Detection of ultra-fast radio bursts from FRB 20121102A

M. P. Snelders◆,1,2, K. Nimmo3, J. W. T. Hessels2,1, Z. Bensellam2, L. P.
Zwaan2, P. Chawla2, O. S. Ould-Boukattine1,2, F. Kirsten4, J. T. Faber5 and V.
Gajjar6,7

◆<EMAIL_ADDRESS>
1 ASTRON, Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4,
7991 PD Dwingeloo, The Netherlands
2 Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science
Park 904, 1098 XH Amsterdam, The Netherlands
3 MIT Kavli Institute for Astrophysics and Space Research, Massachusetts
Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA
4 Department of Space, Earth and Environment, Chalmers University of
Technology, Onsala Space Observatory, 439 92, Onsala, Sweden
5 Cahill Center for Astronomy and Astrophysics, MC 249-17 California Institute
of Technology, Pasadena CA 91125, USA
6 SETI Institute, 339 Bernardo Ave Suite 200 Mountain View, CA 94043, USA
7 Breakthrough Listen, University of California, Berkeley, 501 Campbell Hall
3411, Berkeley, CA, 94720, USA

## Abstract

Fast radio bursts (FRBs) are extragalactic transients with typical durations
of milliseconds. FRBs have been shown, however, to fluctuate on a wide range
of timescales: some show sub-microsecond sub-bursts while others last up to a
few seconds in total. Probing FRBs on a range of timescales is crucial for
understanding their emission physics, how to detect them effectively, and how
to maximize their utility as astrophysical probes. FRB 20121102A is the first-
known repeating FRB source. Here we show that FRB 20121102A is able to produce
isolated microsecond-duration bursts whose total durations are more than ten
times shorter than all other known FRBs to date. The polarimetric properties
of these micro-bursts resemble those of the longer-lasting bursts, suggesting
a common emission mechanism producing FRBs spanning a factor of 1,000 in
duration. Furthermore, this work shows that there exists a population of
ultra-fast radio bursts that current wide-field FRB searches are missing due
to insufficient time-resolution. These results indicate that FRBs occur more
frequently and with greater diversity than initially thought. This could also
influence our understanding of energy, wait time, and burst rate
distributions.

## Introduction

FRBs are broadly divided into two categories: repeating and apparently one-off
FRBs (see Petroff et al. 1, 2 for recent reviews). Although repeaters and one-
offs typically exhibit different observational properties (e.g., repeating
FRBs generally show narrower spectral and wider temporal extents than non-
repeaters 3), it remains unclear whether this is the result of a different
source class, emission physics, propagation effects or beaming geometry4.

While repeating sources represent only a few percent of all known FRBs5, they
are important for understanding the phenomenon. The repeatability allows for
very long baseline interferometry (VLBI) localisations to probe their local
environments 6, 7, 8, 9; long-term monitoring to explore how their properties
vary with time (e.g., scattering timescale10, Faraday rotation measure, RM 11,
12 and dispersion measure, DM13, 14, 15, 16); mapping of their sometimes
periodic activity rate 17; and also multi-epoch, multi-wavelength searches
spanning radio wavelengths 18, 19 to high energies 20, 21.

The first-discovered repeating FRB, FRB 20121102A22, is one of the best-
studied sources. Not only did the repeating nature of FRB 20121102A rule out
cataclysmic models to explain all FRBs 22, but FRB 20121102A was also the
first FRB to be precisely localised to a host galaxy 23, confirming the
extragalactic distances to FRBs implied by their high DMs 24. FRB 20121102A
lives in the outskirts of a star-forming region in a dwarf galaxy at a
luminosity distance of approximately 1 Gpc 23, 25, 6, 26. The FRB emitter is
associated with a compact persistent radio source (PRS), which shows that it
may be in a dense nebula or in the near vicinity of a massive black hole 6.
This is further supported by the discovery that FRB 20121102A is in an extreme
and dynamic magneto-ionic environment, as shown by its exceptionally high and
variable RM11, 19. The RM is variable on a range of timescales27, from days to
years, and has decreased from about $127\times 10^{3}$ rad m-2 to about
$31\times 10^{3}$ rad m-2 (in the observer frame) over the span of $7$
years28, 12, 29.

While most surveys search for FRBs on timescales of milliseconds, in some
cases the raw voltage data are saved, allowing for extremely detailed studies
of FRBs probing the emission on much shorter timescales30. The as-yet non-
repeating FRB 20170827A was shown to exhibit a $0.5$ ms burst envelope, with a
narrow sub-component of duration roughly $30$ $\upmu$s 31. Similarly, FRB
20121102A occasionally shows comparably narrow temporal features within a
broader envelope11. More recently, the repeating sources FRB 20180916B and FRB
20200120E were shown to have temporal sub-structures with durations of
microseconds32 down to tens of nanoseconds33, respectively. Timescales shorter
than tens of microseconds remain unexplored for FRB 20121102A, despite micro
and sub-microsecond burst structure having been observed in other repeating
FRBs 31, 32, 33. To date, these short-timescale variations have only been
observed within broader burst envelopes with durations of at least $0.1$ ms or
longer.

Bursts from FRB 20121102A show a variety of different morphologies, from
simple single Gaussian burst profiles, to complex drifting islands of
emission, now known to be characteristic of repeating FRB morphology14, 3. FRB
20121102A bursts also often appear to be narrowband (roughly 20 % fractional
bandwidth), and have been seen to emit at frequencies between $600$ MHz 15 and
$8$ GHz 19. These $8$ GHz detections with the Green Bank Telescope (GBT) 19
remain the highest-frequency FRB detections, to date. The $4$–$8$ GHz GBT data
presented in Gajjar et al. 19 were searched for bursts at a time-resolution of
$350$ $\upmu$s, discovering a total of $21$ bursts in the first hour of a
$5$-hour observing block, with $18$ bursts occurring within the first $30$
minutes. A re-analysis of this same dataset by Zhang et al. 34, retaining the
same temporal and spectral resolution but using a machine learning search
technique (differing from the standard boxcar matched-filtering technique used
in Gajjar et al. 19), discovered an additional $72$ candidate bursts, with
half of the bursts occurring within the first $30$ minutes.

In this paper, we present a re-analysis of the first $30$ minutes of the
Gajjar et al. 19 GBT dataset, searching for ultra-fast radio bursts on
timescales of microseconds: a parameter space that the previous searches of
this dataset, and current FRB searches in general, are insensitive to.

## Observations and burst search

A $5$-hour observation of FRB 20121102A was carried out on 26 August 2017
using the GBT with the Breakthrough Listen (BL) recording system35. The BL
system recorded from $3.9$ GHz to $9.3$ GHz, fully covering the sensitivity
range of the GBT C-band receiver (Methods). Raw voltages of the dual-linear
receiver were recorded and stored allowing for offline re-processing, coherent
dedispersion, high-time-resolution searches and polarimetric analysis.

Here we have analyzed the voltage data of the first $30$ minutes of the
observation, totalling more than $32$ terabyte (TB) in size (Methods). The
voltage data were coherently dedispersed with a DM of $560.5$ pc cm-3 (from
Hessels et al. 14) using digifil36, removing the intra-channel smearing but
keeping the inter-channel smearing across the band. The resulting total-
intensity filterbanks37 have a time-resolution of $341.\bar{3}$ ns and $4.5$
GHz of bandwidth at a centre frequency of $6126$ MHz.

To save a factor of greater than $10$ in computational costs, we downsample
the filterbanks in time using digifil to create new filterbank data products
with a time-resolution of approximately $2$, $33$ and $524$ $\upmu$s,
respectively. For a hypothetical burst with a duration of $341.\bar{3}$ ns,
this reduces the sensitivity by a factor of $\sqrt{6}$ compared to using the
full available time resolution offered by the data (Methods). Since bursts
from FRB 20121102A often appear narrowband, we choose to do multiple subbanded
searches: dividing the total bandwidth into subbands of widths $4.5$ GHz,
$1.5$ GHz (the spectral width of the widest bursts in Gajjar et al. 19), $750$
MHz, $375$ MHz, and $187.5$ MHz (the spectral width of the bright patches of
the bursts in Gajjar et al. 19) before searching the data for FRBs.

We searched for bursts over a range of DMs centred around $560.5$ pc cm-3
using the PRESTO38 suite of tools and very small step-sizes in trial DM
(Methods). Microsecond-duration bursts require small step-sizes in DM because
their recovered S/N decreases rapidly if the assumed DM is incorrect39. At
these high frequencies the data are relatively clean and thus we did not apply
any radio frequency interference (RFI) masks to avoid the accidental masking
of any bright and/or short-duration bursts. Candidate astrophysical signals
were classified using the machine learning classifier FETCH40. Candidates with
a FETCH-assigned probability $p\geq 0.5$ of being astrophysical or temporal
duration $\leq 500$ $\upmu$s (regardless of the probability given by FETCH)
were manually inspected.

## Results

We find 49 unique bursts within the first $30$ minutes of the observation
(Supplementary Table 1). All of the bursts presented in Gajjar et al. 19 are
re-detected and from the Zhang et al. 34 sample we re-detected $30/46$ of the
candidate bursts, missing only low signal-to-noise (S/N) pulses (where the
astrophysical nature of these events is in any case unclear). We find $19$
bursts that were missed in both previous searches of this dataset. Of the
newly found bursts, $8$ are very short in duration, with the entire burst
envelope lasting no more than about $15$ $\upmu$s (Figures 1 and 2). We refer
to these bursts as ‘ultra-fast radio bursts’, or ultra-FRBs, because of their
microsecond durations.

Bursts B23, B26 and B32 extend up to about $8.4$ GHz, making them the highest-
frequency FRBs detected to date. The emission properties above this frequency
are unknown because above $8.4$ GHz there is a sharp drop in the sensitivity
of the receiver.

The distribution of burst durations (as they were found in the search, Figure
2) may appear by eye to be bimodal, but accounting for Poissonian
uncertainties we find no clear statistical evidence for this. In fact, the
burst durations are marginally consistent with being logarithmically uniform
from microseconds to milliseconds. If we split the burst sample in two at a
duration of $40$ $\upmu$s, we find $8^{+12.7}_{-5.9}$ and $41^{+23.1}_{-16.6}$
bursts below and above this threshold, respectively. Here the uncertainties
represent the $3\sigma$ Poissonian error.

To study the bursts with more precision, we created high-time-resolution,
full-polarisation data products for a selection of bursts; these have been
coherently dedispersed to a DM of $560.105$ pc cm-3 (Methods). These data
products have a time-resolution of $341.\bar{3}$ ns and a frequency resolution
of $2.9296875$ MHz ($\delta\nu\delta t=1$, limited by the uncertainty
principle). It would be difficult to accurately achieve a higher time
resolution by inverting the signal processing step that formed the frequency
channels.

### High-time-resolution

In Figure 1 we present the total intensity (Stokes I) dynamic spectra and
burst profiles for the $8$ shortest-duration bursts in our sample. We
coherently dedisperse all of the bursts to the best-fit DM for burst B30
(Methods and Extended Data Figures 1 and 2), $\mathrm{DM}=560.105$ pc cm-3,
because of its high S/N and extremely narrow burst width. Nonetheless, we note
that there is unavoidable ambiguity in the exact DM of each burst because
their time-frequency morphology is not known a priori.

Bursts B30 (Figure 1ab) and B43 (Figure 1cd) are the highest-S/N bursts in our
ultra-FRB sample. Like other bursts detected in these observations, B30 shows
bright patches in frequency (consistent with the expected Galactic
scintillation of $8$ MHz at $4.7$ GHz and $87$ MHz at $8.0$ GHz19), and is
extended over more than $800$ MHz of bandwidth. The leading edge of the burst
is extremely steep, going from the noise floor to the peak of the burst in
about $1$ $\upmu$s. The entire burst lasts a mere $5$ $\upmu$s, making it the
shortest isolated FRB to date.

Burst B43 is comparatively long-duration: about $15$ $\upmu$s. The shape of
the burst in time-frequency space remains curved after dedispersing: the
remaining curvature does not appear to follow the $\nu^{-2}$ relation for
dispersion and therefore cannot simply be corrected for using a different DM.
Compared to B30, it also shows clear bright patches in frequency, but a less
steep leading edge. Rather, the profile of the burst (Figure 1c) is more
Gaussian shaped. Burst B43 is also found in the Zhang et al. 34 search due to
the high S/N of the burst, but since their data were not coherently
dedispersed and had a sampling time of $350$ $\upmu$s (much larger than the
total burst width), the microsecond duration of this burst was not visible.

Burst B06 (Figures 1ef and 3ef) shows multiple components, and is the only
ultra-FRB that is detected at a central radio frequency $\geq 7$ GHz. The
burst is composed of three components with separations of roughly $15$
$\upmu$s and $25$ $\upmu$s. The first component is the brightest and has a
duration of only about $4$ $\upmu$s.

The remaining ultra-FRBs are weaker and need to be downsampled in time and/or
frequency to be clearly seen in the dynamic spectra. The spectral extent of
the bursts varies between about $120$ MHz (burst B31) and $800$ MHz (bursts
B30 and B38). The ultra-FRBs do not obviously favour any particular frequency-
range nor do they occupy a particular time-range compared to the other bursts
in this dataset (Extended Data Figures 3 and 4).

We calculate the peak flux density, fluence and isotropic-equivalent spectral
luminosity of the 8 ultra-FRBs (Methods and Table 1). We also show these in
the context of the transient phase space diagram (Extended Data Figure 5). The
bursts all have a relatively high spectral luminosity and low transient
duration compared to other bursts from FRB 20121102A and other localised
repeating FRBs. Notably, burst B30 has an inferred brightness temperature that
exceeds $10^{40}$ K. Such high brightness temperatures have so far only been
observed in ‘nanoshots’ from the Crab pulsar41, 42, 43 and in sub-bursts from
FRB 20200120E33.

With the refined DM of $560.105$ pc cm-3, measured using B30 (Methods), we
carefully examined the three highest-S/N bursts discovered in Gajjar et al. 19
on microsecond timescales (Figure 3). No micro-structure was identified in any
of these bursts. A more detailed study of the larger sample at high time-
resolution will be presented in a future paper.

### Polarimetric properties

After calibrating the full polarisation data (Methods), we measure the RM of
our burst sample using a multi-burst joint Stokes QU-fit11 (Figure 4). Similar
to an analysis of bursts from FRB 20121102A from the 305-m Arecibo telescope
by Michilli et al. 11, we find that all of the bursts, including the ultra-
FRBs, can be fit with a single polarisation position angle (PPA), when
averaged over the burst duration and an RM of $93586\pm 4$ rad m-2 (in the
observer frame), which is consistent with previous RM measurements of these
bursts 11, 19, 44. We de-Faraday the Stokes parameters Q and U using the
measured RM and determine the unbiased linear polarisation fraction,
$\mathrm{L}_{\mathrm{unbias}}$ (Methods).

We plot the profiles of Stokes I, $\mathrm{L}_{\mathrm{unbias}}$ and the
circular polarisation (Stokes V) in the bottom panels of Figure 3. We find
that almost all of the ultra-FRBs are consistent with being $100$ % linearly
polarised and show no circular polarisation — similar to the millisecond-
duration bursts11, 19, 44. The exception to this is burst B43, which shows
$31\pm 10$ % circular polarisation and $87\pm 10$ % linear polarisation: still
consistent with being $100$ % polarised overall (Methods).

We compute the time-resolved PPA across the burst profile. Due to the
limitations of our calibration method (Methods), the absolute value of the PPA
is not meaningful. We therefore, per burst, subtract the weighted mean, using
$\mathrm{L}_{\mathrm{unbias}}$ as weights, so the PPA values are centred
around zero. A constant line is then fit to the PPAs and we measure the
reduced-$\chi^{2}$, $\chi^{2}_{\nu}$. The PPA probability distribution45 for
every time bin is shown in the upper panels of Figure 3. We find that at high
time-resolutions $\chi^{2}_{\nu}>1$, indicating that the PPAs are
significantly scattered. This is also observed in a comparable analysis of the
three selected bursts from the Gajjar et al. 19 sample (Figure 3).

Bursts B30 (the shortest-duration burst in our sample, Figure 3ab) and B01
(the longest-duration burst in our sample, Figure 3gh) are intentionally
placed on top of each other to illustrate that, even though their durations
differ by a factor of roughly $600$, their polarimetric properties are very
similar.

## Discussion

Galactic neutron stars, including rotation-powered radio pulsars and radio-
emitting magnetars, are known to produce at least three categories of coherent
radio pulses46: i. canonical pulsar emission from the magnetic polar cap; ii.
microsecond-duration (or shorter) giant pulses from both energetic millisecond
and young pulsars (like the Crab pulsar); and iii. magnetar radio bursts. The
durations, spectra, rotational phase, etc., of these various emission types
strongly suggest that they originate in different parts of the neutron star
magnetosphere and likely via different physical processes. Here we have shown
that FRB sources can also produce isolated microsecond-duration pulses, whose
luminosities are many orders of magnitude brighter than even the giant pulses
from the Crab pulsar (Extended Data Figure 5). The range of timescales we
observe from FRB 20121102A is consistent with what is seen from neutron stars,
and this may indicate that FRBs can also produce multiple burst types. The
similarities between FRB microstructure and Crab giant pulses are discussed in
more detail in, for example, Nimmo et al. 33.

Microsecond (and shorter) timescale variations have previously been observed
in bursts from FRB 20180916B 32 and FRB 20200120E 33. In these cases, the
short timescale variations occur within a broader burst envelope. In contrast,
here we present the discovery of isolated microsecond-duration bursts (without
a broader envelope) from FRB 20121102A. These ultra-FRBs were missed in
previous searches of the same data, where the time resolution was more than
100 times lower 19, 34. As suggested in Nimmo et al. 32, 33, we have shown
here that there indeed exists a population of ultra-FRBs that current FRB
search strategies are missing. The computational expense of searching for
ultra-FRBs, because of the required high time resolution and coherent
dedispersion, has resulted in them being undetected in all other FRB searches
to date. Furthermore, scattering is a significant limitation to resolving such
timescales when observing at low radio frequencies (less than about $1$ GHz).

While FRB 20180916B7, 47, FRB 20200120E8 and FRB 20121102A23, 25, 6, 26 live
in drastically different environments, their burst properties show striking
similarities. For example, similarities in the burst morphologies and
polarimetric properties, including downward-drifting structure in time and
frequency, spectrally narrow emission14, 48, 49, and now microsecond-duration
fluctuations as well32, 33. In addition to the short timescales, the 3 orders-
of-magnitude range of timescales observed in this work resembles the range of
timescales observed for FRB 20180916B32 and FRB 20200120E33. However, it is
possible that microsecond-duration bursts are equally, or even more common in
non-repeating FRBs30 due to them having narrower widths in general3. That
micro-shots have so far been exclusively detected in repeaters is likely
because of the high-time-resolution follow-up observations conducted for these
sources. Alternatively, it could also be because more individual pulses have
been seen and studied in the case of repeating FRBs. There are close to $500$
one-off FRBs in the first CHIME/FRB catalogue50, but they lack the required
time resolution to search for these short-duration sub-bursts. Additional
studies using CHIME/FRB baseband detections of these FRBs will help elucidate
the prevalence of micro-structures in one-off FRBs.

The polarimetric properties of repeating FRBs are often described by high
degrees of linear polarisation, little-to-no circular polarisation, flat PPAs
during the burst duration and only small ($<30^{\circ}$) variations in PPA
between bursts detected at the same observing epoch11, 32, 33. FRB 20180916B
and FRB 20121102A have also been seen to depolarise with decreasing
frequency18, 28, 51, 52. In this work, we additionally show that, at very high
time-resolution, the PPA during the burst duration varies significantly
(Figure 3), resembling the high-time-resolution PPA variations seen for FRB
20180916B32 and FRB 20200120E33.

Recently, Feng et al. 53 showed that FRB 20121102A bursts at $1.25$ GHz are
occasionally circularly polarised (in $<1$ % of bursts) while being $0$ %
linearly polarised (as is the case for all FRB 20121102A bursts at these
frequencies). We observe one ultra-FRB in our sample to be significantly
circularly polarised, burst B43: this corresponds to a rate of circular
polarisation for $4$–$8$ GHz of a few percent which appears to be roughly
consistent with Feng et al. 53, though a larger sample of high-radio-frequency
bursts should be considered in a future study to compare the rates. Feng et
al. 53 suggest that Faraday conversion or the radiation mechanism itself could
be responsible for the circular polarisation, while multi-path propagation in
either the Milky Way or FRB local environment is disfavoured. Our detection of
circular polarisation at much higher radio frequencies supports the idea that
the origin is intrinsic to the emission mechanism, as opposed to being a
propagation effect; we speculate that the rare occurrence of circular
polarisation could be a result of how the emission is beamed towards Earth.
Furthermore, circular polarisation has been detected from the repeating FRB,
FRB 20220912A, which has been shown to live in a relatively clean local
environment, with no changes in DM or RM on month timescales 54, 55. This also
suggests that the circular polarisation is intrinsic to the emission
mechanism.

The distribution of burst durations (Figure 2) is marginally consistent with
being log-uniform between microseconds to milliseconds. The selection bias as
a function of burst duration is not well known, and could swing either in
favour of more ultra-FRBs or more typical-duration FRBs. In our search, the
timescales below $30$ $\upmu$s have only been searched with one trial boxcar,
while timescales $\geq 30$ $\upmu$s have effectively been searched at least
twice due to different combinations of the sampling time and boxcar matched-
filtering widths (Figure 2). If all bursts from FRB 20121102A draw their
fluences, $\mathcal{F}$, from the same underlying distribution, ultra-FRBs
should actually be easier to find since the detection metric is
$\mathcal{F}/\sqrt{W}$, with $W$ the temporal duration of the burst. It is
unclear, however, whether the micro-shots follow the same fluence distribution
as the wider bursts.

The burst spectra and polarimetric properties (polarisation fractions, RM, and
PPA) are consistent between the ultra-FRBs and the wider-bursts, suggesting
that they share a common emission mechanism. It is unclear, however, why this
mechanism sometimes shuts off after only microseconds whereas at other times
it continues for several milliseconds. For FRB 20121102A, the distribution of
timescales is roughly uniform from microseconds to milliseconds. Other sources
may show different temporal distributions and average burst duration; in any
case, FRB searches are currently highly biased against finding ultra-FRBs. FRB
20200120E’s bursts are typically 100 $\upmu$s in duration49, much shorter than
other repeating FRBs3, further supporting the argument for an undetected FRB
population with shorter timescales.

In Nimmo et al. 49 approximately $50$ bursts from FRB 20200120E are analysed
but only one burst has clear microstructure superimposed on a broader
envelope. For that one burst the ratio between the amplitude of the micro-shot
and the broader envelope is only about $3$ — meaning that it would require
fine-tuning of the telescope sensitivity to detect only the micro-shot and not
the broader emission as well. For both FRB 20200120E and FRB 20180916B the
broader bursts were found in searches with a time resolution of $64$
$\upmu$s33 and $82$ $\upmu$s7, respectively. The micro-shots in those data
were only found once the baseband data were both coherently dedispersed and
higher time resolution data products were generated. At $64$ $\upmu$s time
resolution the ultra-FRBs from FRB 20121102A would not have been found. This
supports the idea that the ultra-FRBs from FRB 20121102A are truly isolated
events.

Fluctuations on timescales of tens of nanoseconds have been observed within
the envelopes of bursts originating from FRB 20200120E33. With the current
data, we are unable to probe similar timescales in the FRB 20121102A ultra-
FRBs and wider-bursts. The data used in this study has a limiting time-
resolution of $341.\bar{3}$ ns and for the majority of the ultra-FRBs
discovered in this work the S/N is relatively low. The large scatter observed
in the PPA at high time-resolution, however, is suggestive of the bursts being
made up of narrower shots of emission. The 3 wider bursts from the Gajjar et
al. 19 sample that we also study here show no evidence of micro-structure
within the burst envelopes (Figure 3). This supports the hypothesis that the
ultra-FRBs in our sample are truly isolated events as opposed to being
fluctuations within a wider, dimmer burst component.

The isolated ultra-FRBs thus also provide a better constraint on the size of
the emission region because they are unlikely to be generated by propagation
effects that modulate the burst brightness post-emission56. The extremely
short duration of B30 constrains the emission region to be smaller than a few
kilometres (ignoring any potential relativistic effects). This favours models
in which the bursts are generated near the central engine in a magnetosphere -
as opposed to models in which the bursts are generated much further out in a
relativistic shock.

The existence of ultra-FRBs could influence what one infers about energy, wait
time and burst rate distributions. We therefore encourage that future
experiments should try to search the widest-possible range of timescales. This
should especially be done for repeating FRBs where the known DM and just a
single pointing direction make this computationally easier compared to an
untargeted wide-field search.

The limitations of extremely high-time-resolution FRB searches include:
knowing the DM a priori to be able to coherently dedisperse the data, or
alternatively performing a search using both coherent and incoherent
dedispersion steps; scattering from the Milky Way interstellar medium, which
causes temporal broadening that increases with decreasing frequency;
instrumental bandwidth limiting the sampling rate of the observations; and,
simply, the overall computational cost. The discovery of ultra-FRBs from FRB
20121102A in this work highlights that such transients exist, that they are
being missed in current FRB searches, and that there is reason to invest time
in overcoming the challenges associated with searches for ultra-FRBs in the
future, particularly at giga-Hertz frequencies where scattering from the
Galactic interstellar medium is less than a microsecond for most directions on
the sky. Opening a wider range of timescales for FRB discovery can both lead
to the identification of new FRB source types as well as even more precise
probes of the intervening magnetised plasma and gravitational potential2.

## Methods

### The observation

Observations of FRB 20121102A were conducted with the Robert C. Byrd Green
Bank Telescope (GBT) using the $4$–$8$ GHz (C-band) receiver on 2017 August 26
during a $6$-hour observing block. The first hour was used for calibration
procedures: a noise diode scan was used for flux and polarisation calibration
and a $5$-minute observation of the pulsar PSR B0329+54 was performed to
verify the calibration procedure. The last $5$ hours of the observing block
were split into $30$-minute sections and were spent on FRB 20121102A19.
Observations were conducted with the Breakthrough Listen (BL) digital backend
35, which recorded $8$-bit raw voltage (baseband) data from $3.9$ GHz to $9.3$
GHz — fully covering the C-band receiver bandwidth. The analog down-conversion
system provided four $1500$-MHz passbands configured with central frequencies
of $8563.964844$, $7251.464844$, $5938.964844$, and $4626.464844$ MHz,
respectively. Each dual-polarisation (linear basis) passband was Nyquist-
sampled using $8$-bit digitizers, polyphase channelized to $512$ ‘coarse’
frequency channels, and distributed to a cluster of compute nodes that
recorded these data to disk. In total there are $32$ compute nodes, $8$ for
every passband. Every compute node recorded $187.5$ MHz of bandwidth over $64$
‘coarse’ channels. The passbands have exactly $187.5$ MHz of overlap with
adjacent passbands (Supplementary Table 2). Due to the polyphase filter, the
highest time resolution possible for these data is
$\left(1500\text{\,MHz}/512\right)^{-1}=\left(2.9296875\text{\,MHz}\right)^{-1}=341.\overline{3}$
nanoseconds, where the $\overline{3}$ represents a repeating digit. The nodes
store the data in chunks to keep file sizes manageable. Every chunk consists
of exactly $2^{26}$ time samples, corresponding to $2^{26}\times
341.\overline{3}$ ns $=22.91$ seconds, and is $17.2$ GB in size. Each
$30$-minute scan thus consists of $78$ ‘full’ chunks and one chunk containing
the last $13.29$ seconds. The chunks are stored in GUPPI (Green Bank Ultimate
Pulsar Processing Instrument) raw format (i.e., .raw files). Detailed
information of the observation, the BL project, the BL backend and the GUPPI
raw format can be found in Refs. 19, 57, 35 and 58, respectively.

### Data preparation

The baseband data (.raw files) of the first $30$ minutes were downloaded from
the BL open data archive. The data of $24$ different nodes (see Supplementary
Table 2 for a list of used and available nodes) was retrieved, covering the
frequency range $3876.464844$ MHz – $8376.464844$ MHz. In total, $79\times
24\times 17.2\text{ GB}=32.4$ terabytes (TB) of .raw data was downloaded. For
every chunk, the $24$ .raw files were spliced together in frequency using a
modified version of splicer_raw.py, which is a Python script that is part of
the BL extractor package. The $79$ spliced .raw files, each $412$ GB in size
(the last one is $239$ GB), were converted to a Sigproc37 filterbank file
using digifil. This is the digifil that is part of bl-dspsr. The filterbank
files have been coherently dedispersed within the channels, but not between
the channels, using a dispersion measure (DM) of $560.5$ pc cm-3 (from Hessels
et al. 14). The resulting filterbank files contain $8$-bit intensity data and
have a time-resolution of $341.\overline{3}$ ns. The $4500$-MHz bandwidth of
the files is made out of $1536$ channels with a frequency resolution of
$2.9296875$ MHz.

### The search

Benchmarking showed that searching the data at a time resolution of
$341.\overline{3}$ ns was too computationally expensive. Instead we opted to
further downsample the data in time, using digifil to create new Sigproc
filterbanks at time resolutions of $2.048$, $32.768$ and $524.288$ $\upmu$s,
respectively.

Previous searches19 of these data showed that the bursts are narrowband.
Therefore we create multiple ‘subbands’ out of the $4.5$-GHz bandwidth
filterbank file. We use subbands of $1500$, $750$, $375$ and $187.5$ MHz,
where each subband has a $50$ % overlap with adjacent subbands, e.g., the
$4500$-MHz file is split into five $1500$-MHz files. A total of

$\begin{split}\ &1+\sum_{i}2\times 4500/i-1=87\\\ \ &\mathrm{where}\ \
i\in\left[1500,750,375,187.5\right]\end{split}$ (1)

unique subbands were searched. Combined with the three possible time-
resolutions of the data, there were $87\times 3=261$ passes through the data.

Every filterbank file was dedispersed and summed over all frequency channels
using a range of dispersion measures (Supplementary Table 3) with PRESTO’s
prepsubband. The steps in DM are equal, or smaller, than the step-size
suggested by PRESTO’s DDplan.py. The resulting time series were searched with
PRESTO’s single_pulse_search.py using a S/N threshold of $7$. For the $2$
$\upmu$s and $33$ $\upmu$s data we made use of all the available boxcar
widths, which are logarithmically scaled between $1$ and $300$, while for the
$524$ $\upmu$s data the highest boxcar width that was used is $45$. This made
the search sensitive to timescales between $2$ $\upmu$s and $24$ ms – which is
the maximum duration of a large sample of FRB 20121102A bursts at about $1.4$
GHz as presented in Hewitt et al. 59. At no stage do we apply any radio
frequency interference (RFI) excision, to avoid masking any potential bright
or short-duration bursts.

All the candidates were classified using the machine learning algorithm
FETCH40 (model A). We reject all the candidates that have a temporal width
$>0.5$ ms and probability of being an astrophysical signal $<50$ %. Since
FETCH is trained on (simulated) FRBs with widths between $0.5$ ms and $50$ ms,
we chose to manually inspect every candidate with a width $\leq 0.5$ ms,
regardless of their FETCH-derived probability of being an astrophysical
signal.

We manually inspect the roughly $2300$ candidates that meet our selection
criteria and find that about $2100$ are FRBs. Due to the complex time-
frequency structure of some of the FRBs and the fact that we have $261$ passes
over the data, all of the FRBs are found multiple times — the most extreme
case is burst B01, which is found over $400$ times (Supplementary Table 1).

Bursts are sorted by their arrival times and are clustered using their
proximity in time. A ‘new’ cluster starts if the time difference between
consecutive bursts exceeds $10$ ms. Every cluster is manually checked for any
potential bursts that are close enough in time that they were grouped
together. We keep the clusters if at least one of the burst detections in that
cluster has S/N $\geq 8$.

Given the discovery of the ultra-FRBs, we re-checked all the candidates with a
duration of $2$ $\upmu$s, regardless of their S/N or FETCH probability. After
an additional manual inspection, none of these candidates are believed to be
real astrophysical signals.

### Dispersion measure determination

We use burst B30 to determine the best DM. This burst was chosen because it
has a high S/N, extremely short duration and compared to the other
microsecond-duration bursts it has a relatively large extent in frequency
(Figure 1b and Extended Data Figure 2bd). We coherently dedisperse the burst
to a DM of $560.1$ pc cm-3 and incoherently dedisperse it over a range of
trial DMs — from $559.7$ pc cm-3 to $560.7$ pc cm-3, in steps of $0.0002$ pc
cm-3. For every trial DM we sum the dynamic spectrum over the frequency extent
$5650$–$6525$ MHz. The profile of the burst is normalized such that the off-
burst regions have zero mean and unit standard deviation and the peak S/N is
recorded. We plot the results in Extended Data Figure 1. Individual
measurements are plotted as grey circles and for visual purposes a moving
average is shown as a solid black line. The peak S/N value is not clearly
maximized at one specific DM value. We attribute this to brightness
fluctuations (varying in both time and frequency) of the burst in the dynamic
spectrum (Figure 1b and Extended Data Figure 2bd), as shown in e.g. Hessels et
al. 14. As discussed there, this could be a radius-to-frequency mapping effect
or, in principle, such effects could also arise if the emission at different
frequencies is occurring at different physical distances from Earth. The light
travel distance of $341.\bar{3}$ ns (the highest possible time resolution)
corresponds to about $100$ m (in the absence of relativistic effects). At such
high time resolution, subtle changes in the location of the emission region
may become apparent. The frequency-dependent pulse shapes of pulsars are also
known to complicate DM determination60, 61.

To determine the best DM, we fit a Lorentzian distribution to the data points
close to the peak (shown as a solid green line) and take the best DM as the
center of the fitted distribution, which is at $560.105$ pc cm-3. Due to the
complex shape of the peak S/N profile we estimate the error on the DM to be
$0.05$ pc cm-3.

The same method is repeated for burst B43 (over the frequency extent
$4700$–$5150$ MHz). Extended Data Figure 1 shows individual measurements as
grey pentagons and the dashed black line shows the corresponding moving
average. All the individual data points are fitted with a Gaussian
distribution, which peaks at a DM of $560.308$ pc cm-3. We estimate the error
on this DM to be $0.25$ pc cm-3, given the relatively flat peak S/N profile of
B43. Extended Data Figure 2 illustrates the effects of using a different DM
for bursts B30 and B43. A DM of $560.308$ pc cm-3 would decrease the duration
of burst B43 by about $2$ $\upmu$s (Extended Data Figure 2efgh). However, if a
DM of $560.308$ pc cm-3 is applied to burst B30 it is clearly over-dedispersed
(Extended Data Figure 2cd) and we therefore continue the analysis with a DM of
$560.105$ pc cm-3.

We note that this DM is lower than the one reported in Gajjar et al. 19
($565.0$ pc cm-3) and Hessels et al. 14 ($563.86$ pc cm-3), both of which used
a structure-maximizing DM on burst B01 (called ‘GB-BL’ in Hessels et al. 14
and ‘11A’ in Gajjar et al. 19). However, it should be mentioned that, using
bursts between $1.2$ GHz and $2.3$ GHz, Hessels et al. 14 find a DM of $560.5$
pc cm-3 for FRB 20121102A bursts in general near the epoch of our GBT
observations — giving a better handle on the determined DM because of the
lower radio frequencies.

### Residual temporal smearing

Even though the availability of voltage data allows for coherent dedispersion
there could still be intra-channel smearing due to the usage of an incorrect
value of the DM. Furthermore, intrinsically narrow pulses could also be
broadened by scattering from the Milky Way’s interstellar medium.

In Gajjar et al. 19 they used the Galactic electron density model NE200162 and
estimated a Galactic scattering timescale for FRB 20121102A of $\tau_{s}=20\
\upmu\mathrm{s}\ \nu^{-\alpha}$. Here $\nu$ is the observing frequency in GHz
and $\alpha$ is a scaling parameter which is between $4$ (thin screen model63)
and $4.4$ (Kolmogorov spectrum64). We find that the scattering timescale is
between $2$ ns (best case, $\nu=8.0$ GHz and $\alpha=4.4$) and $41$ ns (worst
case, $\nu=4.7$ GHz and $\alpha=4.0$). Since the highest-possible time
resolution available in our data is $341.\bar{3}$ ns, temporal smearing due to
Galactic scattering is expected to be undetectable in the temporal profiles of
the bursts.

The dispersive sweep can be calculated as46:

$\Delta t=\mathfrak{D}\times(\nu_{1}^{-2}-\nu_{2}^{-2})\times\mathrm{DM}$ (2)

where $\mathfrak{D}$ is the dispersive constant, and $\nu_{1,2}$ are the
upper- and lower bound of the frequency range. Thus, for coherent dispersion,
a channel width of $2.9296875$ MHz, observing frequencies of $4.7$ GHz and
$8.0$ GHz and a dispersive constant of $1/(2.41\times 10^{-4})$ MHz2 pc-1 cm3
s (the same constant that is used in, e.g., dspsr) we find a smearing of $234$
ns and $48$ ns per unit DM within an individual channel. Since the error on
the DM is about $0.05$ pc cm-3 (Extended Data Figure 1) and the highest-
possible time resolution is $341.\bar{3}$ ns, temporal smearing due to
dispersion within individual channels is also insignificant.

### Sensitivity to timescales

The highest time resolution at which the data was searched was $\Delta
t=2.048$ $\upmu$s with a S/N $\geq 8$ detection limit. Any bursts shorter than
$\Delta t$ would thus have a decreased S/N since noise would have been added
into the single time bin in which the burst would occur. The decrease in S/N
scales with the square root of the fraction between the sampling time and the
burst duration. Therefore it is possible that relatively weak and even shorter
duration (less than $2$ $\upmu$s) bursts are missed in the search. However,
individual time samples in the profiles of bursts B30 and B43 reach a S/N
greater than $20$ at a time resolution of $341.\bar{3}$ ns (Figures 1ac and
3bd). A burst with a duration of $341.\bar{3}$ ns and a S/N of $20$ would have
its S/N decreased by a factor $\sqrt{6}$ and would appear as a S/N
$20/\sqrt{6}=8.2$ candidate and would likely have been found in the search,
though just barely.

### Checks for saturation

Bright FRBs can saturate the receiver system and/or recording backend (see
Ikebe et al. 65 and Kirsten et al. 66 for examples). We investigated if burst
B43 is saturated because this burst is relatively bright between $4.9$ GHz and
$4.93$ GHz (Figure 1d). The baseband data of the BL backend are stored as
complex $8$-bit signed integers35, meaning that both the real and the
imaginary part utilize one byte to store their respective signed integers
(i.e., an integer between $-128$ and $+127$, inclusive). The baseband data of
burst B43 was loaded into Python and every sample for all subbands and both
polarisation channels were checked. We found that all the integer values are
within $\pm 105$, suggesting that there is no saturation in the recording
system. It is not surprising that the signal is not saturated: the intensity
of the signal is spread over both polarisation channels and over both the real
and imaginary parts. Furthermore, the signal is also spread out in time in the
baseband data due to dispersive smearing within a subband ($>100$ $\upmu$s for
B43) and only with coherent dedispersion is the microsecond duration of these
bursts revealed. The $8$-bit depth of the baseband data used here is deeper
than cases in which saturation did occur ($4$-bit baseband depth for the burst
shown in Ikebe et al. 65 and $2$-bit baseband depth for the bursts shown in
Kirsten et al. 66).

### Polarimetric calibration

To study the bursts in greater detail, we use digifil to coherently dedisperse
(using a DM of $560.105$ pc cm-3) the baseband data for all the bursts that
are shown in Figures 1, 3 and 4. The resulting files contain the coherence
products and have a time resolution of $341.\bar{3}$ ns, a frequency
resolution of $2.9296875$ MHz and a total bandwidth that varies per burst. The
files are stored in filterbank37 format using $32$-bit floating-point numbers
to avoid potential saturation effects.

In a linear basis with complex sampling, the Stokes I, Q, U and V parameters
can be constructed from the auto- and cross-correlations (i.e., the coherence
products) using67

$\displaystyle\mathrm{I}$
$\displaystyle=\left<\mathrm{AA}^{*}+\mathrm{BB}^{*}\right>$ (3)
$\displaystyle\mathrm{Q}$
$\displaystyle=\left<\mathrm{AA}^{*}-\mathrm{BB}^{*}\right>$ (4)
$\displaystyle\mathrm{U}$ $\displaystyle=\left<2\
\mathrm{Re}\left(\mathrm{AB}^{*}\right)\right>$ (5) $\displaystyle\mathrm{V}$
$\displaystyle=\left<2\ \mathrm{Im}\left(\mathrm{AB}^{*}\right)\right>$ (6)

where $\mathrm{AA}^{*}$ and $\mathrm{BB}^{*}$ are the auto-correlations of the
polarisation channels and $\mathrm{Re}\left(\mathrm{AB}^{*}\right)$ and
$\mathrm{Im}\left(\mathrm{AB}^{*}\right)$ are the real and imaginary parts of
the cross-correlations. An instrumental delay between the polarisation
channels will affect the cross-correlation $\mathrm{AB}^{*}$, which in turn
will affect the Stokes parameters U and V. We make use of the baseband data of
a noise diode scan, taken $2$ minutes before the start of the observation, to
determine the delay between the polarisation channels. The data were folded on
the switching period ($0.04$ s) of the noise diode using dspsr36. An ‘archive’
format file was made for each of the four passband containing the coherence
products. We determine the phase angle (PA) between the cross-products using:

$\mathrm{PA}\left(\nu\right)=\frac{1}{2}\arctan{\left[\mathrm{Re}\left(\mathrm{AB}^{*}\right)/\
\mathrm{Im}\left(\mathrm{AB}^{*}\right)\right]}$ (7)

A slope $\mathcal{S}$ is fit to the $\mathrm{PA}$, taking the wrapping at
$\pm\pi/2$ rad into account. The slope $\mathcal{S}$, which has units of
$\mathrm{rad}/\mathrm{Hz}$, is converted to a delay $\mathcal{D}$ using
$\mathcal{D}=\mathcal{S}/\left(\pi\ \mathrm{rad}\right)$.

The delay between the polarisation channels is, as expected, different per
passband but is on the order of $2.5$ nanoseconds — similar to delays found in
other radio telescopes68, 69, 32.

We correct the cross-correlations for the instrumental delay using:

$\displaystyle\mathrm{Re}\left(\mathrm{AB}^{*}\right)_{\mathrm{corrected}}=\mathrm{Re}\left(\mathcal{Y}\right)$
(8)
$\displaystyle\mathrm{Im}\left(\mathrm{AB}^{*}\right)_{\mathrm{corrected}}=\mathrm{Im}\left(\mathcal{Y}\right)$
(9) $\displaystyle\mathrm{where}\
\mathcal{Y}=\left[\mathrm{Re}\left(\mathrm{AB}^{*}\right)+i\
\mathrm{Im}\left(\mathrm{AB}^{*}\right)\right]\times e^{-2i\pi\nu\mathcal{D}}$
(10)

The Stokes parameters are constructed from the auto-correlations and the
delay-corrected cross correlations using Equations 3–6. Every channel for
every Stokes parameter is normalized by subtracting the mean and dividing by
the standard deviation of an off-burst region of that channel.

We determine the rotation measure (RM) using a multi-burst joint QU-fit
(Figure 4) using the following equations:

$\displaystyle\mathrm{Q}/\mathrm{L}=\cos$
$\displaystyle\left(2\left[c^{2}\mathrm{RM}/\nu^{2}+\phi\right]\right)$ (11)
$\displaystyle\mathrm{U}/\mathrm{L}=\sin$
$\displaystyle\left(2\left[c^{2}\mathrm{RM}/\nu^{2}+\phi\right]\right)$ (12)

where $c$ is the speed of light, $\mathrm{L}$ the quadrature sum of
$\mathrm{Q}$ and $\mathrm{U}$, i.e.
$\mathrm{L}=\sqrt{\mathrm{Q}^{2}+\mathrm{U}^{2}}$ and
$\phi=\phi_{\inf}+\phi_{\mathrm{inst}}$ where $\phi_{\inf}$ is the absolute
angle of the polarisation of the sky referenced to infinite frequency and
$\phi_{\mathrm{inst}}$ is the phase difference between the polarisation hands.
In the fit, the parallactic angle is assumed to be the same for all the
bursts. In reality the parallactic angle changes by $6^{\circ}$ between the
first and the last burst in our dataset. We deviate from the more generalized
form of these equations (see, e.g., Equations 4 and 5 of Nimmo et al. 32) that
incorporate a term for the instrumental delay since we have removed the delay
a priori. We find an observed RM, $\text{RM}_{\text{obs}}$, of $93586\pm 4$
rad m-2, where the error is the $1\sigma$ statistical error. The RM in the
source reference frame11, $\text{RM}_{\text{src}}$, is $(1+z)^{2}\
\text{RM}_{\text{obs}}=1.42\ \text{RM}_{\text{obs}}=1.3\times 10^{5}$ rad m-2,
where $z$ is the redshift of the host galaxy of FRB 20121102A25. This RM is
consistent with previously reported RM values of bursts from the same
dataset27, 19, 44.

The dynamic spectrum of the Stokes parameters Q and U are corrected for
Faraday rotation with the aforementioned RM, using:

$\displaystyle\mathrm{Q}_{\mathrm{corrected}}$
$\displaystyle=\mathrm{Re}\left(\mathcal{W}\right)$ (13)
$\displaystyle\mathrm{U}_{\mathrm{corrected}}$
$\displaystyle=\mathrm{Im}\left(\mathcal{W}\right)$ (14)
$\displaystyle\mathrm{where}\,\mathcal{W}$
$\displaystyle=\left[\mathrm{Q}+i\,\mathrm{U}\right]\times e^{-2i\
c^{2}\nu^{-2}\ \mathrm{RM}}$ (15)

We compute the time series of the Faraday-rotation-corrected Stokes I, Q, U
and V parameters by summing over the frequency extent of the bursts. Each of
the four time series are then again normalized such that the off-burst regions
have zero mean and unit standard deviation. We calculate the measured linear
polarisation by taking the quadrature sum of Stokes Q and U,
$\mathrm{L}_{\mathrm{meas}}=\sqrt{\mathrm{Q}^{2}+\mathrm{U}^{2}}$. Since
$\mathrm{L}_{\mathrm{meas}}$ is derived from squared quantities it has a
positive bias. To correct for this bias we follow the prescription as shown in
Everett & Weisberg 45:

$\mathrm{L}_{\mathrm{unbias}}=\begin{cases}\sigma_{I}\sqrt{\left(\frac{\mathrm{L}_{{\mathrm{meas}}}}{\sigma_{I}}\right)^{2}-1},&\text{if}\
\frac{\mathrm{L}_{{\mathrm{meas}}}}{\sigma_{I}}\geq 1.57\\\
0,&\text{otherwise}\end{cases}$ (16)

where $\sigma_{I}$ is the standard deviation in the off-burst Stokes I.

The large channel widths of our very-high-time-resolution data and the
extremely large RM of FRB 20121102A causes the polarisation angle, $\theta$,
to change significantly within one channel. The intra-channel Faraday rotation
is given by11

$\Delta\theta=\mathrm{RM}_{\mathrm{obs}}c^{2}\nu^{-3}_{c}\Delta\nu,$ (17)

where $c$ is the speed of light, $\nu_{c}$ is the observing frequency, and
$\Delta\nu$ is the channel width. For an RM of $93586$ rad m-2 and a channel
width of $2.9296875$ MHz this results in rotations of $13.6^{\circ}$ and
$2.8^{\circ}$ at $4.7$ GHz and $8.0$ GHz, respectively. The depolarisation
fraction is given by11

$f_{\mathrm{depol}}=1-\left[\sin\left(2\Delta\theta\right)/\
2\Delta\theta\right],$ (18)

resulting in a depolarisation of $3.7$ % and $0.2$ % at these two frequencies,
respectively.

We find that all but one of the bursts are consistent with being $100$ %
linearly polarised (Figure 3). The sole exception is burst B43, which shows
$87\pm 10$ % linear polarisation and $31\pm 10$ % circular polarisation
(Figure 3d). We use a conservative $10$ % error which is a combination of the
statistical error, systematic errors due to the instrumental calibration and
the intra-channel depolarisation. Other bursts at similar frequencies as burst
B43 show no signs of circular polarisation.

We determine the time-resolved PPAs across the burst profiles using:

$\mathrm{PPA}=\frac{1}{2}\arctan\left(\mathrm{U}/\mathrm{Q}\right).$ (19)

We compute the PPA for every sample where $\mathrm{L}_{\mathrm{unbias}}\geq
4$. For every burst we subtract the weighted mean of the PPAs, using
$\mathrm{L}_{\mathrm{unbias}}$ as weights, and plot their probability
distributions45 in the upper panels of Figure 3. We fit the PPAs to a constant
line by minimizing the weighted least-squares. The reduced $\chi^{2}$-value,
$\chi^{2}_{\nu}$, is reported in Figure 3 and often greatly exceeds $1$,
indicating a significant scatter.

### Energetics

To determine the peak flux density, fluence and isotropic-equivalent spectral
luminosity of the 8 ultra-FRBs we make use of the data products described in
subsection ‘Polarimetric calibration’. First, the dynamic spectra of Stokes I
is averaged over a range of frequencies (as illustrated by the dashed
horizontal red lines in Figure 1). Next, the time series are normalized such
that the off-burst regions have zero mean and unit standard deviation. To
convert from S/N units to physical units we make use of the radiometer
equation46 assuming a system temperature of $26$ K and an antenna gain of $2$
K Jy-1, i.e. a system equivalent flux density (SEFD) of $13$ Jy. These
quantities are expected to have fractional uncertainties of at most $20$ %. To
determine the fluence we integrate the profile over a specific time range that
is indicated with a magenta bar in Figure 1. The spectral luminosity is
calculated assuming a luminosity distance of $972$ Mpc25 to FRB 20121102A. The
results are tabulated in Table 1 and plotted in Extended Data Figure 5.

## Data availability

The data that support the plots within this paper and other findings of this
study are available from https://doi.org/10.5281/zenodo.8112803 or from the
corresponding author upon reasonable request. The voltage data are available
through the Breakthrough Initiatives Open Data Portal:
https://breakthroughinitiatives.org/opendatasearch, and are explained in
detail in: http://seti.berkeley.edu:8000/frb-data/.

## Code availability

The pulsar package dspsr is available at https://dspsr.sourceforge.net/ and a
modified version of dspsr, bl-dspsr, that is able to read voltage data from
the Breakthrough Listen backend is available at
https://github.com/UCBerkeleySETI/bl-dspsr. Code to splice and extract voltage
data is available at https://github.com/greghell/extractor. FETCH can be found
at https://github.com/devanshkv/fetch. The PRESTO suite of tools is available
at https://github.com/scottransom/presto.

## Acknowledgements

We would like to thank the Breakthrough Listen project for keeping the raw
baseband data from these observations and making it publicly available.
Breakthrough Listen is funded by the Breakthrough Initiatives
(https://breakthroughinitiatives.org/). We thank the referees for their
constructive comments that improved the manuscript. We thank J. Weisberg for
useful discussions about radio astronomy and polarimetry. A.D. Seymour is
thanked for tips regarding the GBT BL data. Research by the AstroFlash group
at University of Amsterdam, ASTRON and JIVE is supported in part by an NWO
Vici grant (PI Hessels; VI.C.192.045). K.N. is an MIT Kavli Fellow.

## Author contributions

M.P.S. led the burst search, data analysis, and made the figures and tables.
He wrote the majority of the manuscript. K.N. made significant contributions
to the writing and provided guidance on the data analysis. J.W.T.H. supervised
the work, guided the overall approach, and made significant contributions to
the writing. All co-authors provided input on the scientific interpretation.

## Competing interests

The authors declare no competing interests.

Table 1: Burst properties of the 8 microsecond duration bursts. Burst | Peak S/Nα | Peak Flux Densityα,γ | Fluenceγ | Spectral Luminosityγ,δ | Frequency Extentζ | Time Extentψ
---|---|---|---|---|---|---
|  | [Jy] | [mJy ms] | [$10^{33}$ erg s-1 Hz-1] | [MHz] | [$\upmu$s]
B06 | 11.8 | 9.3 | 65 | 1.5 | 199.2 | 48.5
B07 | 8.8 | 4.2 | 32 | 2.9 | 178.7 | 12.3
B10 | 7.8 | 3.0 | 31 | 1.6 | 275.4 | 22.5
B30 | 27.4 | 14.7 | 41 | 7.2 | 858.4 | 6.5
B31 | 7.2 | 3.0 | 89 | 1.2 | 120.1 | 86.0
B38 | 7.5 | 1.2 | 27 | 0.8 | 867.2 | 36.9
B43 | 23.8 | 17.7 | 161 | 7.7 | 448.2 | 23.6
B44 | 8.6 | 3.5 | 27 | 1.6 | 492.2 | 19.5
α Determined for the time resolutions indicated in Figure 1.
γ We estimate a conservative $20$ % error on these measurements, arising due
to the uncertainty
in the system equivalent flux density (SEFD) of the GBT.
δ Isotropic-equivalent and using a luminosity distance of $972$ Mpc25.
ζ The frequency range over which the dynamic spectra was averaged,
i.e., the difference between the red horizontal dashed lines in Figure 1.
ψ The time range over which the profile was integrated to determine the
fluence,
i.e., the length of the magenta bar in Figure 1.

Figure 1: Total intensity dynamic spectra and profiles of the 8 ultra-FRBs. We
plot the dynamic spectrum (bottom panels) of the total-intensity data (Stokes
I) and their corresponding frequency-averaged time series (upper panels) for
the 8 bursts with a detection duration $<30$ $\upmu$s, sorted by their peak
S/N. The time and frequency resolutions of the dynamic spectra are indicated
in the top-left corners and the burst identifiers are shown in the top-right
corners. The dashed red lines indicate the frequency range that was averaged
over to create the burst profiles in the upper panels and the magenta bars
indicate the time range over which the profile was integrated to determine the
fluence of each burst (Table 1). All the bursts have been coherently
dedispersed to a DM of $560.105$ pc cm-3 (Methods). For visual purposes the
limits of the colour map have been capped at the 99th percentiles of the
dynamic spectrum.

Figure 2: Histogram of the bursts durations, as they were found in the search.
The majority of the bursts have a typical duration of roughly $0.5$ ms, but 8
bursts have extremely short durations of less than about $15$ $\upmu$s, with
some of them as short as roughly $4$ $\upmu$s. The coloured triangles indicate
the possible detection durations, i.e., they are the boxcar widths used in
PRESTO’s single_pulse_search.py (Methods). The colours yellow, magenta and
cyan represent boxcars where the underlying sampling time of the data was $2$,
$33$ and $524$ $\upmu$s, respectively. The black asymmetric error bars are
centred on the number of counts per bin and represent the $90$ % Poisson
uncertainty range of counts in each bin. Only one burst with a temporal width
between about $20$ $\upmu$s and $100$ $\upmu$s was found and none of the
bursts have a temporal width greater than about $4$ ms, even though the search
was sensitive to those timescales.

Figure 3: Full-polarisation, frequency-averaged profiles and polarisation
position angles (PPAs) for a selection of bursts. The top row shows the three
brightest ultra-FRBs and the bottom row shows the three highest S/N
millisecond-duration bursts (Supplementary Table 1). The profiles of the
bursts (bottom panels) show the total intensity (Stokes I) in black, circular
polarisation (Stokes V) in blue and the Faraday-rotation-corrected unbiased
linear polarisation in red (Methods). The colour-scale of the PPA (top panels)
represents the probability distribution per time sample, with the darker
colours representing a higher S/N in the unbiased linear polarisation
($\mathrm{L}_{\mathrm{unbias}}$). PPAs are only shown for time samples where
$\mathrm{L}_{\mathrm{unbias}}\geq 4$. Per burst, the PPAs were shifted such
that their weighted mean was centred around zero and they were fit with a
constant line. The reduced-$\chi^{2}$-values, $\chi^{2}_{\nu}$, for those fits
are displayed in the top left corners. Every burst has been coherently
dedispersed to a dispersion measure of $560.105$ pc cm-3 (Methods).

Figure 4: Multi-burst joint QU-fit. The Stokes parameters Q (panel A) and U
(panel B) oscillate due to Faraday rotation. The Stokes parameters have been
normalised by the total measured linear polarisation
($\mathrm{L_{\mathrm{meas}}}=\sqrt{\mathrm{Q}^{2}+\mathrm{U}^{2}}$). Different
colours represent different bursts. Diamonds represent bursts that were
previously found by Gajjar et al. 19 and circles represent newly discovered
microsecond-duration bursts. Panel C shows the difference between the measured
and modeled PPAs and the $1\sigma$ uncertainty on each PPA measurement. Only
data points with a S/N $\geq 5$ are shown.

Figure 1: Peak S/N of the $\mathbf{341.\bar{3}}$-ns profiles of bursts B30 and
B43, as a function of DM. The bursts were first coherently dedispersed to a DM
of $560.1$ pc cm-3 and thereafter incoherently dedispersed to a range of
nearby trial DMs with a step size of $0.0002$ pc cm-3. The peak S/N is
determined for every DM and is shown with grey circles for B30 and grey
pentagons for B43. For visual purposes, moving averages are shown with solid
and dashed black lines for B30 and B43, respectively. A Lorentzian
distribution is fit (solid green line) to the individual data points close to
the peak of the profile of B30 and the best DM, $560.105$ pc cm-3, is
determined to be the centre of the fitted distribution (solid vertical line).
Similarly, all the data points of B43 are fitted with a Gaussian distribution
(solid orange line), which peaks at a DM of $560.308$ pc cm-3 (dashed vertical
line).

Figure 2: Dispersion measure (DM) comparison for the two brightest ultra-FRBs.
Burst B30 (panels a, b, c and d) and burst B43 (panels e, f, g and h) are
coherently dedispersed to a DM of $560.105$ pc cm-3 (left column) and
$560.308$ pc cm-3 (right column). The dynamic spectra (panels b, d, f and h)
have a time resolution of $341.\bar{3}$ ns and a frequency resolution of
$2.9296875$ MHz. The dashed red lines indicate the frequency range that was
averaged over to create the burst profiles, which also have a time resolution
of $341.\bar{3}$ ns (panels a, c, e and g). Using a DM of $560.308$ pc cm-3
slightly increases the peak S/N of burst B43 and the width of burst B43
decreases by about $2$ $\upmu$s. However, using a DM of $560.308$ pc cm-3,
burst B30 is clearly over-dedispersed (panel d).

Figure 3: Detection duration as a function of time. Every FRB is found in
multiple subbands and/or trial boxcar widths (Methods). Both the colour
shading and the size of the data points correspond to the S/N of the burst
detection, with larger/darker dots indicating a higher S/N. Colours indicate
whether the bursts have been found before by Gajjar et al. 19 and Zhang et al.
34.

Figure 4: Spectral extent as a function of time. Every FRB is found in
multiple subbands and/or trial boxcar widths (Methods). For every burst
detection we plot a rectangle showing the frequency range of the corresponding
subband and a $2$-second time interval around the burst arrival time. Every
rectangle has the same transparency and the colours become darker as multiple
patches are plotted on top of each other. Colours indicate whether the bursts
have been found before by Gajjar et al. 19 and Zhang et al. 34.

Figure 5: The transient phase space for coherent radio emission. The transient
duration (width times the central frequency of the burst) and the isotropic-
equivalent spectral luminosity of the 8 ultra-FRBs shown in Figure 1 are
plotted as red stars. A selection of published localised repeating FRBs are
plotted as crosses. Radio bursts from the Galactic magnetar SGR 1935+2154 are
shown as purple plusses. The pulsar and Rotating RAdio Transient (RRAT)
population are shown as pink and magenta circles, respectively. Giant radio
pulses (GRPs) and ‘nanoshots’ of the Crab pulsar are shown as orange and
yellow circles, respectively. The diagonal grey lines represent levels of
constant brightness temperature. The three data points from FRB 20200120E that
have a brightness temperature of $>10^{40}$ K are from bursts that show short
temporal structure within a broader burst envelope33. Based on a figure
presented in Nimmo et al. 33 (see references therein).

Table 1: Basic detection properties of the bursts. The time of arrival (TOA),
signal-to-noise (S/N), duration, probability and frequency extent are with
respect to the detection with the highest S/N of that burst.

Burst | TOAα | PRESTO | Duration | Probabilityγ | Total |  $\nu_{\mathrm{low}}$δ |  $\nu_{\mathrm{high}}$δ | Gajjar et al. 19 | Zhang et al. 34
---|---|---|---|---|---|---|---|---|---
| [s] | S/N | [$\upmu$s] | $\times 100\,\%$ | detections | [MHz] | [MHz] | identifier | identifier
B01 | 16.2701 | 74.05 | 458.752 | 100 | 424 | 6970.00 | 7157.50 | 11A | 1
B02 | 18.4898 | 9.54 | 294.912 | 100 | 6 | 6970.00 | 7157.50 |  | 2
B03 | 30.5842 | 10.15 | 1572.864 | 100 | 13 | 6126.25 | 6501.25 |  | 4
B04 | 133.7138 | 8.36 | 196.608 | 100 | 2 | 5563.75 | 5938.75 |  |
B05 | 170.5020 | 8.31 | 196.608 | 100 | 2 | 7063.75 | 7438.75 |  |
B06 | 195.1935 | 14.11 | 4.096 | 100 | 16 | 6970.00 | 7157.50 |  |
B07 | 197.3795 | 10.58 | 6.144 | 0 | 5 | 4720.00 | 4907.50 |  |
B08 | 204.7666 | 9.53 | 458.752 | 100 | 7 | 5001.25 | 5751.25 |  | 6
B09 | 244.1764 | 8.82 | 196.608 | 100 | 3 | 4626.25 | 4813.75 |  |
B10 | 248.3161 | 12.47 | 12.288 | 0 | 15 | 5001.25 | 5751.25 |  |
B11 | 249.9378 | 8.84 | 61.440 | 100 | 6 | 4813.75 | 5188.75 |  |
B12 | 263.4282 | 16.01 | 294.912 | 100 | 42 | 5376.25 | 6126.25 | 11B | 8,9
B13 | 277.4008 | 12.65 | 3145.728 | 100 | 11 | 5376.25 | 6126.25 |  | 10
B14 | 281.8991 | 12.47 | 196.608 | 100 | 13 | 4813.75 | 5001.25 |  |
B15 | 285.4547 | 23.44 | 458.752 | 100 | 48 | 6970.00 | 7157.50 | 11C | 11
B16 | 315.0567 | 12.90 | 983.040 | 82 | 5 | 6970.00 | 7157.50 |  | 12
B17 | 323.3700 | 99.22 | 1048.576 | 95 | 154 | 6876.25 | 7251.25 | 11D | 13
B18 | 334.9762 | 8.84 | 294.912 | 100 | 20 | 5376.25 | 6876.25 |  |
B19 | 344.7870 | 41.62 | 1048.576 | 69 | 105 | 6970.00 | 7157.50 | 11E | 14
B20 | 356.0529 | 55.87 | 1048.576 | 50 | 47 | 6970.00 | 7157.50 | 11F | 15
B21 | 415.5190 | 8.76 | 458.752 | 100 | 6 | 4626.25 | 5376.25 |  |
B22 | 580.6662 | 14.18 | 294.912 | 100 | 34 | 5376.25 | 6126.25 | 11G | 19
B23 | 597.6259 | 95.94 | 294.912 | 100 | 128 | 6876.25 | 7626.25 | 11H | 20
B24 | 622.5689 | 17.49 | 196.608 | 100 | 42 | 6876.25 | 8376.25 |  |
B25 | 652.5810 | 9.52 | 458.752 | 100 | 12 | 5376.25 | 6876.25 |  | 21
B26 | 662.2081 | 9.45 | 294.912 | 100 | 8 | 5751.25 | 6501.25 |  | 22
B27 | 691.0620 | 10.96 | 294.912 | 100 | 18 | 5751.25 | 6501.25 | 11I | 23
B28 | 691.8646 | 25.21 | 458.752 | 100 | 77 | 5376.25 | 6126.25 |  |
B29 | 704.0925 | 21.31 | 294.912 | 100 | 49 | 6126.25 | 6501.25 | 11J | 24
B30 | 731.5633 | 33.13 | 4.096 | 20 | 56 | 5751.25 | 6501.25 |  |
B31 | 766.6877 | 11.15 | 12.288 | 5 | 25 | 5001.25 | 5751.25 |  |
B32 | 769.8707 | 21.22 | 655.360 | 100 | 51 | 6876.25 | 8376.25 | 11K | 25
B33 | 804.5907 | 8.24 | 655.360 | 100 | 4 | 5563.75 | 5938.75 |  | 26
B34 | 808.8735 | 8.93 | 458.752 | 100 | 7 | 4813.75 | 5188.75 |  |
B35 | 840.9772 | 9.55 | 983.040 | 100 | 14 | 5563.75 | 5938.75 | 11L | 27,28,29
B36 | 842.1029 | 8.52 | 458.752 | 100 | 3 | 5001.25 | 5751.25 |  | 31
B37 | 878.2482 | 8.80 | 131.072 | 100 | 6 | 5001.25 | 5751.25 |  |
B38 | 993.2905 | 14.50 | 12.288 | 100 | 77 | 5657.50 | 5845.00 | 11M | 33
B39 | 994.9033 | 10.35 | 294.912 | 100 | 11 | 4813.75 | 5188.75 |  |
B40 | 1036.4686 | 35.74 | 196.608 | 100 | 102 | 5001.25 | 5751.25 | 11N | 34
B41 | 1142.4320 | 28.03 | 458.752 | 100 | 71 | 5376.25 | 6876.25 | 11O | 37
B42 | 1257.4688 | 8.32 | 1474.560 | 100 | 3 | 5001.25 | 5376.25 | 11P | 39
B43 | 1280.6520 | 46.14 | 12.288 | 100 | 47 | 4813.75 | 5188.75 |  | 40
B44 | 1327.5579 | 12.91 | 12.288 | 0 | 12 | 5001.25 | 5751.25 |  |
B45 | 1440.8676 | 10.10 | 983.040 | 100 | 5 | 5751.25 | 6501.25 |  | 42
B46 | 1454.5774 | 55.57 | 196.608 | 100 | 129 | 5751.25 | 6501.25 | 11Q | 43
B47 | 1630.8575 | 19.32 | 458.752 | 100 | 46 | 4626.25 | 5001.25 |  | 45
B48 | 1637.2393 | 12.24 | 196.608 | 100 | 22 | 5657.50 | 5845.00 |  |
B49 | 1789.3961 | 30.26 | 196.608 | 100 | 55 | 5376.25 | 5751.25 | 11R | 46
α Seconds since the start of the observation (MJD $57991.57760417$).
γ The FETCH40 probability of the candidate being of astrophysical origin
(model A)
indicating the importance of manually checking the candidates with duration
$\leq 500$ $\upmu$s.
δ For clarity $0.214844$ MHz has been subtracted from every number.

Table 2: Frequency coverage per compute node. Node |  | Passbandγ
---
| $\nu_{\textrm{low}}^{\alpha}$
---
[MHz]
| $\nu_{\textrm{center}}^{\alpha}$
---
[MHz]
| $\nu_{\textrm{high}}^{\alpha}$
---
[MHz]
BLP00 | 0 | 9126.25 | 9220.00 | 9313.75
BLP01 | 0 | 8938.75 | 9032.50 | 9126.25
BLP02 | 0 | 8751.25 | 8845.00 | 8938.75
BLP03 | 0 | 8563.75 | 8657.50 | 8751.25
BLP04 | 0 | 8376.25 | 8470.00 | 8563.75
BLP05† | 0 | 8188.75 | 8282.50 | 8376.25
BLP06† | 0 | 8001.25 | 8095.00 | 8188.75
BLP07△ | 0 | 7813.75 | 7907.50 | 8001.25
BLP10†△ | 1 | 7813.75 | 7907.50 | 8001.25
BLP11† | 1 | 7626.25 | 7720.00 | 7813.75
BLP12† | 1 | 7438.75 | 7532.50 | 7626.25
BLP13† | 1 | 7251.25 | 7345.00 | 7438.75
BLP14† | 1 | 7063.75 | 7157.50 | 7251.25
BLP15† | 1 | 6876.25 | 6970.00 | 7063.75
BLP16† | 1 | 6688.75 | 6782.50 | 6876.25
BLP17△ | 1 | 6501.25 | 6595.00 | 6688.75
BLP20†△ | 2 | 6501.25 | 6595.00 | 6688.75
BLP21† | 2 | 6313.75 | 6407.50 | 6501.25
BLP22† | 2 | 6126.25 | 6220.00 | 6313.75
BLP23† | 2 | 5938.75 | 6032.50 | 6126.25
BLP24† | 2 | 5751.25 | 5845.00 | 5938.75
BLP25† | 2 | 5563.75 | 5657.50 | 5751.25
BLP26† | 2 | 5376.25 | 5470.00 | 5563.75
BLP27△ | 2 | 5188.75 | 5282.50 | 5376.25
BLP30†△ | 3 | 5188.75 | 5282.50 | 5376.25
BLP31† | 3 | 5001.25 | 5095.00 | 5188.75
BLP32† | 3 | 4813.75 | 4907.50 | 5001.25
BLP33† | 3 | 4626.25 | 4720.00 | 4813.75
BLP34† | 3 | 4438.75 | 4532.50 | 4626.25
BLP35† | 3 | 4251.25 | 4345.00 | 4438.75
BLP36† | 3 | 4063.75 | 4157.50 | 4251.25
BLP37† | 3 | 3876.25 | 3970.00 | 4063.75
γ The passbands were numbered 0–3 and were tuned to a
centre frequency of $8563.964844$, $7251.464844$,
$5938.964844$ and $4626.464844$ MHz, respectively.
α For clarity $0.214844$ MHz has been subtracted from
every value.
† The nodes that were used to create high-time-resolution
filterbanks for the burst search.
△ These nodes have exactly $187.5$ MHz overlap
with an adjacent node of a different passband.

Table 3: Search strategy. Dispersion measure (DM) range, DM step-size
($\Delta$DM) and number of ‘subbands’ used in PRESTO’s prepsubband as a
function of sampling time ($\Delta t$), bandwidth (BW) and frequency ($\nu$)
range. For clarity we have subtracted $0.214844$ MHz from all the shown
frequencies. Full table available online

.  $\Delta t$ BW Channels $\nu_{\mathrm{low}}$ $\nu_{\mathrm{center}}$
$\nu_{\mathrm{high}}$ DMlow DMhigh Trial DMs $\Delta$DM Subbands
$\left[\upmu\mathrm{s}\right]$ [MHz] N [MHz] [MHz] [MHz] [pc cm-3] [pc cm-3] N
[pc cm-3] N 2.048 4500.0 1536 3876.25 6126.25 8376.25 552.90 566.90 701 0.02
128 2.048 1500.0 512 3876.25 4626.25 5376.25 552.91 566.89 467 0.03 64 2.048
1500.0 512 4626.25 5376.25 6126.25 552.90 566.90 281 0.05 64 2.048 1500.0 512
5376.25 6126.25 6876.25 552.90 566.90 281 0.05 64 2.048 1500.0 512 6126.25
6876.25 7626.25 552.90 566.90 141 0.10 64 2.048 1500.0 512 6876.25 7626.25
8376.25 552.90 566.90 141 0.10 64 2.048 750.0 256 3876.25 4251.25 4626.25
552.90 566.90 281 0.05 32 2.048 750.0 256 4251.25 4626.25 5001.25 552.90
566.90 281 0.05 32 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

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