et al. (2010), Cenko et al. (2007), Perley & Cenko
(2015))).

Figure 6: The 5 GRBs on the northern hemisphere within $0.59\leq z\leq 0.62$.  GRB name | z | d (Mpc) | l | b
---|---|---|---|---
GRB150323A | 0.593 | 2209 | 174.802 | 36.297
GRB100213B | 0.604 | 2244 | 176.928 | 33.339
GRB050525A | 0.6063 | 2251 | 54.953 | 15.545
GRB070612 | 0.617 | 2285 | 183.626 | 30.051
GRB110106B | 0.618 | 2288 | 172.908 | 40.449
Table 1: This table shows the 5 GRBs on the northern hemisphere in the
$0.59\leq z\leq 0.62$ redshift range. GRB ID, redshift (z), co-moving distance
(d), galactic coordinates (l and b).

The second suspicious area is around $41\leq n\leq 47$, $1.6\leq A\leq 1.9$,
with K being typically $29\leq K\leq 32$. Although the bootstrap probabilities
are close to be significant in a bigger (n, A) area ($36\leq n\leq 57$,
$1.57\leq A\leq 1.95$), but only two neighbouring points reach the minimal
significant limit (5%); $n=43$, $A=1.7$, $p=0.038$ and $n=43$, $A=1.728$,
$p=0.048$. This cluster is found in the redshift range $0.9\leq z\leq 1.3$.
Because it is only marginally significant ($p\leq 0.04-0.08$), with only two
points having small significance, we consider this cluster as only rising to
the level of being suspicious. However, as we will see, the third area
includes this part of the analysed parameter space in both redshift and sky
position. It is worth noting that the sky area of this cluster corresponds to
the position of the Hercules–Corona Borealis Great Wall (Horváth et al., 2014,
2015; Horvath et al., 2020).

The third area indicates anisotropies on a large angular scale, about
$n=97-113$ and $1.9\leq A\leq 2.1$, where the probabilities are less than
$1-5\%$. Table 2 shows the four most significant points in the ($n$, $A$)
parameter space. For the $n=104$, $A=1.948$ we run another 3000 bootstraps and
only 71 runs (from the total 4000 runs) reached $K=69$, indicating that the
probability of getting this fluctuation by chance is only 0.01775. Nineteen
checked points had $p$ less than $3\%$ (red squares in Figure 5).

Fig 7 shows the 104 GRBs distribution on celestial sphere. Sixty nine of them
are concentrated in $31\%$ of the hemisphere area. The redshift range of this
cluster is about $0.9\leq z\leq 2.1$. This volume contains the previously-
identified Hercules–Corona Borealis Great Wall (HCBGW) along with the second
suspicious region (around $n=43-47$, $A=1.6-1.7$, $0.9\leq z\leq 1.3$), which
was discussed in this section previously.

n | A | p | K | z1 | z2
---|---|---|---|---|---
101 | 1.948 | 0.02 | 67 | 0.959 | 2.1
104 | 1.948 |  red0.017 | 69 | 0.938 | 2.1
104 | 2.011 | 0.021 | 70 | 0.938 | 2.1
108 | 1.948 | 0.021 | 71 | 0.885 | 2.1
Table 2: This table shows the 4 highest probabilities on the zone of
$n=97-113$ and $A=1.9-2.1$. $n$ is the number of GRBs in the redshift range,
$A$ is the circle size in steradian, $p$ is the bootstrap frequency, $K$ is
the maximal GRB’s number in the circle, and $z1$ and $z2$ are the minimal and
maximal redshifts. Figure 7: The northern hemisphere’s (extended) Hercules-
Corona Borealis Great Wall. The redshift range is $0.938\leq z\leq 2.101$.

###  3.2 Southern galactic hemisphere

The results for the southern galactic hemisphere can be seen in Table 8,
namely the $K$s. The entire results of the analysis can be found in the
supplement materials. The southern galactic hemisphere probabilities are shown
in Table 9. Fig. 8 also demonstrates the bootstrap probabilities.

Figure 8: The bootstrap probabilities of the southern hemisphere on the ($n$,
$A$) plane. White $p\geq 0.08$, light blue $0.08>p\geq 0.05$, blue $0.05>p\geq
0.03$, red $0.03>p\geq 0.01$. Only blue and red dots are significant.

The first statistically interesting clustering is identified by $n=19$,
$A=0.4396$, and $K=10$. The redshift range for this group is $0.748-0.8595$.
This structure has been already discovered by Balázs et al. (2015), and is
shown in Fig. 9. In Balázs et al. (2015) the redshift range was $0.78<z<0.86$.
Nine GRBs out of the 15 were in a small area. Now we have 10 GRBs out of the
19 in the same area. The significance is $96.3\%$, see the blue dot in Fig. 8,
and the blue number in Table 8 (0.037 in $n=19$, $A=0.4396$). It is still
significant, however not that much as Balázs et al. (2015) found.

Figure 9: The figure shows the 19 GRBs in the $0.748-0.8595$ redshift range
(red dots) in the southern hemisphere. This grouping was previously identified
as the Giant GRB Ring.

The second statistically interesting southern hemisphere structure is found
for $n=26$, $A=0.2199$, and $K=9$. The redshift range is $1.171<z<1.444$ and
the significance is 96.9%. The second most significant value in this area is
$p=0.035$ for $n=26$, $A=0.0942$, and $K=7$. The redshift range is the same.
Notice that $n=26$ in both cases. The sky distribution of this redshift range
is shown in Fig. 10. There is no previous reference to this cluster in the
literature. However, due to the marginal significance and the small number of
concentration (7 and 9) this could be only a statistical fluctuation.

The next significant area is about $n=38-49$ and $A=0.69-0.88$. The two most
significant part (the two red dots in Fig. 8) are $n=41$, $A=0.691$, $K=17$,
$p=0.024$ and $n=41$, $A=0.785$, $K=19$, $p=0.026$. In both cases the redshift
range is $0.553<z<0.8595$ (see Fig. 11). As one can see this area contains the
first significant area which was about $0.748-0.8595$ redshift (the Giant GRB
Ring), see red and salmon dots in Fig. 11.

Finally, the southern hemisphere contains a large significant concentration of
GRBs with n in the range $71\leq n\leq 77$ and most of the concentration
within an angular area of around 0.7 steradian. The most significant points in
this area of the $p(n,A)$ plane are listed in Table 3. Also see the 7 red dots
in the left middle in Fig. 8. For the sky distribution of this cluster see
Fig. 12

Figure 10: 26 GRBs in the southern hemisphere, within $1.171\leq z\leq 1.444$. Figure 11: The southern hemisphere GRB sky distribution within $0.553\leq z\leq 0.8595$, c.f. Fig. 9. The red dots are the same as in Fig. 9, the salmon dots are within $0.553\leq z\leq 0.748$. A grouping around the Giant GRB Ring is growing.  Figure 12: An even larger grouping around the Giant GRB Ring within $0.553<z<1.35$. The area is about 0.7 steradian. C.f. Figs. 9. and 11.  n | A | K | p | z1 | z2
---|---|---|---|---|---
71 | 0.6908 | 24 |  red0.022 | 0.553 | 1.155
72 | 0.6908 | 24 | 0.027 | 0.553 | 1.170
76 | 0.6908 | 25 |  red0.023 | 0.553 | 1.233
76 | 0.8792 | 29 | 0.025 | 0.553 | 1.233
77 | 0.6908 | 25 | 0.027 | 0.553 | 1.25
77 | 0.8792 | 29 | 0.031 | 0.553 | 1.25
82 | 0.6908 | 26 | 0.027 | 0.553 | 1.31
83 | 0.6908 | 26 | 0.035 | 0.553 | 1.325
83 | 0.8792 | 30 | 0.047 | 0.553 | 1.325
87 | 0.6908 | 27 | 0.031 | 0.553 | 1.350
Table 3: This table shows some points from the largest area of significant
field of the Figure 8. $n$ is the number of GRBs in the redshift range, $A$ is
the circle size in steradian, $K$ is the maximum number of GRBs in the circle,
$p$ is the bootstrap frequency, and $z1$ and $z2$ are the minimal and maximal
redshifts.

##  4 North–South differences

The isotropies of the two hemispheres can be compared to one another by
subtracting Table 8 from Table 6. The result, shown in Fig. 13, demonstrates
that the two hemispheres exhibit similar, large-scale isotropy over most of
the ($n$, $A$) plane, as indicated by the area colored white, where the
differences are between -2 and 6. This means the observed K distributions in
the ($n$, $A$) plane are similar in both hemispheres. However, there is a
large region in which the difference between the distributions is three times
larger than anywhere else. This suggests that, in the northern hemisphere,
there is a large area in which there are significantly more GRBs than one
would expect for a homogeneous, isotropic distribution.

The bootstrap probabilities can also be studied. For example, in the case of
$n=27$ and $A=0.503$, the number $K$ is 12 for both hemispheres. The bootstrap
probabilities are very similar. One can make 100 bootstrap runs and study the
$K$ distribution. Every bootstrap run has a largest number, which we call $K$
(see Sect. 2.2). This 100 $K$s distribution can be seen in Figure 14. In this
plot, blue refers to northern hemisphere data while red refers to southern
hemisphere data. In both cases, we plot 4 distributions (each having 100
runs). These runs do not differ very much from each other. This is also the
case in the whole white area in Fig. 13. The K distributions are similar in
most of the white area in the northern and southern hemispheres. Over most of
the sky, the differences between the northern and southern hemisphere $K$
distributions are minimal. However, that is not the case in the green and gray
areas in Fig. 13.

For example, the $K$ distributions are very different for $n=104$ and
$A=1.95$, as seen in Figure 15. In this case, for the observed 542 GRBs, $K$
is 69 in the northern hemisphere and 52 in the southern hemisphere. The
bootstrap frequency (reaching 69) in the northern hemisphere is 106 out of
5000 ($p=0.0212$). However, in the southern hemisphere, the bootstrap
frequency $K$ never reaches 62, therefore the frequency for 69 is zero. This
demonstrates that the two hemisphere distributions differ very much around the
2-steradian scale, the northern hemisphere having about 17 more GRBs in this
part of the sky.

Figure 13: The difference between the $K_{n}$ northern and $K_{s}$ southern
hemisphere $K(n,A)$ functions. Blue $K_{n}-K_{s}=-3$, red $K_{n}-K_{s}=-2$,
white $-2<K_{n}-K_{s}<6$, yellow $5<K_{n}-K_{s}<11$, green
$10<K_{n}-K_{s}<15$, grey $14<K_{n}-K_{s}$. Figure 14: The difference between
the northern (reds) and southern (blue) hemisphere K distributions in case
$n=27$ and $A=0.503$. Each step function represents a hundred bootstrap runs.
Figure 15: The difference between the northern (right) and southern (left)
hemisphere K distributions in a case $n=104$ and $A=1.95$.

##  5 Discussion

The discovery of the Sloan Great Wall (with a size of $\sim$ 0.4 Gpc) using
galaxy locations (Gott et al., 2005) ushered in a new age of identifying
large-scale universal structures via luminous objects. The association of
luminous quasars with galaxies has allowed even larger structures to be
identified, including the Huge Large Quasar Group (Clowes et al., 2013) and
the Giant Quasar Arc (Lopez et al., 2022) (each with a size of 1.2 Gpc). GRBs,
with their high luminosities, have extended this capability to even larger
structures. Studying the spatial distribution of GRBs, both a great wall
(Horváth et al., 2014) and a Giant GRB Ring (Balázs et al., 2015) were
identified.

Because it is hard to collect unbiased survey data for galaxies and quasars,
searches for large-scale groupings of these astronomical objects have relied
on studies limited to specific spatial regions. However, since GRBs are mostly
identified from large-scale surveys, it is easier to use them for examining
large-scale deviations from isotropy and homogeneity over the full sky. In
this manuscript, we have shown how deviations from large-scale universal
homogeneity can be studied using all available information pertaining to GRBs
with known redshifts. Although the search for GRBs has taken place with great
uniformity across the sky, a bias exists against measuring GRB redshifts close
to the galactic plane. Only a few GRBs with redshifts have been detectable in
this region. To minimize this known bias, our approach is to separately study
spatial distributions in the northern and southern galactic hemispheres.

Using the method described in Sect. 2.2 we find two suspicious anisotropic
regions in each hemisphere.

###  5.1 Northern hemisphere

The northern hemisphere contains a cluster of five GRBs in the narrow redshift
range ($0.59\leq z\leq 0.62$) (see Table 1). Of these five, four are found in
a small area (see Fig. 6).

The second suspicious redshift range is about $0.9-1.3$, which contains
$n=43-47$ GRBs, out of them $30-33$ is in an $A=1.6-1.7$ steradian size
circle. These GRBs are in the same area of the sky, where the Hercules–Corona
Borealis Great Wall has been identified, except the HCBGW is farther away
($1.6<z<2.1$). However, the significance of this GRB group is weak in our
present analysis.

The third grouping contains the second group ($0.9<z<1.3$) since it spans the
$0.9-2.1$ redshift range. The sky position is the same for both case. The
bootstrap method shows significance in a huge area of the n, A parameter
space. $97<n<113$ and $1.9<A<2.1$. For example, 69 out of the 104 GRBs are
concentrated in the area around the Hercules and the Corona Borealis
constellations in the sky. The size of this volume (in co-moving distance) is
$2.5\times 3\times 3$ Mpc or $8\times 10\times 10$ Gly, which contains 40%
more GRBs than expected.

In conclusion, in the northern hemisphere we find a group of four GRBs and an
extended space of the Hercules–Corona Borealis Great Wall. Based on data
collected several years ago, the HCBGW appeared $3-4$ times smaller in radial
size than we now measure it to be. Now we find the radial size to be
comparable with the other two dimensions.

###  5.2 Southern hemisphere

There are four suspicious ranges in the southern hemisphere (see Fig. 8).

One is $n=26$, $A=0.2199$, and $K=9$. The redshift range is $1.171<z<1.444$
and the significance is $96.9\%$. See Fig. 10 for this group which contains 9
GRBs. However, because of the small group number (9) and a weak significance
($p=0.031$) this could happened by chance.

The other three significant ranges are boxed into each other. Therefore these
refer only to one GRB grouping.

As we discussed in Sect. 3.2 the redshift ranges are $0.748-0.8595$ ($K=10$),
$0.553<z<0.8595$ ($K=17$) and $0.553<z<1.2-1.3$ ($K=24-29$).

One, which was previously found by (Balázs et al., 2015), with a redshift
$0.75-0.86$, containing 10 GRBs out of 19 on 0.4396 steradian in the sky. The
second contains 17 GRBs (out of 41) including the previous 10. The third group
contains $24-29$ GRBs out of $71-77$ on a redshift range from 0.55 to about
$1.2-1.3$ covering 0.7 steradian (see Table 3). This volume contains the
previous two volumes.

The techniques described in this manuscript have identified probable large-
scale anisotropic cosmological structures using luminous GRBs. As the GRB
sample size has increased, these structures have become more pronounced and
new structures are beginning to appear. These results support the continued
monitoring of the sky for GRBs using survey satellites.

###  5.3 Binomial probabilities and number of tries

The Binomial Theorem can be used as an alternate approach to obtaining
clustering probabilities. For data tested many times, the overall binomial
probability is found by multiplying the individual binomial probability
obtained for a measurement by the number of independent trials. However, it is
hard to directly use the number of bootstrap trials to calculate a binomial
probability because a great many of these trials are not independent. We might
avoid this problem by employing Monte-Carlo analysis and re-analyzing the data
instead of identifying independent trials from our bootstrap procedure, but
the latter approach is preferred because uncertainty in the sky exposure
function makes the Monte Carlo approach a less reliable technique for this
analysis.

The southern sky contains 280 GRBs with measured redshifts. For every circle
size containing $n=100$ points, we made trials for 180 intervals. Since these
intervals overlap, only a handful of them are independent. For example, the
$k=1$ and $k=2$ intervals contain 100 GRBs of which 99 are the same. For $n=5$
there are 275 intervals, of which only 56 are totally independent. Similarly,
we tried 116 different $n$ values ($3\leq n\leq 118$), but these were not
independent because every trial with $n=16$ included two trials with $n=15$
and three trials with $n=14$, etc. Since there are only 280 southern
hemisphere GRBs in our sample, we can estimate the number of trials as being a
few hundred.

Due to overlap, we must similarly estimate the number of independent spherical
cap trials. We made 10,000 position trials representing every size of an
angular circle. For a circle with area 0.1 steradian, some 50 could be
categorized as being independent. For a circle of area 0.5 steradian, only
9-10 can be independent, and these are not independent from many previous
trials like the 0.44 or the 0.56 steradian circle sizes. Therefore, we can
estimate the number of trials also as a few hundred. Multiplying this with the
number obtained in the previous paragraph (also a few hundred) the number of
trials is estimated as being between $5\times 10^{4}$ and $2\times 10^{5}$.

With this information we can estimate the binomial probabilities for our
prominent clusters. In the northern hemisphere, the first significant result
occurs for $n=5$, $A=0.0628$, and $K=4$. The binomial probability with 5, 4
and $p=0.01$ (because $A=0.0628$ is one hundredth of the half sky) is $5\times
10^{-8}$. Since we made $5\times 10^{4}$ or $2\times 10^{5}$ trials, the
probability is still no larger than $p=0.01$. For the case of $n=43$, $A=1.7$,
and $K=30$, the individual $p=0.27$ and the binomial probability (for 43, 30)
is $6.9\times 10^{-9}$. Multiplying this by $5\times 10^{4}$ or $2\times
10^{5}$ results in $p=(4-14)\times 10^{-4}.$ For the case of $n=104$, $K=69$,
and $A=1.948$, we find $p=0.31$, for which the binomial probability is
$10^{-14}$, which is still smaller than $10^{-8}$. Table 4 contains the
estimated binomial and bootstrap probabilities for each of these significant
clusters.

Table 5 contains the similarly-calculated binomial estimated probabilities for
the southern hemisphere. It can be seen the binomial probabilities are
generally similar to or slightly less than than the bootstrap probabilities.

The last four entries in Table 4 identify binomial probabilities which are
much more smaller than the corresponding bootstrap probabilities. As the table
and Fig. 5 show, there are 87 more trials that show this very low estimated
probabilities.

The Cosmological Principle implies that the redshift and sky distribution of
GRBs in the northern and southern galactic hemispheres should be similar.
However, Fig. 13 demonstrates that they are not. Although more southern
galactic GRBs have been detected at present, many of the northern sky GRBs are
found within the large cluster located at $1\leq z\leq 2$. Bootstrapping
events from the northern sky causes us to choose many GRBs from this
concentration, which means most of the bootstrapped samples contain many GRBs
from this group. Fig. 15 demonstrates this effect, where the average bootstrap
numbers are larger (in average) with 10 in the northern hemisphere than in the
southern hemisphere. Other than this, Fig. 14 demonstrates that most samples
in the two hemispheres that are bootstrapped $K$ (with the same $n$ and $A$)
do not differ by much. This suggests that the bootstrap probabilities tend to
overestimate the significance of features relative to the values that are
obtained using the binomial estimate.

n | K | Arel | $p_{binom}$ | $p_{est}$ | num | $p_{bstr}$
---|---|---|---|---|---|---
5 | 4 | 0.01 | $5\times 10^{-8}$ | 0.0025-0.01 | 1 | 0.021
43 | 30 | 0.27 | $7\times 10^{-9}$ | 0.0004-0.0014 | 2 | 0.038
101 | 67 | 0.31 | $2\times 10^{-13}$ | $(1-4)\times 10^{-8}$ | 91 | 0.02
104 | 69 | 0.31 | $1\times 10^{-14}$ | < $10^{-8}$ | 91 | 0.017
104 | 70 | 0.32 | $5\times 10^{-14}$ | <$10^{-8}$ | 91 | 0.021
108 | 71 | 0.31 | $5\times 10^{-14}$ | <$10^{-8}$ | 91 | 0.021
Table 4: Binomial probabilities and the estimated probabilities (multiplied by the number of tries) for the northern hemisphere ($p_{est}$). K and n are the same as described in Sect. 2.2. Here, Arel is the relative area of the hemisphere, $p_{binom}$ is the binomial probability with p=Arel, n tries, and K successes. $p_{est}$ is the binomial probability multiplied by the number of independent tries, ’num’ is the number of significant signs in the neighboring area shown in Fig. 5, and $p_{bstr}$ is the bootstrap probability calculated in Sect. 3.1. Some of these are shown in Table 2.  n | K | Arel | $p_{binom}$ | $p_{est}$ | num | $p_{bstr}$
---|---|---|---|---|---|---
19 | 10 | 0.07 | $1.45\times 10^{-7}$ | 0.007-0.03 | 1 | 0.037
26 | 9 | 0.035 | $1.43\times 10^{-7}$ | 0.007-0.03 | 5 | 0.031
26 | 7 | 0.015 | $9\times 10^{-8}$ | 0.005-0.018 | 5 | 0.035
41 | 19 | 0.125 | $1.06\times 10^{-7}$ | 0.005-0.02 | 13 | 0.026
71 | 24 | 0.11 | $2.8\times 10^{-7}$ | 0.014-0.05 | 60 | 0.022
72 | 24 | 0.11 | $3.8\times 10^{-7}$ | 0.019-0.07 | 60 | 0.027
76 | 25 | 0.11 | $2.9\times 10^{-7}$ | 0.015-0.06 | 60 | 0.023
77 | 25 | 0.11 | $3.88\times 10^{-7}$ | 0.019-0.07 | 60 | 0.027
82 | 26 | 0.11 | $3.87\times 10^{-7}$ | 0.019-0.07 | 60 | 0.027
83 | 26 | 0.11 | $5.04\times 10^{-7}$ | 0.025-0.1 | 60 | 0.035
87 | 27 | 0.11 | $3.79\times 10^{-7}$ | 0.019-0.07 | 60 | 0.031
Table 5: Binomial probabilities and the estimated probabilities (multiplied by
the number of tries) for the southern hemisphere ($p_{est}$). K and n are the
same as described in Sect. 2.2. Here, Arel is the relative area of the
hemisphere, $p_{binom}$ is the binomial probability with p=Arel, n tries, and
K successes. $p_{est}$ is the binomial probability multiplied by the number of
tries, ’num’ is the number of significant signs in the neighboring areas shown
in Fig. 8. $p_{bstr}$ is the bootstrap probabilities which were calculated in
Sect. 3.2. Some of these are shown in Table 3.

##  6 Summary

Our method investigated the 542 GRBs having observed redshift. We vary the
redshift, the group size, and the area of the sky. In this 3D parameter space
we found several suspicious regions. One in the northern galactic sky contains
4 GRBs with $1.2\%$ probability of having this by chance (see Fig. 6). In the
southern hemisphere there is another small group having 7 or 9 GRBs, but the
significance in both cases does not reach even the $3\%$ level (see Fig. 10).

However, there were two groups (one in each hemisphere), where the clustering
yielded several significant results in the ($n,A$) plane (red blocks in
Figures 5 and 8). In the southern hemisphere there are three areas in the
($n$, $A$) space. From these areas one area’s small part was previously
identified as the Giant GRB Ring (Balázs et al., 2015, 2018) (see Figures 9,
11, and 12). In the northern hemisphere a huge part of the ($n$, $A$) plane
suggested a group (having $69-71$ GRBs out of the $104-108$, see Table 2) in a
$31\%$ area of the hemisphere (Fig. 7). A part of this structure was
identified previously in the literature as the Hercules–Corona Borealis Great
Wall (Horváth et al., 2014, 2015).

## Acknowledgements

The authors thank the Hungarian TKP2021-NVA-16 and OTKA K134257 program for
their support. The authors also thank Dr Robert J. Nemiroff for useful
comments that improved the paper.

## Data Availability

The data underlying this paper are available in Gamma-Ray Burst Online Index
(GRBOX) database published by the Caltech Astronomy Department
(http://www.astro.caltech.edu/grbox/grbox.php), Joachim Greiner’s table
(https://www.mpe.mpg.de/~jcg/grbgen.html), and relevant Gamma-ray Coordination
Network (https://gcn.gsfc.nasa.gov/gcn3_archive.html) messages. The code and
data used for this study are available at
https://github.com/zbagoly/GRBSphericalCapStat.

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##  Appendix A Tables

n A | 0.0628 | 0.1256 | 0.188 | 0.251 | 0.314 | 0.376 | 0.439 | 0.502 | 0.565 | 0.628 | 0.691 | 0.754 | 0.816 | 0.879 | 0.942 | 1.005 | 1.068
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
3 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3
4 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4
5 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5
6 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5
7 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6
8 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 7
9 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 7 | 7
10 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 8
11 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 7 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8
12 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 7 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 9
13 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 7 | 7 | 7 | 8 | 8 | 8 | 9 | 9 | 9 | 9
14 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 7 | 7 | 8 | 8 | 9 | 9 | 9 | 9 | 9 | 10
15 | 4 | 5 | 6 | 6 | 6 | 7 | 7 | 8 | 8 | 8 | 9 | 9 | 10 | 10 | 10 | 10 | 10
16 | 4 | 5 | 6 | 6 | 6 | 7 | 7 | 8 | 8 | 9 | 9 | 10 | 10 | 10 | 10 | 10 | 11
17 | 4 | 5 | 6 | 6 | 6 | 7 | 7 | 8 | 8 | 9 | 9 | 10 | 10 | 10 | 10 | 10 | 11
18 | 4 | 5 | 6 | 6 | 7 | 8 | 8 | 9 | 9 | 9 | 10 | 10 | 11 | 11 | 11 | 11 | 12
19 | 4 | 5 | 6 | 6 | 7 | 8 | 8 | 9 | 9 | 10 | 10 | 11 | 11 | 11 | 11 | 12 | 12
20 | 4 | 5 | 6 | 6 | 7 | 8 | 8 | 9 | 9 | 10 | 10 | 11 | 12 | 12 | 12 | 12 | 13
21 | 4 | 5 | 7 | 7 | 7 | 8 | 8 | 9 | 9 | 10 | 10 | 11 | 12 | 12 | 12 | 12 | 13
22 | 4 | 5 | 7 | 7 | 8 | 8 | 9 | 10 | 10 | 11 | 11 | 12 | 12 | 13 | 13 | 13 | 13
23 | 4 | 5 | 7 | 7 | 8 | 8 | 9 | 10 | 10 | 11 | 11 | 12 | 13 | 13 | 13 | 13 | 14
24 | 4 | 5 | 7 | 7 | 8 | 9 | 9 | 10 | 10 | 11 | 11 | 12 | 13 | 13 | 13 | 14 | 14
25 | 4 | 5 | 7 | 7 | 8 | 9 | 9 | 10 | 10 | 11 | 11 | 12 | 13 | 13 | 14 | 15 | 15
Table 6: This table shows the number K depending on n and the area of the circle (A, in steradian) for the northern hemisphere. See Sect. 3.1 for more details. See the complete table as a supplement material. n A | 0.0628 | 0.094 | 0.1256 | 0.157 | 0.188 | 0.22 | 0.251 | 0.283 | 0.314 | 0.346 | 0.377 | 0.408 | 0.439 | 0.471 | 0.502 | 0.534 | 0.565
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
3 | 1 | 1 | 1 | 1 | 1 | 0.974 | 0.99 | 0.996 | 0.998 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
4 | 0.608 | 0.867 | 0.978 | 0.995 | 0.999 | 1 | 1 | 1 | 1 | 1 | 0.763 | 0.838 | 0.89 | 0.935 | 0.957 | 0.971 | 0.985
5 |  red0.021 | 0.09 | 0.197 | 0.347 | 0.489 | 0.629 | 0.772 | 0.871 | 0.94 | 0.97 | 0.991 | 0.997 | 1 | 1 | 1 | 1 | 1
6 |  blue0.062 | 0.205 | 0.399 | 0.622 | 0.798 | 0.907 | 0.969 | 0.989 | 0.995 | 0.999 | 1 | 1 | 1 | 1 | 1 | 1 | 0.947
7 | 0.122 | 0.347 | 0.634 | 0.838 | 0.946 | 0.984 | 1 | 1 | 1 | 1 | 0.855 | 0.919 | 0.959 | 0.981 | 0.989 | 0.995 | 0.999
8 | 0.2 | 0.503 | 0.8 | 0.947 | 0.989 | 0.999 | 1 | 1 | 0.862 | 0.932 | 0.98 | 0.989 | 0.998 | 1 | 1 | 0.741 | 0.82
9 | 0.296 | 0.635 | 0.906 | 0.986 | 0.999 | 1 | 1 | 1 | 0.963 | 0.993 | 0.997 | 0.999 | 1 | 1 | 0.874 | 0.931 | 0.966
10 | 0.407 | 0.773 | 0.961 | 0.998 | 1 | 1 | 1 | 0.983 | 0.997 | 0.598 | 0.727 | 0.832 | 0.908 | 0.968 | 0.985 | 0.994 | 0.996
11 | 0.497 | 0.116 | 0.277 | 0.53 | 0.75 | 0.894 | 0.979 | 0.999 | 1 | 0.774 | 0.886 | 0.944 | 0.972 | 0.533 | 0.636 | 0.728 | 0.82
12 | 0.617 | 0.17 | 0.398 | 0.683 | 0.871 | 0.964 | 0.996 | 1 | 1 | 0.906 | 0.965 | 0.989 | 0.997 | 0.745 | 0.833 | 0.893 | 0.948
13 | 0.714 | 0.239 | 0.511 | 0.796 | 0.95 | 0.99 | 1 | 0.787 | 0.917 | 0.974 | 0.997 | 1 | 1 | 0.886 | 0.948 | 0.978 | 0.988
14 | 0.781 | 0.319 | 0.634 | 0.877 | 0.977 | 0.999 | 1 | 0.878 | 0.97 | 0.994 | 0.999 | 1 | 1 | 0.954 | 0.984 | 0.994 | 0.997
15 | 0.843 | 0.401 | 0.74 | 0.938 | 0.481 | 0.686 | 0.842 | 0.944 | 0.989 | 0.698 | 0.834 | 0.931 | 0.978 | 0.608 | 0.721 | 0.832 | 0.91
16 | 0.901 | 0.476 | 0.817 | 0.971 | 0.598 | 0.795 | 0.932 | 0.984 | 0.996 | 0.82 | 0.929 | 0.98 | 0.995 | 0.763 | 0.87 | 0.936 | 0.969
17 | 0.937 | 0.566 | 0.888 | 0.993 | 0.707 | 0.877 | 0.973 | 0.999 | 1 | 0.903 | 0.976 | 0.998 | 0.999 | 0.88 | 0.95 | 0.982 | 0.992
18 | 0.959 | 0.639 | 0.927 | 0.998 | 0.799 | 0.932 | 0.994 | 0.745 | 0.887 | 0.967 | 0.606 | 0.751 | 0.881 | 0.458 | 0.585 | 0.708 | 0.825
19 | 0.978 | 0.715 | 0.967 | 0.999 | 0.874 | 0.968 | 0.999 | 0.837 | 0.94 | 0.988 | 0.735 | 0.864 | 0.951 | 0.609 | 0.732 | 0.842 | 0.928
20 | 0.987 | 0.778 | 0.979 | 0.999 | 0.919 | 0.984 | 0.999 | 0.907 | 0.975 | 0.996 | 0.851 | 0.944 | 0.99 | 0.733 | 0.85 | 0.933 | 0.974
21 | 0.993 | 0.833 | 0.987 | 0.999 | 0.384 | 0.641 | 0.861 | 0.963 | 0.995 | 0.79 | 0.933 | 0.988 | 0.998 | 0.844 | 0.928 | 0.973 | 0.988
22 | 0.996 | 0.877 | 0.994 | 0.999 | 0.461 | 0.732 | 0.921 | 0.984 | 0.998 | 0.885 | 0.973 | 0.996 | 0.828 | 0.447 | 0.577 | 0.711 | 0.83
Table 7: This table shows the bootstrap probabilities (frequencies) for a certain n and the area of the circle (A, in steradian) for the northern hemisphere. See Sect. 3.1 for more details. Here we are using the same color code as Fig. 5 used. See the complete table as a supplement material. n A | 0.0628 | 0.1256 | 0.188 | 0.251 | 0.314 | 0.376 | 0.439 | 0.502 | 0.565 | 0.628 | 0.691 | 0.754 | 0.816 | 0.879 | 0.942 | 1.005 | 1.068
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
3 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3
4 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4
5 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5
6 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5
7 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6
8 | 3 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6
9 | 3 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 7 | 7 | 7 | 7
10 | 3 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 7 | 7 | 7 | 7
11 | 3 | 4 | 5 | 5 | 5 | 6 | 6 | 7 | 7 | 7 | 7 | 7 | 7 | 8 | 8 | 8 | 8
12 | 4 | 4 | 5 | 6 | 6 | 6 | 7 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 9 | 9
13 | 4 | 4 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 7 | 8 | 8 | 8 | 9 | 9 | 9 | 9
14 | 4 | 4 | 6 | 6 | 6 | 7 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 9 | 9 | 10 | 10
15 | 4 | 5 | 6 | 6 | 6 | 7 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 9 | 9 | 10 | 10
16 | 4 | 5 | 6 | 6 | 6 | 8 | 9 | 9 | 9 | 9 | 10 | 10 | 10 | 10 | 10 | 10 | 10
17 | 4 | 5 | 6 | 6 | 7 | 8 | 9 | 9 | 9 | 9 | 10 | 10 | 10 | 10 | 11 | 11 | 11
18 | 5 | 5 | 6 | 6 | 7 | 8 | 9 | 9 | 9 | 9 | 10 | 10 | 10 | 10 | 11 | 11 | 11
19 | 5 | 5 | 6 | 7 | 7 | 9 | 10 | 10 | 10 | 10 | 11 | 11 | 11 | 11 | 11 | 12 | 12
20 | 5 | 5 | 6 | 7 | 7 | 9 | 10 | 10 | 10 | 10 | 11 | 11 | 11 | 11 | 12 | 12 | 12
21 | 5 | 5 | 6 | 7 | 7 | 9 | 10 | 10 | 10 | 10 | 11 | 11 | 11 | 11 | 12 | 12 | 12
22 | 5 | 6 | 7 | 8 | 8 | 9 | 10 | 10 | 10 | 10 | 11 | 11 | 12 | 12 | 12 | 13 | 13
23 | 5 | 6 | 7 | 8 | 8 | 9 | 10 | 10 | 10 | 10 | 11 | 11 | 12 | 12 | 12 | 13 | 13
24 | 5 | 6 | 7 | 8 | 8 | 9 | 10 | 11 | 11 | 11 | 12 | 12 | 12 | 13 | 13 | 13 | 14
25 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 11 | 11 | 11 | 12 | 12 | 13 | 13 | 14 | 14 | 15
26 | 6 | 7 | 8 | 9 | 9 | 10 | 11 | 11 | 11 | 11 | 12 | 13 | 13 | 13 | 14 | 14 | 15
27 | 6 | 7 | 8 | 9 | 10 | 10 | 11 | 12 | 12 | 12 | 13 | 13 | 14 | 14 | 15 | 15 | 16
28 | 6 | 7 | 8 | 9 | 10 | 10 | 11 | 12 | 12 | 12 | 13 | 13 | 14 | 14 | 15 | 15 | 16
29 | 6 | 7 | 8 | 9 | 10 | 10 | 11 | 12 | 12 | 12 | 13 | 13 | 14 | 14 | 15 | 15 | 16
Table 8: This table shows the number K depending on n and the area of the circle (A, in steradian) for the southern hemisphere. See Sect. 3.2 for more details. See the complete table as a supplement material. n A | 0.0628 | 0.094 | 0.1256 | 0.157 | 0.188 | 0.22 | 0.251 | 0.283 | 0.314 | 0.346 | 0.377 | 0.408 | 0.439 | 0.471 | 0.502 | 0.534 | 0.565
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
3 | 1 | 1 | 0.691 | 0.824 | 0.912 | 0.957 | 0.982 | 0.996 | 0.999 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
4 | 0.587 | 0.833 | 0.951 | 0.987 | 0.997 | 0.215 | 0.291 | 0.372 | 0.476 | 0.566 | 0.648 | 0.723 | 0.78 | 0.849 | 0.891 | 0.929 | 0.952
5 | 0.801 | 0.966 | 0.999 | 1 | 0.408 | 0.583 | 0.712 | 0.805 | 0.061 | 0.092 | 0.119 | 0.145 | 0.178 | 0.233 | 0.287 | 0.349 | 0.399
6 | 0.937 | 0.997 | 1 | 1 | 0.691 | 0.835 | 0.923 | 0.965 | 0.211 | 0.298 | 0.378 | 0.453 | 0.533 | 0.635 | 0.716 | 0.784 | 0.836
7 | 0.986 | 0.328 | 0.53 | 0.728 | 0.867 | 0.952 | 0.987 | 0.337 | 0.459 | 0.59 | 0.702 | 0.807 | 0.873 | 0.919 | 0.953 | 0.232 | 0.281
8 | 0.996 | 0.456 | 0.7 | 0.88 | 0.964 | 0.992 | 0.999 | 0.554 | 0.065 | 0.099 | 0.154 | 0.206 | 0.269 | 0.361 | 0.443 | 0.515 | 0.609
9 | 0.999 | 0.594 | 0.838 | 0.968 | 0.297 | 0.459 | 0.625 | 0.777 | 0.163 | 0.238 | 0.323 | 0.416 | 0.521 | 0.627 | 0.729 | 0.804 | 0.87
10 | 0.999 | 0.736 | 0.925 | 0.99 | 0.446 | 0.657 | 0.82 | 0.923 | 0.289 | 0.403 | 0.532 | 0.638 | 0.742 | 0.824 | 0.908 | 0.948 | 0.968
11 | 1 | 0.842 | 0.973 | 0.997 | 0.607 | 0.817 | 0.924 | 0.98 | 0.446 | 0.587 | 0.729 | 0.823 | 0.897 | 0.337 | 0.423 | 0.509 | 0.593
12 | 0.52 | 0.908 | 0.99 | 0.523 | 0.75 | 0.195 | 0.305 | 0.444 | 0.618 | 0.771 | 0.872 | 0.928 | 0.388 | 0.497 | 0.595 | 0.715 | 0.794
13 | 0.598 | 0.943 | 0.997 | 0.639 | 0.144 | 0.288 | 0.437 | 0.62 | 0.773 | 0.89 | 0.95 | 0.981 | 0.576 | 0.69 | 0.783 | 0.874 | 0.931
14 | 0.683 | 0.964 | 0.998 | 0.746 | 0.216 | 0.391 | 0.552 | 0.745 | 0.865 | 0.958 | 0.475 | 0.607 | 0.155 | 0.226 | 0.308 | 0.404 | 0.496
15 | 0.765 | 0.294 | 0.57 | 0.842 | 0.298 | 0.509 | 0.699 | 0.864 | 0.944 | 0.478 | 0.628 | 0.751 | 0.262 | 0.372 | 0.468 | 0.584 | 0.685
16 | 0.83 | 0.397 | 0.692 | 0.911 | 0.421 | 0.634 | 0.815 | 0.94 | 0.977 | 0.629 | 0.758 | 0.264 | 0.052 | 0.088 | 0.132 | 0.189 | 0.262
17 | 0.867 | 0.468 | 0.765 | 0.95 | 0.518 | 0.731 | 0.888 | 0.382 | 0.563 | 0.734 | 0.272 | 0.385 | 0.098 | 0.156 | 0.227 | 0.312 | 0.398
18 | 0.142 | 0.554 | 0.841 | 0.971 | 0.614 | 0.833 | 0.949 | 0.501 | 0.687 | 0.848 | 0.373 | 0.513 | 0.156 | 0.244 | 0.337 | 0.443 | 0.538
19 | 0.176 | 0.631 | 0.895 | 0.984 | 0.714 | 0.272 | 0.42 | 0.621 | 0.792 | 0.923 | 0.477 | 0.146 |  blue0.037 | 0.064 | 0.101 | 0.147 | 0.205
20 | 0.215 | 0.695 | 0.938 | 0.993 | 0.794 | 0.345 | 0.52 | 0.736 | 0.886 | 0.965 | 0.592 | 0.206 | 0.061 | 0.101 | 0.158 | 0.217 | 0.285
21 | 0.263 | 0.758 | 0.315 | 0.621 | 0.851 | 0.427 | 0.612 | 0.819 | 0.935 | 0.984 | 0.699 | 0.284 | 0.092 | 0.153 | 0.234 | 0.32 | 0.405
22 | 0.322 | 0.141 | 0.396 | 0.709 | 0.304 | 0.073 | 0.16 | 0.297 | 0.465 | 0.658 | 0.799 | 0.379 | 0.125 | 0.207 | 0.303 | 0.405 | 0.511
23 | 0.36 | 0.17 | 0.464 | 0.771 | 0.355 | 0.11 | 0.221 | 0.379 | 0.573 | 0.766 | 0.874 | 0.484 | 0.189 | 0.284 | 0.397 | 0.532 | 0.646
24 | 0.414 | 0.198 | 0.522 | 0.833 | 0.418 | 0.141 | 0.269 | 0.454 | 0.68 | 0.844 | 0.932 | 0.588 | 0.259 | 0.365 | 0.503 | 0.213 | 0.294
25 | 0.461 | 0.26 | 0.598 | 0.871 | 0.491 | 0.192 | 0.361 | 0.562 | 0.229 | 0.384 | 0.542 | 0.217 | 0.066 | 0.122 | 0.191 | 0.28 | 0.381
26 |  blue0.046 |  blue0.035 | 0.12 | 0.312 | 0.108 |  blue0.031 | 0.071 | 0.157 | 0.303 | 0.48 | 0.635 | 0.288 | 0.105 | 0.169 | 0.254 | 0.368 | 0.484
27 | 0.054 |  blue0.048 | 0.156 | 0.366 | 0.134 |  blue0.045 | 0.104 | 0.206 | 0.058 | 0.135 | 0.24 | 0.36 | 0.147 | 0.221 | 0.332 | 0.124 | 0.183
28 | 0.063 | 0.055 | 0.192 | 0.427 | 0.169 | 0.062 | 0.139 | 0.253 | 0.086 | 0.172 | 0.305 | 0.446 | 0.19 | 0.287 | 0.417 | 0.181 | 0.254
Table 9: This table shows the bootstrap probabilities (frequencies) for a
certain n and the area of the circle (in steradian) for the southern
hemisphere. Here we are using the same color code as Fig. 8 used. See Sect.
3.2 for more details. See the complete table as a supplement material.


Present address:]Department of Physics, Taiyuan Normal University, Jinzhong
030619, People’s Republic of China ††thanks: R.C., X.H., and J.L.M. made equal
contributions to this work.

# AC polarizability and photoionization cross-section measurements in an
optical lattice

Ryan Cardman<EMAIL_ADDRESS>Xiaoxuan Han  [  Jamie L. MacLennan  Alisher
Duspayev  Georg Raithel  Department of Physics, University of Michigan, Ann
Arbor, MI 48109

(September 3, 2024)

###### Abstract

We use double-resonant two-photon laser spectroscopy to measure the dynamic
scalar polarizability of the rubidium 5$D_{3/2}$ level,
$\alpha^{S}_{5D_{3/2}}$, at a wavelength of $\lambda=1064$ nm. Since $\lambda$
is shorter than the photoionization (PI) limit of the Rb 5$D_{3/2}$ level,
$\alpha^{S}_{5D_{3/2}}$ depends on both bound-bound and bound-free transition
matrix elements. The level also undergoes significant broadening due to PI.
The $1064$-nm field is applied in the form of a deep optical lattice ($\sim
10^{5}$ photon recoils) generated by an in-vacuum field-enhancement cavity. In
our spectroscopic method, we use known dynamic polarizabilities to eliminate
the need to measure the light intensity. Our method yields, in atomic units,
$\alpha^{S}_{5D_{3/2}}=-524(17)$, in agreement with estimates. Additionally,
we extract the $5D_{3/2}$ photoionization cross section $\sigma$ at $1064$ nm
from spectral linewidths; we find $\sigma=44(1)$ Mb.

For some time, neutral atoms have been trapped by off-resonant optical fields
for the purpose of redefining the second with unparalleled precision Boulder
Atomic Clock Optical Network Collaboration*  _et al._(2021)Boulder Atomic
Clock Optical Network (BACON) Collaboration*, Beloy, Bodine, Bothwell, Brewer,
Bromley, Chen, Deschênes, Diddams, Fasano, Fortier, Hassan, Hume, Kedar,
Kennedy, Khader, Koepke, Leibrandt, Leopardi, Ludlow, McGrew, Milner, Newbury,
Nicolodi, Oelker, Parker, Robinson, Romisch, Schäffer, Sherman, Sinclair,
Sonderhouse, Swann, Yao, Ye, and Zhang (BACON), simulating theoretical models
Le Kien  _et al._ (2013); Guardado-Sanchez  _et al._ (2018); Hu  _et al._
(2019), and constructing quantum computing protocols Saffman (2016); Morgado
and Whitlock (2021). An atom in an electric field with a frequency far from
resonance of an electric-dipole transition undergoes an energy shift due to
the AC Stark effect, which is proportional to the field intensity. In
applications of optical-dipole traps in optical clocks and in spectroscopy,
differential AC Stark shifts of the relevant atomic states must be either very
well known or eliminated using carefully determined “magic” wavelengths
Safronova  _et al._ (2003); Flambaum  _et al._ (2008); Lundblad  _et al._
(2010). The AC shifts follow $\Delta W=-\alpha(\omega_{L})E_{L}^{2}/4$, with
laser electric field $E_{L}$ and angular frequency $\omega_{L}$, and state-
dependent dynamic polarizabilities $\alpha$. Following the intensity-gradient
force on the atomic CM coordinate, $-\nabla_{\bf{R}}\Delta W({\bf{R}})$,
states with positive (negative) $\alpha$ are attracted to (repelled from)
locations of high field intensity. Also, the AC shifts of spectral lines of
atomic transitions scale with the difference between the $\alpha$-values of
the relevant states.

For an atomic state $\left|n,l,j,m_{j}\right\rangle$ in a field with
polarization unit vector $\hat{\epsilon}$, the dynamic polarizability is

$\alpha_{n,l,j,m_{j}}(\omega_{L})=\frac{2}{\hbar}\bigg{[}\sum_{n^{\prime},l^{\prime},j^{\prime},m^{\prime}_{j}}|\left\langle
n^{\prime},l^{\prime},j^{\prime},m^{\prime}_{j}\right|\hat{\epsilon}\cdot\hat{\textbf{d}}\left|n,l,j,m_{j}\right\rangle|^{2}\\\
\times\frac{\omega^{n^{\prime}l^{\prime}j^{\prime}}_{nlj}}{(\omega^{n^{\prime}l^{\prime}j^{\prime}}_{nlj})^{2}-\omega^{2}_{L}}+\sum_{l^{\prime},j^{\prime},m^{\prime}_{j}}\int_{0}^{\infty}d\epsilon^{\prime}\rho(\epsilon^{\prime})\\\
\times|\left\langle\epsilon^{\prime},l^{\prime},j^{\prime},m^{\prime}_{j}\right|\hat{\epsilon}\cdot\hat{\textbf{d}}\left|n,l,j,m_{j}\right\rangle|^{2}\frac{\omega^{\epsilon^{\prime}}_{nlj}}{(\omega^{\epsilon^{\prime}}_{nlj})^{2}-\omega^{2}_{L}}\bigg{]},$
(1)

where $\hat{\textbf{d}}$ is the electric-dipole moment operator,
$\omega^{n^{\prime}l^{\prime}j^{\prime}}_{nlj}=(W_{n^{\prime}l^{\prime}j^{\prime}}-W_{nlj})/\hbar$
for bound state energies $W_{nlj}$ and $W_{n^{\prime}l^{\prime}j^{\prime}}$,
and $\omega^{\epsilon^{\prime}}_{nlj}=(\epsilon^{\prime}-W_{nlj})/\hbar$ for
free-electron-state (FES) energies $\epsilon^{\prime}$. The quantity
$\rho(\epsilon^{\prime})$ is the density of FESs with energy
$\epsilon^{\prime}$ and is equal to one per unit energy, for energy-normalized
states Friedrich (2017). When the photon energy $\hbar\omega_{L}$ is large
enough to photoionize the atom, the second term in Eq. 1 may become
significant because the integrand has a pole at
$\omega_{L}=\omega_{nlj}^{\epsilon^{\prime}}$. In a linearly polarized field,
the $m_{j}$-dependent polarizabilities $\alpha_{n,l,j,m_{j}}(\omega_{L})$
depend on $m_{j}$-independent scalar and tensor polarizabilities,
$\alpha^{S}_{n,l,j}(\omega_{L})$ and $\alpha^{T}_{n,l,j}(\omega_{L})$,

$\alpha_{n,l,j,m_{j}}(\omega_{L})=\alpha^{S}_{n,l,j}(\omega_{L})+\frac{3m^{2}_{j}-j(j+1)}{j(2j-1)}\alpha^{T}_{n,l,j}(\omega_{L}),$
(2)

where the second term vanishes for $j<1$. A third term, proportional to the
vector polarizability $\alpha^{V}_{n,l,j}(\omega_{L})$, is added to Eq. 2 only
when the field polarization is not linear.

In this study, we measure $\alpha^{S}_{5D_{3/2}}$ for rubidium in an optical
lattice of laser wavelength $\lambda=1064$ nm. Its photon energy $hc/\lambda$
suffices to photoionize $5D_{3/2}$-atoms trapped in the optical lattice,
making a case in which the FES-term in Eq. 1 may become significant. While no
theoretical or experimental estimate has been made for the $5D_{3/2}$ dynamic
polarizability, the static polarizabilities have been investigated Davydkin
and Zon (1982); Kamenski and Ovsiannikov (2006); Kondrat’ev  _et al._ (2008);
Snigirev  _et al._ (2014), as well as the dynamic polarizability at 778.1 nm
for 5$D_{5/2}$ Martin  _et al._ (2019a).

The Rb $5D_{3/2}$-levels in a 1064-nm field are also broadened by
photoionization (PI), as estimates based on measured Rb $5D_{5/2}$ Duncan  _et
al._ (2001) and calculated Duncan  _et al._ (2001); Aymar  _et al._ (1984) Rb
$5D$ PI cross sections show. These estimates exhibit that the PI-induced level
broadening at 1064 nm is on the same order of magnitude as the AC Stark shift.
In light of this fact, a comprehensive spectroscopic study of Rb $5D$-states
in a 1064-nm optical lattice appears particularly worthwhile, as both the
dynamic scalar polarizability and the PI cross section can be measured
simultaneously. From an applications point of view, the Rb 5$D$ states are
appealing to study for several reasons. First, there has been continued
interest ranging from earlier decades Nez  _et al._ (1993); Touahri  _et al._
(1997); Hilico  _et al._ (1998) to recent years Terra and Hussein (2016);
Rathod and Natarajan (2017); Martin  _et al._ (2018) in using the strong and
narrow (natural linewidth $<$ 1 MHz) two-photon transition
5$S_{1/2}\rightarrow$5$D_{J}$ as an optical frequency reference, which
necessitates precise calculation or cancellation of the relevant light shifts
Gerginov and Beloy (2018); Martin  _et al._ (2019b). This two-photon
transition’s wavelength (778 nm) can be directly generated by a semiconductor
laser, or via second-harmonic-generation of a laser at 1556 nm, which falls
within the telecommunication band. Furthermore, the
5S${}_{1/2}\rightarrow$5D5/2 transition is one of the recommended transitions
by the Consultative Committee of Length (CCL) for the practical definition of
the meter Quinn (2003).

Figure 1: (a) Sketch of the experimental apparatus, and (b) energy level
diagram with outlined AC Stark shifts as a function of radial position with
respect to the axis of the 1064 nm optical lattice (not to scale).

Main components of the apparatus, atomic energy levels, and optical beam
geometries are exhibited in Fig. 1. We load 85Rb atoms from a MOT, initially
prepared in the $F=3$ ground-state hyperfine level, into the near-perfect
TEM00 mode of a near-concentric field-enhancement cavity with a finesse of
$\approx 600$ at $\lambda=1064\leavevmode\nobreak\ $nm Chen  _et al._ (2014,
2015); MacLennan  _et al._ (2019). After loading, the lattice is adiabatically
ramped to a maximum depth of $\sim 10^{5}E_{r}$, where $E_{r}=h\times 2.076$
kHz is the photon recoil for 85Rb at 1064 nm. In order to observe the lattice-
shifts affecting the D1 line, we pulse a $795$-nm probe laser for a duration
of $\sim 15\leavevmode\nobreak\ \mu$s and scan it from -36 MHz to $1344$ MHz
with respect to the frequency of the
$\left|5S_{1/2},F=3\right\rangle\rightarrow\left|5P_{1/2},F^{\prime}=2\right\rangle$
transition. A co-propagating 762-nm probe laser is pulsed on for $\sim 500$ ns
and is independently scanned from 57 MHz to -843 MHz with respect to the
lattice-field-free frequency difference between the
$\left|5P_{1/2},F^{\prime}=2\right\rangle$ and
$\left|5D_{3/2},F^{\prime\prime}=3\right\rangle$ states. Lattice-induced PI of
atoms excited into $5D_{3/2}$ yields a count of atoms in this state when the
resulting ions are guided onto a micro-channel plate detector (MCP) with an
extraction voltage. This procedure yields a two-dimensional map of ion counts
as a function of the two probe laser frequencies, as shown in Fig. 2(a).

Figure 2: (a) Results of two-photon spectroscopy as a function of 795-nm and
762-nm probe laser frequencies ($\nu_{795}$ and $\nu_{762}$, respectively).
The reference frequencies $\nu_{1}$ and $\nu_{2}$ correspond to splittings
between the lattice-field-free $5S_{1/2},F=3$ and $5P_{1/2},F^{\prime}=2$
levels, and between the lattice-field-free $5P_{1/2},F^{\prime}=2$ and
$5D_{3/2},F^{\prime\prime}=3$ levels, respectively. Data are averaged over 30
experimental cycles. (b) Numerical simulation (see Supplement) for input
parameters $\alpha^{S}_{5D_{3/2}}=-524$, $\alpha^{T}_{5D_{3/2}}=0$ and
$\sigma=40$ Mb.

In Fig. 2(a) we display ion counts as a function of the frequencies of both
probe lasers. The figure exhibits two diagonally-aligned branches of
$5D_{3/2}$ atomic signals. The branches correspond to the intermediate
hyperfine states $F^{\prime}=2$ and 3, which present two pathways through
which the atoms can be excited into $5D_{3/2}$ via step-wise, double-resonant
two-photon excitation. For the two pathways $F^{\prime}=2$ and 3, we define
detunings $\Delta^{F^{\prime}}_{795}$ and $\Delta^{F^{\prime}}_{762}$ of the
excitation lasers from their respective lattice-field-free resonances (see
Fig. 1(b) and axis insets in Fig. 2(a)).

The AC shifts evident from the slopes of the signal branches in Fig. 2 are
caused by the dynamic polarizabilities of all relevant atomic states,
$5S_{1/2},F=3$, and $5P_{1/2},F^{\prime}=2$ or 3, and $5D_{3/2},F^{\prime}=1$
to $4$. For the ground and intermediate-state polarizabilities at 1064 nm, we
use the theoretical values $\alpha^{S}_{5S_{1/2}}=687.3(5)$ Arora and Sahoo
(2012) and $\alpha^{S}_{5P_{1/2}}=-1226(18)$ Neuzner  _et al._ (2015) (in
atomic units, and applicable to all hyperfine sublevels). The objective of the
measurement then is, in principle, to extract the scalar and tensor
polarizabilities, $\alpha^{S}_{5D_{3/2}}$ and $\alpha^{T}_{5D_{3/2}}$, from
the slopes of the signal branches in Fig. 2. From an estimate given below, it
is predicted that the magnitude of $\alpha^{T}_{5D_{3/2}}$ is at or below the
level of uncertainty of the experimental method. In the following, we
therefore restrict our analysis to the case $\alpha^{T}_{5D_{3/2}}=0$.

The two signal branches in Fig. 2 yield two measurements for the slopes,
$d\Delta^{F^{\prime}}_{762}/d\Delta^{F^{\prime}}_{795}$, associated with the
two intermediate states $5P_{1/2},F^{\prime}=2$ and $5P_{1/2},F^{\prime}=3$.
The scalar polarizability $\alpha^{S}_{5D_{3/2}}$ then follows from

$\alpha^{S}_{5D_{3/2}}(F^{\prime})=\alpha^{S}_{5_{P_{1/2}}}-\frac{d\Delta^{F^{\prime}}_{762}}{d\Delta^{F^{\prime}}_{795}}(\alpha^{S}_{5S_{1/2}}-\alpha^{S}_{5P_{1/2}})\quad.$
(3)

This measurement method is similar to a method used in Chen  _et al._ (2015).
Because Fig. 2(a) yields two readings for the slope
$d\Delta^{F^{\prime}}_{762}/d\Delta^{F^{\prime}}_{795}$, we obtain two
measurements for $\alpha^{S}_{5D_{3/2}}$ that correspond with the intermediate
hyperfine pathways $F^{\prime}=2$ and $F^{\prime}=3$. The method is self-
calibrating in the sense that a direct measurement of the lattice intensity in
the atomic sampling region is unnecessary.

Table 1:  Summary of quantities used to extract $\alpha^{S}_{5D_{3/2}}$, as well as a summary of results (in atomic units). Quantity | Value | Source
---|---|---
$\alpha^{S}_{5S_{1/2}}$ | 687.3(5) | Arora and Sahoo (2012)
$\alpha^{S}_{5P_{1/2}}$ | -1226(18) | Neuzner  _et al._ (2015)
$d\Delta^{F^{\prime}=2}_{762}/d\Delta^{F^{\prime}=2}_{795}$ | -0.36(1) | Experimental data, this work
$d\Delta^{F^{\prime}=3}_{762}/d\Delta^{F^{\prime}=3}_{795}$ | -0.371(6) | Experimental data, this work
$\alpha^{S}_{5D_{3/2}}(F^{\prime}=2)$ | -537(27) | Eq. 3
$\alpha^{S}_{5D_{3/2}}(F^{\prime}=3)$ | -516(22) | Eq. 3
$\alpha^{S}_{5D_{3/2}}$ | -524(17) | Weighted average

The experimental data are acquired by scanning the detuning of the 762-nm
laser, $\nu_{762}-\nu_{2}$, for a set of values for the detuning
$\nu_{795}-\nu_{1}$ of the 795-nm laser. A vertical slice of the map in Fig.
2(a) at a value of $\nu_{795}-\nu_{1}=0.984$ GHz is provided in the inset of
Fig. 3(a) as an example. We find that the spectral lines along each frequency
setting of the 795-nm laser follow near-perfect Lorentzian shapes, down to the
noise level several linewidths away from the line centers. This observation
serves as experimental evidence that the line profiles are the result of a
level decay mechanism, which in our case is the PI of the $5D_{3/2}$ levels.
For each of the individual spectra at fixed $\nu_{795}-\nu_{1}$, we obtain the
line centers and linewidths using Lorentzian fits. Depending on whether the
signal has contributions from both signal branches $F^{\prime}=2$ and
$F^{\prime}=3$ in Fig. 2 or from just one, we employ double- or single-
Lorentzian fits, respectively. The line centers yield two sets of data points
$\Delta^{F^{\prime}}_{762}(\Delta^{F^{\prime}}_{795})$, for $F^{\prime}=2$ and
$3$. We present the sets for the $F^{\prime}=3$ branch in Fig. 3(b). The
slopes $d\Delta^{F^{\prime}}_{762}/d\Delta^{F^{\prime}}_{795}$ from respective
weighted linear fits yield a pair of measured values of
$\alpha^{S}_{5D_{3/2}}$ via Eq. 3. Essential inputs from other sources and our
results are summarized in Table 1. The uncertainties in the scalar
polarizability measurements are dominated by the statistical errors of the
linear fits and the given uncertainty in $\alpha^{S}_{5P_{1/2}}$. Systematic
effects from stray electromagnetic fields, atomic collisions, and laser
frequency linearity are negligible. The weighted average over the pair of
measured dynamic scalar polarizabilities is $\alpha^{S}_{5D_{3/2}}=-524(17)$
atomic units.

Expanding upon previous results from other works, theoretical estimates for
$\alpha^{S}_{5D_{3/2}}$ and $\alpha^{T}_{5D_{3/2}}$ can be obtained. In Refs.
Safronova  _et al._ (2004); Safronova and Safronova (2011), the listing of
either the dipole matrix elements themselves or the contributions of each
transition to other polarizabilities allow the calculation of the equivalent
contribution for the dynamic polarizability at an arbitrary frequency, using
the first term in Eq. 1. For the $(n\geq 7)F_{5/2}$ and $(n\geq 9)P_{J}$
contributions, we use dipole matrix elements from an online atom calculator
Šibalić  _et al._ (2017) or from our own calculations Reinhard  _et al._
(2007). In this way, we estimate the bound-bound contributions to the dynamic
polarizability (discrete sum in Eq. 1). To estimate the continuum integral in
Eq. 1, we compute bound-free matrix elements using model potentials given in
Marinescu  _et al._ (1994) on a dense FES energy grid (spacing $\sim h\times
40$ GHz) over a range from $\epsilon^{\prime}=0$ up to $\sim 20$ eV, yielding
continuum contributions of 18 for $\alpha^{S}_{5D_{3/2}}$ and -3 for
$\alpha^{T}_{5D_{3/2}}$. The core contribution to $\alpha^{S}_{5D_{3/2}}$, not
shown in Eq. 1, is estimated at a value of 9, provided in Martin  _et al._
(2019a) for DC and for 778 nm. Our resulting estimates for the net scalar and
tensor polarizabilities, including all mentioned contributions, are
$\alpha^{S}_{5D_{3/2}}=-516$ and $\alpha^{T}_{5D_{3/2}}=21$. From our
simulation described in the Supplementary Material, we estimate that the
presence of a tensor polarizability of 21 would cause a positive shift of
$\sim 12$ in the deduced scalar polarizability, when using Eq. 3. Therefore,
we conclude that the estimated systematic uncertainty in
$\alpha^{S}_{5D_{3/2}}$ related to the tensor polarizability lies within the
uncertainty of $\pm 17$ for the measured $\alpha^{S}_{5D_{3/2}}$ reported in
Table 1.

Figure 3: (a) $\tilde{\sigma}$ vs $I$ at the atoms’ locations. The data and a
fit result depicted in red yield a $5D_{3/2}$ photoionization cross section of
$\sigma=45(1)$ Mb. The inset shows a typical spectral peak (a single, vertical
slice from Fig. 2(a)) characterized by the 762-nm laser detuning at a fixed
795-nm frequency setting ($\nu_{795}-\nu_{1}=0.984$ GHz), along with $\Gamma$
and $\Delta^{F^{\prime}=3}_{762}$ obtained from its Lorentzian fitting
parameters. We average ion counts over 30 experimental cycles. (b) Lorentzian
peak centers in $\nu_{762}-\nu_{2}$ and their uncertainties versus
$\nu_{795}-\nu_{1}$ for the $F^{\prime}=3$ signal branch in Fig. 2 (a). The
slope of the weighted linear fit to these data is
$d\Delta^{F^{\prime}}_{762}/d\Delta^{F^{\prime}}_{795}=-0.371(6)$. We repeat
this procedure for the $F^{\prime}=2$ branch.

In the following, we discuss a measurement of the $5D_{3/2}$ PI cross section.
For a given detuning $\Delta^{F^{\prime}}_{795}$, the 1064-nm light intensity
$I$ at the locations of atoms contributing to the peak in the ion signal in
the $F^{\prime}$ branch is given by
$I=2h\Delta^{F^{\prime}}_{795}c\epsilon_{0}/(\alpha^{S}_{5S_{1/2}}-\alpha^{S}_{5P_{1/2}})$,
with an uncertainty arising from the polarizabilities and the natural
linewidth of the rubidium D1 line (5.75 MHz Steck (2021)). The linewidth of
ion spectra for fixed $\Delta^{F^{\prime}}_{795}$ and scanned
$\Delta^{F^{\prime}}_{762}$, $\Gamma/2\pi$, obtained from Lorentzian fits as
shown in Fig. 3(a), then provides an upper limit of the PI decay rate
$\Gamma_{PI}$ at intensity $I$. Using

$\Gamma_{PI}=\frac{\sigma I}{\hbar\omega_{L}},$ (4)

for each measured decay rate $\Gamma$(I) we obtain an upper bound
$\tilde{\sigma}(I)=\hbar\omega_{L}\Gamma(I)/I$ for the Rb $5D_{3/2}$ PI cross
section $\sigma$. For atoms located near the bottoms of the lattice wells,
corresponding to the largest $\Delta^{F^{\prime}}_{795}$ and the largest
intensities $I$, the broadening is near-exclusively given by PI. In contrast,
at the lowest $\Delta^{F^{\prime}}_{795}$ and intensities $I$ other
mechanisms, such as the excited-state hyperfine coupling and residual off-
resonant two-photon signals, are principal. As a result,
$\tilde{\sigma}(I)\approx\sigma$ at large $I$ and $\tilde{\sigma}(I)>>\sigma$
at low $I$. This effect is exhibited in Fig. 3(a), where the quantity
$\tilde{\sigma}$ converges to $\sigma$ (and $\Gamma$ to $\Gamma_{PI}$) at the
high-intensity end of the $I$-axis.

Quantitatively, we obtain $\sigma$ from the data in Fig. 3(a) using two
methods. In Method A, we take the arithmetic average of $\tilde{\sigma}$ in
the asymptotic region $I\gtrsim 45$ GW/m2 and utilize the standard error of
the mean as its uncertainty. Method A yields $\sigma=45(1)$ Mb. In Method B,
we apply a fit function $\tilde{\sigma}=\sqrt{\sigma^{2}+\gamma^{2}/I^{2}}$,
where $\gamma$ accounts for broadening mechanisms other than PI. This fit also
gives $\sigma=45(1)$ Mb. Combining the two methods, we have $\sigma=45(1)$ Mb.
Our simulations, explained in the Supplement, show that the measurement method
likely overestimates the PI cross section of an isotropic atom sample by
$\approx 1.7\%$, leading to our slightly corrected final result of
$\sigma=44(1)$ Mb. Based on the good qualitative agreement of measured and
simulated strength ratios between the $F^{\prime}=2$ and $F^{\prime}=3$ signal
bars in Fig. 2, where the atom sample in the simulation is isotropic, we do
not believe that optical pumping causes a significant deviation of our
measured PI cross section from what it would be under perfectly isotropic
conditions.

In a calculation based on fine-structure-less model potentials from Marinescu
_et al._ (1994), we have found a total shell-averaged PI cross section of 32.4
Mb at 1064 nm and 43.7 Mb at the PI threshold wavelength, which is close to
calculations in Duncan  _et al._ (2001) and Aymar  _et al._ (1984). Trap-loss
measurements in Duncan  _et al._ (2001) for Rb $5D_{5/2}$ gave a result of 18
Mb at 1064 nm and an estimate of 25 Mb at threshold. The discrepancies between
the results for the Rb $5D$ PI cross sections await a future explanation.

In summary, we have spectroscopically measured the dynamic scalar
polarizability of the rubidium $5D_{3/2}$ state in a 1064 nm optical lattice
using two probe lasers at 795 nm and 762 nm. We report
$\alpha^{S}_{5D_{3/2}}=-524(17)$ and estimate
$|\alpha^{T}_{5D_{3/2}}|<<|\alpha^{S}_{5D_{3/2}}|$. The observed PI-induced
line broadening has yielded a PI cross section of $\sigma=44(1)$ Mb. Future
experimental directions involve the measurement of $\alpha^{S}_{5P_{1/2}}$
with a Rydberg excitation field probing the 1064-nm lattice shifts of the
intermediate $5P_{1/2}$ state and auxiliary Rydberg levels Chen  _et al._
(2015). Our measurement will also aid in the preparation of recently-predicted
Rydberg-atom-ion molecules Duspayev  _et al._ (2021) and of novel high-
angular-momentum Rydberg states in a deep, 1064-nm Rydberg-atom optical
lattice Younge  _et al._ (2010); Cardman  _et al._ (2021) using step-wise
excitation via an intermediate Rb $5D$ state. It would be also desirable to
obtain theoretical estimates of $\alpha^{S}_{5D_{3/2}}$ and improved
theoretical values for $\sigma$ to compare with experimental results.

## ACKNOWLEDGMENTS

This work was supported by NSF grant No. PHY-1806809. X.H. acknowledges
support from the program of China Scholarships Council (No. 201808140193). We
would like to thank Yun-Jhih Chen for initial experimental work.

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