the UVW1 light curve shows a zoomed-in view of a light curve of the only flare
for which there were multiple detections in the same filter, showing the
timescale of the event more clearly. See Fig. 36 for the original (PIII)
TUVOpipe detection.

Information extracted in PIV showed that matches to this source appeared in
several M-dwarf catalogues (e.g. SPECULOOS, Sebastian et al., 2021, where this
source is denoted as Sp2254+1606). In addition, using the PIV information
obtained for match in the Gaia catalogue, we noticed the very red colour of
the source (its Gaia magnitude in the G band is 13.6, compared with $\sim$20.5
in UVW1) and its inferred Gaia temperature of 4200K. These characteristics
confirm it as a cool, red source, consistent with an M-dwarf. These stars are
known to exhibit flares that can be highly energetic, brightening by several
magnitudes in the optical and UV (see e.g. Welsh et al., 2007; also note the
flare spectrum may peak in the UV, see Paudel et al., 2021), and lasting from
seconds to hours. Relatively few photometric studies of M-dwarf flares have
been performed in the UV (for an example, see Welsh et al. 2007).

Interestingly, despite similar numbers of data points between all filters,
fewer flares are detected in the optical bands compared with the UV filters
(e.g. only one flare each in V and B, while at least a few to several events
were seen in U, UVW1, UVM2, and UVW2). Additionally, the increase in
brightness for the flares detected in V and B bands is relatively small
($\sim$0.75 and $\sim$0.5 in V and B, respectively) compared with brightenings
of up to several magnitudes in some of the U-band and UV flares (for example,
the earliest flare seen in the UVM2 causes an increase in brightness by 4
magnitudes). These characteristics might result from a combination of flares
intrinsically causing stronger brightenings in the UV than in the optical,
along with the fact that the UV emission from M-dwarfs in quiescence is very
low compared to the optical. The latter point may suggest that while in the
optical only the strongest flares are bright enough to be detected above the
optical emission of the star, the lower stellar emission in the UV allows for
even weaker flare events to be detected, hence the more numerous detections in
the UV filters. This may therefore make the case that the UV is the optimal
band in which to search for flares.

For only one of the detected flares, UVOT data in multiple filters had been
obtained. This event (at MJD $\sim$57000) was seen in the B, U, UVW1, and UVW2
filters (the V-band flare seen around this time does not correspond to the
same event; it occurs a few weeks later). The optical and UV
(quasi-)simultaneous data may indicate that the flare causes a more
significant brightening in the U and UV bands than in the optical, with the
peak brightening observed in the U band: we measure the amplitude with respect
to the quiescent level of around 0.43, 3.1, 1.3, and 1.9 magnitudes in the B,
U, UVW1, and UVW2 filters, respectively. However, we note the caveat that the
images were taken a few minutes apart, and since flare timescales can range
from seconds to hours, the differences in the brightenings observed between
filters may also depend in part on the stage of the outburst that happened to
be observed with each filter.

The flares we detected have relatively short durations, based on the cadence
of the Swift observations. For some of the events, the data points in
quiescence immediately prior and subsequent to flares are separated from the
flare by down to around 1.5 hours, which gives an upper limit for the flare
duration (for other events this upper limit is much weaker, up to several
days). Additionally, one of the UVW1 brightenings does have coverage during a
possible rise stage of the flare event (see the inset axes in Fig. 37). This
shows a rise from 20.94 $\pm$ 0.23 to 18.99 $\pm$ 0.13 magnitudes in just over
an hour. Overall, the constraints we obtained on both the timescales and
outburst amplitudes of the observed brightenings are consistent with the known
properties of flare stars, so we suggest that this source is very likely a
flare star.

Due to a few past and ongoing surveys targeting fields in the Galactic centre
with Swift (Degenaar & Wijnands, 2010; Degenaar et al., 2013, 2015), there are
several thousand archival UVOT images available covering the innermost region
around the Galactic centre. We are currently undertaking two projects in which
we make use of this large data set. However, due to the very long times
associated with processing all the data, they are currently works in progress,
which will be described in detail in separate publications. Here, we briefly
describe our plans and preliminary results.

One of the projects consists of examining persistent sources. Using PV, we can
create long-term light curves of all the persistent sources that are detected
in these fields (i.e. by using one of the images as a reference image in which
to detect the sources). This method does not allow for the discovery of
transients with no quiescent counterparts detected in the reference image.
However, the PV light curves could still reveal interesting variable behaviour
of the persistent sources, since we can use them to characterise the long-term
UV behaviour of many sources with relatively high sampling cadences and
spanning over a decade. As a preliminary study, we focused on one Galactic
centre field (around the inner 12’), creating PV light curves of all the 459
sources detected at 5-$\sigma$ in one U-band image. The selected image for
source detection was chosen based on a relatively high exposure time of a few
hundred seconds as well as inspecting it by eye to ensure there were no major
artefacts in the image (ObsID 00095660216; extension 1). In this preliminary
search, we did not find any significantly variable sources (see Fig. 38 for an
example of one of the PV light curves).

To determine the distance towards the Galactic centre, which we are probing
with this study, we queried the Gaia catalogue for counterparts to all the
sources. For each source, if there was match in the Gaia catalogue (see Sect.
3.5.3 for how we determine if a catalogue source is a match with a source we
detect) in which the error on the parallax was small (significance of at least
5.0), we retained the associated distance measurement from Gaia. We created a
histogram of the results in order to examine the distribution of distances of
the sources that the UVOT can detect towards the Galactic centre (shown in
Fig. 39). From the distribution, we can see that when looking towards the
Galactic centre with the UVOT at 5-$\sigma$ in the U-band, we can see up to
only around 2.5 kpc. We also notice a clear bimodality in the distribution,
with peaks around 1.1 and 1.8 kpc.

Figure 38: Long-term light curve created with PV of a persistent source in the
Galactic centre, at RA=17:45:44.00, Dec=-29:07:01.1 in the U, UVW1, UVM2, and
UVW2 filters (we do not show the V and B data points here because only a
handful of observations of this field were performed with those filters). The
few data points straying from the stable levels are likely the result of
artefacts in the science images or imperfect alignments. Figure 39:
Distribution of distances for sources detected at 5-$\sigma$ in the U-band in
a Galactic centre field. Distances were obtained by querying the Gaia
catalogue for counterparts to the UVOT sources we detect (and retaining only
those with at least a 5.0 significance in the parallax, to ensure the
distances we retain are reliable). The data is grouped in 40 equally sized
bins.

We emphasise that this is still a preliminary study. We are currently
expanding it to sources detected in all filters (i.e. not just the U-band),
down to 3-$\sigma$, and to a larger region around the Galactic centre; this
may affect the preliminary results mentioned here. In the second planned
project, we will run TUVOpipe in the archival mode to look for transients in
all the UVOT observations available of these Galactic centre fields. It is
also worth noting that the long-term light curves we already created show
exceptionally stable photometry over the course of the observations (spanning
over a decade), indicating the excellent long-term calibration of the UVOT
(see Fig. 38).

##  5 Summary and concluding remarks

In this paper we introduce TUVOpipe, a specialised pipeline we constructed in
order to automatically discover serendipitous transients in the UV wavelength
range in data obtained with UVOT aboard Swift. UVOT is the primary focus of
the TUVO project (see Wijnands et al., in prep.), a recently launched
investigation in which we aim to explore transients in the relatively
understudied UV regime. Every day, the pipeline is run to search for
variability in all recent ($<$1 day old) images obtained with UVOT. Between a
few hours and one day after the observations are performed, transient
candidates are displayed to users. This includes, for each transient: light
curves, image stamps, parameters derived from basic analysis of the light
curves, and information regarding any known counterparts or previous
classifications from astronomical catalogues.

The first year of running the pipeline has yielded promising results. Having
processed over 75 000 images, a few tens of thousands of candidate UV
transients have been detected serendipitously, currently at a rate of a few
tens of real transients per day. Most of these are known variable sources or
previously discovered transients, though a fraction of these represent sources
with no clear previous identifications, indicating that TUVOpipe can indeed
discover new transients in the UV. This ‘high-interest’ category included the
detection of a new outburst from a known candidate CV (Verberne et al., 2020),
the discovery of a strong UV outburst of a previously unknown CV (Modiano et
al., in prep.), and the discovery of a new unidentified transient (Modiano &
Wijnands, 2021). The pipeline also successfully recovers many previously known
transients such as DNe, XRB outbursts, novae, SNe, and AGN.

We are considering further upgrades to TUVOpipe. One enhancement to be
integrated in PV is a tool we are currently developing to run Lomb-Scargle
methods on the PV light curves to attempt to automatically find periods. We
also plan to develop this aspect of the pipeline further, with additional
period-searching tools as well as further statistical methods to characterise
variability in the light curves. We are also investigating the option of
probing additional information from online databases: we will automatically
collect light curves of the counterparts of our sources determined in PIV
(e.g. from ZTF, ASAS-SN, Gaia, ATLAS) to display alongside our PIV and PV
products. Finally, we are currently investigating the possibility of
automatically querying XRT light
curves414141https://www.swift.ac.uk/user_objects/API/pyDocs/README.md and
images; this would allow us to immediately verify whether there is any X-ray
detection at the position of our transients (currently this is done manually).
Some of these further improvements may be implemented in a Part VI of
TUVOpipe.

We note that TUVOpipe is designed to detect UV transient sources that are of
the highest interest to the TUVO project (relatively high amplitude
transients; see Wijnands et al., in prep., for details), and is not intended
to produce a comprehensive picture of all true variable and transient sources
present in the UVOT data. The complexity of the data analysis, in particular
the image subtraction and source detection stages, inevitably introduces
imperfections. This means that TUVOpipe cannot securely detect all real
variability exhibited by all sources detected using the UVOT, nor can it
automatically remove all false positives. Therefore, we would like to stress
that the pipeline is not constructed in order to be complete, so it cannot be
used for comprehensive population and statistical studies of UV transients.
Rather, it is a method used to maximise the potential for efficiently
discovering as many interesting UV transients as possible in real-time and
archival UVOT data.

After discovering UV transients, the next, and crucial, stage for our TUVO
project consists of ground- and space-based photometric and spectroscopic
follow-up. These observations target the transients that we find highly
interesting based on the output of PIV and PV (see Wijnands et al., in prep.,
for details on how we are currently pursuing this strategy with several
facilities, and see Modiano et al. (in prep.) for an example).

###### Acknowledgements.

DM is partly supported by the Netherlands Research School for Astronomy
(NOVA). We acknowledge the use of public data from the Swift data archive.
This research has made use of the VizieR catalogue access tool, CDS,
Strasbourg, France (DOI : 10.26093/cds/vizier). This research has made use of
NASA’s Astrophysics Data System Bibliographic Services. We thank Aaron
Tohuvavohu for pointing out a previous error in our calculation of the number
of UVOT images processed, which led us to fix a bug in our code and obtain the
correct numbers.

## References

  * Abbott et al. (2016) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016, Phys. Rev. Lett., 116, 061102
  * Abbott et al. (2017) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017, ApJ, 848, L12
  * Ahumada et al. (2020) Ahumada, R., Prieto, C. A., Almeida, A., et al. 2020, ApJS, 249, 3
  * Alard (2000) Alard, C. 2000, A&AS, 144, 363
  * Alard & Lupton (1998) Alard, C. & Lupton, R. H. 1998, ApJ, 503, 325
  * Andrae et al. (2018) Andrae, R., Fouesneau, M., Creevey, O., et al. 2018, A&A, 616, A8
  * Andreoni & Cooke (2019) Andreoni, I. & Cooke, J. 2019, in Southern Horizons in Time-Domain Astronomy, ed. R. E. Griffin, Vol. 339, 135–138
  * Andreoni et al. (2017) Andreoni, I., Jacobs, C., Hegarty, S., et al. 2017, PASA, 34, e037
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., SipHocz, B. M., et al. 2018, aj, 156, 123
  * Bailer-Jones et al. (2018) Bailer-Jones, C. A. L., Rybizki, J., Fouesneau, M., Mantelet, G., & Andrae, R. 2018, AJ, 156, 58
  * Barthelmy et al. (2005) Barthelmy, S. D., Barbier, L. M., Cummings, J. R., et al. 2005, Space Sci. Rev., 120, 143
  * Becker (2015) Becker, A. 2015, HOTPANTS: High Order Transform of PSF ANd Template Subtraction, Astrophysics Source Code Library
  * Becker et al. (2020) Becker, T. M., Cunningham, N., Molyneux, P., et al. 2020, The Planetary Science Journal, 1, 53
  * Bellm et al. (2019) Bellm, E. C., Kulkarni, S. R., Graham, M. J., et al. 2019, PASP, 131, 018002
  * Berthier et al. (2006) Berthier, J., Vachier, F., Thuillot, W., et al. 2006, in Astronomical Society of the Pacific Conference Series, Vol. 351, Astronomical Data Analysis Software and Systems XV, ed. C. Gabriel, C. Arviset, D. Ponz, & S. Enrique, 367–+
  * Bertin & Arnouts (1996) Bertin, E. & Arnouts, S. 1996, A&AS, 117, 393
  * Bianchi et al. (2017) Bianchi, L., Shiao, B., & Thilker, D. 2017, ApJS, 230, 24
  * Bloemen et al. (2016) Bloemen, S., Groot, P., Woudt, P., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9906, Ground-based and Airborne Telescopes VI, ed. H. J. Hall, R. Gilmozzi, & H. K. Marshall, 990664
  * Breeveld et al. (2010) Breeveld, A. A., Curran, P. A., Hoversten, E. A., et al. 2010, MNRAS, 406, 1687
  * Breeveld et al. (2011) Breeveld, A. A., Landsman, W., Holland, S. T., et al. 2011, in American Institute of Physics Conference Series, Vol. 1358, American Institute of Physics Conference Series, ed. J. E. McEnery, J. L. Racusin, & N. Gehrels, 373–376
  * Buckley et al. (2006) Buckley, D. A. H., Swart, G. P., & Meiring, J. G. 2006, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 6267, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, 62670Z
  * Burrows et al. (2005) Burrows, D. N., Hill, J. E., Nousek, J. A., et al. 2005, Space Sci. Rev., 120, 165
  * Campana et al. (2008) Campana, S., Stella, L., & Kennea, J. A. 2008, ApJ, 684, L99
  * Cao et al. (2012) Cao, Y., Kasliwal, M. M., Neill, J. D., et al. 2012, ApJ, 752, 133
  * Cao et al. (2016) Cao, Y., Nugent, P. E., & Kasliwal, M. M. 2016, PASP, 128, 114502
  * Carmo et al. (2020) Carmo, A., Ferreira Lopes, C. E., Papageorgiou, A., et al. 2020, MNRAS, 498, 2833
  * Cenko et al. (2012) Cenko, S. B., Krimm, H. A., Horesh, A., et al. 2012, ApJ, 753, 77
  * Chambers & et al. (2017) Chambers, K. C. & et al. 2017, VizieR Online Data Catalog, II/349
  * Côte et al. (2012) Côte, P., Scott, A., Balogh, M., et al. 2012, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 8442, Space Telescopes and Instrumentation 2012: Optical, Infrared, and Millimeter Wave, ed. M. C. Clampin, G. G. Fazio, H. A. MacEwen, & J. Oschmann, Jacobus M., 844215
  * Cutri et al. (2003) Cutri, R. M., Skrutskie, M. F., van Dyk, S., et al. 2003, 2MASS All Sky Catalog of point sources.
  * Cutri et al. (2021) Cutri, R. M., Wright, E. L., Conrow, T., et al. 2021, VizieR Online Data Catalog, II/328
  * Degenaar et al. (2013) Degenaar, N., Miller, J. M., Kennea, J., et al. 2013, ApJ, 769, 155
  * Degenaar & Wijnands (2010) Degenaar, N. & Wijnands, R. 2010, A&A, 524, A69
  * Degenaar et al. (2015) Degenaar, N., Wijnands, R., Miller, J. M., et al. 2015, Journal of High Energy Astrophysics, 7, 137
  * Downes et al. (2001) Downes, R. A., Webbink, R. F., Shara, M. M., et al. 2001, PASP, 113, 764
  * Gaia Collaboration et al. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1
  * Gehrels et al. (2004) Gehrels, N., Chincarini, G., Giommi, P., et al. 2004, ApJ, 611, 1005
  * Gezari et al. (2013) Gezari, S., Martin, D. C., Forster, K., et al. 2013, ApJ, 766, 60
  * Giovannelli (2008) Giovannelli, F. 2008, Chinese Journal of Astronomy and Astrophysics Supplement, 8, 237
  * Groot et al. (2019) Groot, P., Bloemen, S., & Jonker, P. 2019, in The La Silla Observatory - From the Inauguration to the Future, 33
  * Guizar-Sicairos et al. (2008) Guizar-Sicairos, M., Thurman, S. T., & Fienup, J. R. 2008, Optics Letters, 33, 156
  * Heinze et al. (2018) Heinze, A. N., Tonry, J. L., Denneau, L., et al. 2018, AJ, 156, 241
  * Helou et al. (1991) Helou, G., Madore, B. F., Schmitz, M., et al. 1991, The NASA/IPAC extragalactic database., ed. M. A. Albrecht & D. Egret, Vol. 171, 89–106
  * Hoffman et al. (2021) Hoffman, A. M., Murakami, Y. S., Zheng, W., Stahl, B. E., & Filippenko, A. V. 2021, MNRAS, 502, 818
  * Hosenie et al. (2021) Hosenie, Z., Bloemen, S., Groot, P., et al. 2021, Experimental Astronomy [arXiv:2104.13950]
  * Hu et al. (2021) Hu, L., Wang, L., & Chen, X. 2021, arXiv e-prints, arXiv:2109.09334
  * IceCube Collaboration et al. (2018) IceCube Collaboration, Aartsen, M. G., Ackermann, M., et al. 2018, Science, 361, 147
  * Kennea et al. (2018) Kennea, J. A., Coe, M. J., Evans, P. A., Waters, J., & Jasko, R. E. 2018, ApJ, 868, 47
  * Kulkarni et al. (2021) Kulkarni, S. R., Harrison, F. A., Grefenstette, B. W., et al. 2021, arXiv e-prints, arXiv:2111.15608
  * Marchini et al. (2021) Marchini, A., Aceti, P., Banfi, M., et al. 2021, The Astronomer’s Telegram, 14328, 1
  * Masci et al. (2019) Masci, F. J., Laher, R. R., Rusholme, B., et al. 2019, PASP, 131, 018003
  * Middleton et al. (2017) Middleton, M. J., Casella, P., Gandhi, P., et al. 2017, New A Rev., 79, 26
  * Modiano et al. (2020) Modiano, D., Parikh, A. S., & Wijnands, R. 2020, A&A, 634, A132
  * Modiano et al. (2021) Modiano, D., Verberne, S., & Wijnands, R. 2021, The Astronomer’s Telegram, 14762, 1
  * Modiano & Wijnands (2021) Modiano, D. & Wijnands, R. 2021, The Astronomer’s Telegram, 14997, 1
  * Monet et al. (2003a) Monet, D. G., Levine, S. E., Canzian, B., et al. 2003a, AJ, 125, 984
  * Monet et al. (2003b) Monet, D. G., Levine, S. E., Canzian, B., et al. 2003b, AJ, 125, 984
  * Neustroev et al. (2017) Neustroev, V. V., Marsh, T. R., Zharikov, S. V., et al. 2017, MNRAS, 467, 597
  * Ochsenbein et al. (2000) Ochsenbein, F., Bauer, P., & Marcout, J. 2000, A&AS, 143, 23
  * Page et al. (2019) Page, M. J., Brindle, C., Talavera, A., et al. 2019, VizieR Online Data Catalog, II/356
  * Page et al. (2014) Page, M. J., Yershov, V., Breeveld, A., et al. 2014, in Proceedings of Swift: 10 Years of Discovery (SWIFT 10, 37
  * Patruno et al. (2016) Patruno, A., Maitra, D., Curran, P. A., et al. 2016, ApJ, 817, 100
  * Patterson et al. (2019) Patterson, M. T., Bellm, E. C., Rusholme, B., et al. 2019, PASP, 131, 018001
  * Paudel et al. (2021) Paudel, R. R., Barclay, T., Schlieder, J. E., et al. 2021, ApJ, 922, 31
  * Pence et al. (2010) Pence, W. D., Chiappetti, L., Page, C. G., Shaw, R. A., & Stobie, E. 2010, A&A, 524, A42
  * Poole et al. (2008) Poole, T. S., Breeveld, A. A., Page, M. J., et al. 2008, MNRAS, 383, 627
  * Roming et al. (2005) Roming, P. W. A., Kennedy, T. E., Mason, K. O., et al. 2005, Space Sci. Rev., 120, 95
  * Russell et al. (2019) Russell, T. D., Tetarenko, A. J., Miller-Jones, J. C. A., et al. 2019, ApJ, 883, 198
  * Sagiv et al. (2014) Sagiv, I., Gal-Yam, A., Ofek, E. O., et al. 2014, AJ, 147, 79
  * Sai et al. (2021) Sai, H., Wang, X., Wu, J., et al. 2021, MNRAS, 504, 4226
  * Samus et al. (2017) Samus, N. N., Kazarovets, E. V., Durlevich, O. V., Kireeva, N. N., & Pastukhova, E. N. 2017, Astronomy Reports, 61, 80
  * Sebastian et al. (2021) Sebastian, D., Gillon, M., Ducrot, E., et al. 2021, A&A, 645, A100
  * Shafter et al. (2011) Shafter, A. W., Darnley, M. J., Hornoch, K., et al. 2011, ApJ, 734, 12
  * Shappee et al. (2014) Shappee, B. J., Prieto, J. L., Grupe, D., et al. 2014, ApJ, 788, 48
  * Siegel et al. (2015) Siegel, M. H., Porterfield, B. L., Balzer, B. G., & Hagen, L. M. Z. 2015, The Astronomical Journal, 150, 129
  * Singer et al. (2021) Singer, L. P., Cenko, S., & Dorado Science Team. 2021, in American Astronomical Society Meeting Abstracts, Vol. 53, American Astronomical Society Meeting Abstracts, 309.05
  * Smith et al. (2011) Smith, H. A., Catelan, M., & Kuehn, C. 2011, in RR Lyrae Stars, Metal-Poor Stars, and the Galaxy, ed. A. McWilliam, Vol. 5, 17
  * Stalin et al. (2005) Stalin, C. S., Gopal-Krishna, Sagar, R., et al. 2005, Bulletin of the Astronomical Society of India, 33, 401
  * Stroh & Falcone (2013) Stroh, M. C. & Falcone, A. D. 2013, ApJS, 207, 28
  * Tonry et al. (2018) Tonry, J. L., Denneau, L., Heinze, A. N., et al. 2018, PASP, 130, 064505
  * Verberne et al. (2020) Verberne, S., Modiano, D., Wijnands, R., & TUVO Project. 2020, Research Notes of the American Astronomical Society, 4, 233
  * Watson et al. (2006) Watson, C. L., Henden, A. A., & Price, A. 2006, Society for Astronomical Sciences Annual Symposium, 25, 47
  * Webb et al. (2020) Webb, N. A., Coriat, M., Traulsen, I., et al. 2020, A&A, 641, A136
  * Welsh et al. (2005) Welsh, B. Y., Wheatley, J. M., Heafield, K., et al. 2005, AJ, 130, 825
  * Welsh et al. (2007) Welsh, B. Y., Wheatley, J. M., Seibert, M., et al. 2007, ApJS, 173, 673
  * Wenger et al. (2000) Wenger, M., Ochsenbein, F., Egret, D., et al. 2000, A&AS, 143, 9
  * Wheatley et al. (2008) Wheatley, J. M., Welsh, B. Y., & Browne, S. E. 2008, AJ, 136, 259
  * Whitmore et al. (2016) Whitmore, B. C., Allam, S. S., Budavári, T., et al. 2016, AJ, 151, 134
  * Wijnands et al. (2001) Wijnands, R., Méndez, M., Markwardt, C., et al. 2001, ApJ, 560, 892
  * Zhang et al. (2017) Zhang, G., Gelfand, J. D., Russell, D. M., et al. 2017, MNRAS, 469, 4236
  * Zhang & Gao (2021) Zhang, M. & Gao, X. 2021, Transient Name Server Discovery Report, 2021-1937, 1

##  Appendix A Appendix

Database or facility |  Probed catalog description |  Catalog reference
---|---|---
Simbad111 |  Database of astronomical objects published in astronomy journal papers |  Wenger et al. (2000)
Gaia222 |  Catalog of all Gaia sources (DR 2); Catalog of distances to 1.33 billion Gaia stars |  Gaia Collaboration et al. (2018); Bailer-Jones et al. (2018)
NASA/IPAC Extragalactic Database (NED)333 |  Database of multi-wavelength data for extragalactic objects |  Helou et al. (1991)
Sloan Digital Sky Survey (SDSS)444 |  Catalog of all Sloan Digital Sky Survey sources (DR 16) |  Ahumada et al. (2020)
Hubble Space Telescope (HST)555 |  Large UV-optical-infrared high-resolution space telescope source catalog |  Whitmore et al. (2016)
Galaxy Evolution Explorer (GALEX)666 |  Database of all GALEX UV sources |  Bianchi et al. (2017)
XMM-Newton777 |  X-ray mission point-source catalog (4XMM-DR9) |  Webb et al. (2020)
XMM-Newton Optical Monitor (XMM-OM)888 |  Optical monitor aboard XMM-Newton source catalog (SUSS4.1) |  Page et al. (2019)
Zwicky Transient Facility (ZTF) MARS alert stream999 |  Ground-based optical transient searching facility alert stream |  Patterson et al. (2019)
Transient Name Server (TNS)101010 |  Transient database |
All Sky Automated Survey for SuperNovae (ASAS-SN; transients) 111111 |  Ground-based optical transient searching facility - transient catalog |  Shappee et al. (2014)
All Sky Automated Survey for SuperNovae (ASAS-SN; variables)121212 |  Ground-based optical transient searching facility - variables catalog |  Shappee et al. (2014)
The Panoramic Survey Telescope and Rapid Response System (PanSTARRS)131313 |  Ground-based optical transient searching facility catalog |  Chambers & et al. (2017)
Wide-field Infrared Survey Explorer (WISE)141414 |  Infrared space survey mission catalog |  Cutri et al. (2021)
USNO-B151515 |  All-sky optical catalog |  Monet et al. (2003b)
The International Variable Star Index (VSX)161616 |  Variable star database |  Watson et al. (2006)
The Two Micron All Sky Survey (2MASS)171717 |  All-sky infrared catalog |  Cutri et al. (2003)
General Catalog of Variable Stars (GCVS)181818 |  Variable star database |  Samus et al. (2017)
A Catalog And Atlas of Cataclysmic Variables191919 |  Cataclysmic variable database |  Downes et al. (2001)
Asteroid Terrestrial-impact Last Alert System (ATLAS)202020 |  Ground-based optical asteroid and transient searching facility catalog |  Tonry et al. (2018)
SkyBot212121 |  Solar system object database |  Berthier et al. (2006)
Vizier222222 |  Meta-compilation of astronomical catalogs |  Ochsenbein et al. (2000)
Table 2: All catalogs which are probed automatically by TUVOpipe in PIVa.

a The first column shows the database or facility which produces the catalog.
The second column describes the catalog we probe. The references in the third
column are to the relevant catalog/data release papers. Footnotes point to the
web pages where the catalogs we probe are hosted.

1http://simbad.u-strasbg.fr/simbad/
2https://gea.esac.esa.int/archive/
3https://ned.ipac.caltech.edu/
4http://skyserver.sdss.org/dr16/en/home.aspx
5https://hla.stsci.edu/
6https://galex.stsci.edu/GR6/
7http://xmm-catalog.irap.omp.eu/
8https://www.cosmos.esa.int/web/xmm-newton/xsa
9https://mars.lco.global/?page=1
10https://www.wis-tns.org/
11http://www.astronomy.ohio-state.edu/asassn/transients.html
12https://asas-sn.osu.edu/variables
13https://catalogs.mast.stsci.edu/panstarrs/
14https://wise2.ipac.caltech.edu/docs/release/allsky/
15http://tdc-www.harvard.edu/catalogs/ub1.html
16https://www.aavso.org/vsx/
17https://irsa.ipac.caltech.edu/Missions/2mass.html
18http://www.sai.msu.su/gcvs/gcvs/
19https://archive.stsci.edu/prepds/cvcat/
20https://atlas.fallingstar.com/home.php
21https://vo.imcce.fr/webservices/skybot/
22https://vizier.u-strasbg.fr/viz-bin/VizieR


# Generalized multichannel optical theorem: Coherent control of the total
scattering cross section

Adrien Devolder1, Timur V. Tscherbul2, and Paul Brumer1 1Chemical Physics
Theory Group, Department of Chemistry, and Center for Quantum Information and
Quantum Control, University of Toronto, Toronto, Ontario, M5S 3H6, Canada
2Department of Physics, University of Nevada, Reno, NV, 89557, USA

###### Abstract

The optical theorem is a fundamental aspect of quantum scattering theory.
Here, we generalize this theorem to the case where the incident scattering
state is a superposition of internal states of the collision partners,
introducing additional interference contributions and, e.g., providing a route
to control the total integral cross section. As in its standard form, forward
scattering plays an essential role in the generalized optical theorem, but
with interference terms being related to the inelastic forward scattering
amplitudes _between_ states in the initial superposition. Using the resultant
control index, we show that extensive control is possible over ultracold
collisions of oxygen molecules in their rovibrational ground states, and of
85Rb-85Rb collisions, promising systems for the first experimental
demonstration of the quantum interference control of the total scattering
cross section.

##  I Introduction

The multichannel optical theorem is a remarkable result in scattering theory
that relates the imaginary part of the elastic forward amplitude to the total
scattering cross section [1, 2], a consequence of probability conservation
during the scattering. No matter how complicated the scattering, all
information about the total cross section is contained in the elastic forward
scattering. Discovered first for light by Sellmeier [3] and Rayleigh [4], the
optical theorem was subsequently extended to quantum mechanics by Feenberg [5]
and Bohr et al.[6], and has been used in many areas, such as atomic, molecular
and optical physics [7, 8], plasma physics [9], astrophysics [8], atmospheric
physics [7, 8], nuclear physics[10, 11, 12] and high-energy physics [13].

The standard optical theorem assumes an initial pure state of fixed energy.
Here we extend this theorem to a broader class of initial states, a
superposition at a fixed energy that generates new and interesting
interference contributions. We then demonstrate the utility of the generalized
theorem in controlling the total scattering cross section. Examples of
efficient control of ultracold atomic and molecular collisions are provided.

Over the past decades, progress in cooling techniques has enabled the creation
of ultracold gases of atoms and molecules[14]. Nevertheless, loss of coherence
is caused by collisions between the atoms/molecules and limits their use in
quantum information science [15, 16, 17, 18]. On the other hand, such
collisions are essential for understanding ultracold chemistry [19, 20].
Fortunately, some ultracold collisions can be controlled due to the ability to
fully define the internal states of the atoms/molecules, and due to the small
value of the kinetic energy relative to the perturbations induced by external
fields. As a consequence, the vast majority of ultracold control scenarios are
based on external fields (magnetic, electric or optical) [21, 22], with
molecules/atoms prepared in a well-defined internal state. In these
strategies, the control knobs are the field parameters (strength, frequency,
etc.).

However, this approach, as noted below, has deficiencies, motivating a
different strategy. That is, the control of the internal degree of freedom at
ultracold temperature enables the preparation of quantum superpositions of
internal states that can be used to induce interferences between scattering
amplitudes, in analogy with the double slit experiments [23, 24]. Then,
instead of control via variation of field parameters, ultracold collisions are
controlled by changing the nature of the initial superposition. The resultant
effect on the system is the principle of coherent control [23].

In previous work, we demonstrated that the ultracold regime is ideal for
coherent control and that control can be achieved for resonant processes such
as collisional spin exchange [25], which can be completely suppressed (or
activated), via destructive (constructive) interference. Control is achieved
without persistent application of external perturbations. In particular, the
collision partners (atoms or molecules) need not have electric and(/or)
magnetic dipole moments to be coherently controlled. For example, collisions
of nonmagnetic homonuclear molecules, like H2 or Sr2, could be manipulated.
Moreover, the absence of external field could be important for high-precision
measurements (for example with Sr2 molecules [26, 27]) where suppressing
external perturbations is significant.

At present, contrary to the unimolecular processes [28], the observation of
coherent control of bimolecular processes is still an open experimental
challenge due to a number of issues. First, the preparation of the initial
superposition is experimentally challenging. Conditions on the coherent
control of scattering events require either entanglement of the external and
internal degrees of freedom [29, 30, 31, 32], or superposition of degenerate
magnetic sublevels (an $m$-superposition) [33]. However, recently, some
progress has been made in preparing $m$-superpositions in H2 molecules and its
isotopologues [34, 35, 36]. Secondly, our previous study of coherent control
at ultralow temperatures [25] focused on state-to-state cross sections, and
indicated a need for complicated coincidence measurements of the two scattered
molecules [37, 38]. On the other hand, the total cross section, considered
here, could be experimentally easier to measure and control.

Below we demonstrate the utility of the generalized theorem in controlling the
total scattering cross section. The theorem allows to answer the following
fundamental questions: How does the standard optical theorem generalize when
the scattering is of an initial superposition of internal states? What are the
new insights for coherent control of the total cross section arising from this
generalization? And, with these new insights, can we identify atomic and/or
molecular collisions promising for a first experimental demonstration of
coherent control of the total scattering cross section? Answering these
questions is important for the development of this completely new control
strategy for ultracold collisions and for applications in other scattering
scenarios.

The structure of the paper is as follows. We first derive the generalized
multichannel optical theorem (Section II) and, in the rest of the paper, use
the resultant theorem to address the control related issues raised above. In
Section III, we compare the derived formulas with the standard form of the
optical theorem. The generalized optical theorem allows us to define a
coherent control index for the total integral cross section (ICS) in Section
IV, which we then use to analyze promising systems for experiments in Section
V. Exact scattering calculations are presented which show impressive control
over the total cross section for the experimentally realizable ultracold
85Rb-85Rb and 17O2-17O2 collisions. We conclude in section VI.

##  II Derivation

Consider an initial superposition of $N_{sup}$ degenerate internal states of
the scattering partners denoted $\ket{i}=\ket{\nu_{A}}\ket{\nu_{B}}$. Here
$\nu_{A,B}$ are the quantum numbers characterizing the internal states of the
molecular or atomic collision partners A+B. The initial state is:

$\Psi_{in}(\vec{r},\xi)=\textrm{e}^{ikz}\sum_{i=1}^{N_{sup}}a_{i}\ket{i},$ (1)

where the $z$-axis is defined along the initial relative momentum $\vec{k}$.
Here $a_{i}$ are the superposition coefficients, $\vec{r}=(r,\theta,\phi)$ is
the relative position between the two collision partners, and $\xi$ is
composed of all internal coordinates contained in $\ket{i}$.

After the collision, the system is in a superposition of scattered spherical
waves in various open channels [1]:

$\Psi_{out}(\vec{r},\xi)=\sum_{j}f_{sup\rightarrow
j}(\theta,\phi)\frac{\textrm{e}^{ik_{j}r}}{r}\ket{j},$ (2)

where $k_{j}$ is the final relative momentum in the state $\ket{j}$ and

$f_{sup\rightarrow j}(\theta,\phi)=\sum_{i=1}^{N_{sup}}a_{i}f_{i\rightarrow
j}(\theta,\phi)$ (3)

is the scattering amplitude from the initial superposition to the final state
$\ket{j}$. Here, $f_{i\rightarrow j}$ is the scattering amplitude from state
$\ket{i}$ to $\ket{j}$. The overall wavefunction therefore obeys the boundary
condition:

$\Psi(\vec{r},\xi)\xrightarrow[r\to\infty]{}\textrm{e}^{ikz}\sum_{i=1}^{N_{sup}}a_{i}\ket{i}+\sum_{j}f_{sup\rightarrow
j}(\theta,\phi)\frac{\textrm{e}^{ik_{j}r}}{r}\ket{j}.$ (4)

The optical theorem can be derived by imposing conservation of probability via
the continuity equation:

$\oint\vec{j}(\vec{r})\cdot\hat{e}_{r}r^{2}d\Omega=0,$ (5)

where
$\vec{j}(\vec{r})=\int\textrm{d}\xi\frac{\hbar}{2i\mu}\Psi^{*}(\vec{r},\xi)\vec{\nabla}\Psi(\vec{r},\xi)+\textrm{c.c}$
is the current density, $\mu$ is the reduced mass, $\hat{e}_{r}$ is the unit
radial vector, $d\Omega=\,\sin\theta\textrm{d}\theta\textrm{d}\phi$ is the
solid angle, and c.c denotes complex conjugate. The continuity equation states
that the scattering flux through any closed surface must vanish, which is also
the case for a sphere with radius $r\rightarrow\infty$ where the boundary
condition (4) is imposed. This flux, defined as
$I\equiv\lim_{r\rightarrow\infty}\oint\vec{j}(\vec{r}).\hat{e}_{r}r^{2}\textrm{d}\Omega$,
can be expanded into three terms: the incoming contribution $I_{in}$, the
outgoing contribution $I_{out}$ and the interference contribution $I_{int}$:

$I_{in}+I_{out}+I_{int}=0.$ (6)

The contributions $I_{ini}$, $I_{out}$ and $I_{int}$ are derived in the
appendices A, B and C, respectively, and their final values are:

$I_{in}=0,$ (7) $I_{out}=\frac{\hbar k}{\mu}\sigma_{sup}^{tot},$ (8)
$I_{int}=\frac{-4\pi\hbar}{\mu}\sum_{i,i^{\prime}=1}^{N_{sup}}\textrm{Im}[a_{i}^{*}a_{i^{\prime}}f_{i^{\prime}\rightarrow
i}(\theta=0)].$ (9)

Introducing (7),(8) and (9) in Eq. (6), one obtains the total integral cross
section (ICS):

$\sigma_{sup}^{tot}=\frac{4\pi}{k}\sum_{i,i^{\prime}=1}^{N_{sup}}\textrm{Im}[a_{i}^{*}a_{i^{\prime}}f_{i^{\prime}\rightarrow
i}(\theta=0)].$ (10)

Consider then the dependence of the total cross section on the relative phases
$\beta_{i}$ between the states of the initial superposition, which can be
explicitly illustrated by writing the superposition coefficients in their
polar form: $a_{i}=|a_{i}|\textrm{e}^{i\beta_{i}}$:

$\begin{split}\sigma_{sup}^{tot}=&\frac{4\pi}{k}\sum_{i=1}^{N_{sup}}|a_{i}|^{2}\textrm{Im}[f_{i\rightarrow
i}(\theta=0)]\\\ +&\frac{4\pi}{k}\sum_{i}\sum_{i^{\prime}\neq
i}|a_{i}||a_{i^{\prime}}|\textrm{Im}\left[\textrm{e}^{i(\beta_{i^{\prime}}-\beta_{i})}f_{i^{\prime}\rightarrow
i}(\theta=0)\right].\end{split}$ (11)

Finally, the symmetric relation between the scattering amplitudes
$f_{i^{\prime}\rightarrow i}=f_{i\rightarrow i^{\prime}}$ can be exploited to
obtain the generalized optical theorem:

$\begin{split}\sigma_{sup}^{tot}=&\frac{4\pi}{k}\sum_{i=1}^{N_{sup}}|a_{i}|^{2}\textrm{Im}[f_{i\rightarrow
i}(\theta=0)]\\\
+&\frac{8\pi}{k}\sum_{i=1}^{N_{sup}}\sum_{i^{\prime}>i}|a_{i}||a_{i^{\prime}}|\textrm{Im}[f_{i\rightarrow
i^{\prime}}(\theta=0)]\,\cos(\beta_{i^{\prime}}-\beta_{i}).\end{split}$ (12)

The symmetric form is valid when the time-reversal symmetry applies. When it
does not, Eq. (11) should be used.

Equations (11) and (12) are the central result of this work. They establish a
relation between the magnitude of the total ICS and the preparation
coefficients of the internal superposition, enabling coherent control of the
total ICS. Note that they are of the standard coherent control form [23],
indirect scattering terms plus interference between pairs of states.

##  III A generalization of the standard optical theorem

The generalized optical theorem (Eq. 12) establishes the total ICS as composed
of an incoherent contribution and a coherent contribution:

$\sigma_{incoh}=\frac{4\pi}{k}\sum_{i=1}^{N_{sup}}|a_{i}|^{2}\textrm{Im}[f_{i\rightarrow
i}(\theta=0)],$ (13)
$\sigma_{coh}=\frac{8\pi}{k}\sum_{i=1}^{N_{sup}}\sum_{i^{\prime}>i}|a_{i}||a_{i^{\prime}}|\textrm{Im}[f_{i\rightarrow
i^{\prime}}(\theta=0)]\,\cos(\beta_{i^{\prime}}-\beta_{i}).$ (14)

The incoherent contribution is related to the elastic forward scattering
amplitudes of individual states in the superposition, weighted by their
populations. This contribution would be the same for a classical mixture with
$|a_{i}|$ given, for example by Boltzmann populations. The standard optical
theorem without superposition follows from the incoherent contribution (13) in
the limit $N_{sup}=1$, $a_{1}=1$:

$\sigma_{1}^{tot}=\frac{4\pi}{k}\textrm{Im}[f_{1\rightarrow 1}(\theta=0)].$
(15)

For the purpose of control, the significance of the coherent contribution (14)
is that it allows control over the total ICS by varying the amplitude product
$|a_{i}||a_{i^{\prime}}|$ and the relative phases
$(\beta_{i^{\prime}}-\beta_{i})$ between the states in the initial
superposition. The coherent contribution (14) is related to the inelastic
forward scattering amplitudes _between_ the states in the initial
superposition, and is the primary attribute of the generalized optical
theorem, corresponding to interference between scattering events in different
channels. _It indicates that inelastic scattering between the channels
involved in the initial superposition is a prerequisite for coherent control
of total ICS._ Note that whatever the system, the coherent terms oscillate as
$\cos(\beta_{i^{\prime}}-\beta_{i})$ and, that contrary to the elastic
component $\textrm{Im}[f_{i\rightarrow i}(\theta=0)]$, the inelastic component
$\textrm{Im}[f_{i\rightarrow i^{\prime}}(\theta=0)]$ can be negative.

Consider now the key quantity that determine the magnitude of the coherent
contribution to the total ICS, (14), the imaginary part of the inelastic
forward scattering amplitude:

$\textrm{Im}[f_{i\rightarrow
i^{\prime}}(\theta=0)]=\frac{1}{4k}\sum_{j}\sum_{\ell^{\prime},m^{\prime}}\tilde{T}_{i\rightarrow
j,\ell^{\prime},m^{\prime}}\tilde{T}_{i^{\prime}\rightarrow
j,\ell^{\prime},m^{\prime}}^{*},$ (16)

where $\tilde{T}_{i\rightarrow
j,\ell^{\prime},m^{\prime}}=\sum_{\ell}i^{\ell}\sqrt{2\ell+1}T_{i,\ell,0\rightarrow
j,\ell^{\prime},m^{\prime}}$, $\ell$ and $\ell^{\prime}$ are the initial and
final partial wave, and $T_{i,\ell,0\rightarrow j,\ell^{\prime},m^{\prime}}$
are the $T$-matrix elements. Equation (16) provides an important perspective,
that is: _interference is a result of scattering of states $\ket{i}$ and
$\ket{i^{\prime}}$ into the same states $j,\ell^{\prime},m^{\prime}$_.
Further, the sum
$\sum_{j}\sum_{\ell^{\prime},m^{\prime}}\tilde{T}_{i\rightarrow
j,\ell^{\prime},m^{\prime}}\tilde{T}_{i^{\prime}\rightarrow
j,\ell^{\prime},m^{\prime}}^{*}$ is seen to be real. Equation (16) is required
by conservation of probability and the generalized optical theorem can also be
proven via this relation.

##  IV Coherent control of the total cross section

###  IV.1 Two-state superpositions

The generalized optical theorem allows for considerable new insights into the
coherent control of the total ICS. First, consider the case where the initial
superposition of the scattering partners is composed of two states,
$\Psi_{in}(\vec{r},\xi)=\textrm{e}^{ikz}\left(\cos\eta\ket{1}+\sin\eta\textrm{e}^{i\beta}\ket{2}\right)$.
Here, control is achieved by changing the amplitude and phase of the
superposition by varying $\eta$ and $\beta$, respectively. The generalized
optical theorem [Eq. (12)] takes the form:

$\begin{split}\sigma_{sup}^{tot}=&\frac{4\pi}{k}\\{\cos^{2}(\eta)\textrm{Im}[f_{1\rightarrow
1}]+\,\sin^{2}(\eta)\textrm{Im}[f_{2\rightarrow 2}]\\}\\\
+&\frac{8\pi}{k}\,\cos\eta\,\sin\eta\ \textrm{Im}[f_{1\rightarrow
2}]\,\cos\beta.\end{split}$ (17)

Insight is afforded by minimization and maximization of this expression. The
optimization with respect to $\beta$ is straightforward; maximal for
$\cos\beta=1$ ($\beta_{max}=0$) and minimal for $\cos\beta=-1$
($\beta_{max}=\pi$) if $\textrm{Im}[f_{1\rightarrow 2}(\theta=0)]$ is
positive. It is the opposite if $\textrm{Im}[f_{1\rightarrow 2}(\theta=0)]$ is
negative. The optimization with respect to $\eta$ gives:

$\eta_{min}=\frac{1}{2}\arctan\left(\frac{2\left|\textrm{Im}[f_{1\rightarrow
2}]\right|}{\textrm{Im}[f_{2\rightarrow 2}]-\textrm{Im}[f_{1\rightarrow
1}]}\right),$ (18)
$\eta_{max}=-\frac{1}{2}\arctan\left(\frac{2\left|\textrm{Im}[f_{1\rightarrow
2}]\right|}{\textrm{Im}[f_{2\rightarrow 2}]-\textrm{Im}[f_{1\rightarrow
1}]}\right),$ (19)

Introducing $\eta_{min}$,$\beta_{min}$, $\eta_{max}$ and $\beta_{max}$ in the
Eq. (17), one obtains the minimal and maximal values of the total ICS:

$\begin{split}\sigma_{min}^{tot}=&\frac{2\pi}{k}\left(\textrm{Im}[f_{1\rightarrow
1}]+\textrm{Im}[f_{2\rightarrow 2}]\right)\\\
-&\frac{2\pi}{k}\sqrt{\left(\textrm{Im}[f_{2\rightarrow
2}]-\textrm{Im}[f_{1\rightarrow
1}]\right)^{2}+4\textrm{Im}^{2}[f_{1\rightarrow 2}]}\end{split},$ (20)

$\begin{split}\sigma_{max}^{tot}=&\frac{2\pi}{k}\left(\textrm{Im}[f_{1\rightarrow
1}]+\textrm{Im}[f_{2\rightarrow 2}]\right)\\\
+&\frac{2\pi}{k}\sqrt{\left(\textrm{Im}[f_{2\rightarrow
2}]-\textrm{Im}[f_{1\rightarrow
1}]\right)^{2}+4\textrm{Im}^{2}[f_{1\rightarrow 2}]}\end{split}.$ (21)

The extent of the control is determined by the magnitude of
$\textrm{Im}[f_{1\rightarrow 2}]$, a quantity bounded between zero and
$\sqrt{\textrm{Im}[f_{1\rightarrow 1}]\textrm{Im}[f_{2\rightarrow 2}]}$. That
leads us to define a control index:

$R_{c}=\frac{|\textrm{Im}[f_{1\rightarrow
2}]|}{\sqrt{\textrm{Im}[f_{1\rightarrow 1}]\textrm{Im}[f_{2\rightarrow 2}]}},$
(22)

which ranges from zero to one. When $R_{c}=1$, the Schwartz equality
($\left|\textrm{Im}[f_{1\rightarrow
2}]\right|=\sqrt{\textrm{Im}[f_{1\rightarrow 1}]\textrm{Im}[f_{2\rightarrow
2}]}$) holds. In this case, the minimum value of the total cross section
$\sigma_{min}^{tot}$ vanishes while the maximal value is the sum of the total
cross sections in absence of superposition,
$\sigma_{max}^{tot}=\frac{4\pi}{k}\left(\textrm{Im}[f_{1\rightarrow
1}]+\textrm{Im}[f_{2\rightarrow 2}]\right)=\sigma_{1}^{tot}+\sigma_{2}^{tot}$.
Hence, the value of $R_{c}$ allows us to quantify the extent of coherent
control of the total ICS and interpret when systems display the maximum
possible degree of control (which is realized for $R_{c}$=1). An example of
complete control is the case of ideal resonance, [32], where the resonance
occurs for all final states. Another favorable situation is when the number of
channels significantly populated in a collision is equal to (or less than) the
number of states in the superposition[39]. The latter case is illustrated
below in the section V.

###  IV.2 $N_{sup}$-state superpositions ($N_{sup}$>2)

Figure 1: Coherent control of the total integral cross section for the cold
17O2\+ 17O2 collisions at 10 $\mu$K from the initial superposition $\Psi_{E}$
(upper panels) and $\Psi_{NE}$ (lower panels). (a) and (c): Control landscape;
(b) and (d): Control by the relative phase $\beta$ with $\eta=\pi/4$ fixed.

The optimization can be generalized to a superposition of $N_{sup}$ states via
a procedure similar to that introduced in Ref. [39]. We define the matrix
$\Sigma_{ij}=\frac{4\pi}{k}\textrm{Im}[f_{i\rightarrow j}(\theta=0)]$ and
rewrite the generalized optical theorem (Eq. 12) as:

$\sigma_{sup}^{tot}=\bm{a^{\dagger}\Sigma a},$ (23)

where $\bm{a}$ is a vector with the components $a_{i}$.

The optimization problem of finding $\sigma_{min}^{tot}$ and
$\sigma_{max}^{tot}$ transforms to the solution of an eigenvalue equation for
$\bm{\Sigma}$:

$\bm{\Sigma a}=\sigma_{opt}^{tot}\bm{a},$ (24)

where the optimized coefficients are the corresponding eigenvectors. The
matrix $\bm{\Sigma}$ is block diagonal with respect to the symmetry of the
scattering. For example, a superposition of states with different projections
$M_{int}$ of the total internal angular momentum does not display interference
since the imaginary part of the inelastic scattering amplitude between these
states is zero. Therefore, they occupy different blocks of the matrix
$\bm{\Sigma}$. The resulting eigenvectors only contain the states with the
same value of $M_{int}$ and correspond to entangled superpositions thereof.

The best system controllability is obtained if the lowest eigenvalue of
$\bm{\Sigma}$ is equal to zero; i.e. if the determinant of $\bm{\Sigma}$ is
null. As the matrix $\bm{\Sigma}$ is semi-infinite
($\textrm{det}(\bm{\Sigma})\geq 0$), we can define a generalized control index
$R_{c}$:

$R_{c}=\sqrt[N]{\frac{\textrm{det}(\bm{\Sigma})_{-}}{\textrm{det}(\bm{\Sigma})_{+}}},$
(25)

where $\textrm{det}(\bm{\Sigma})_{\pm}$ are the sum of positive and negative
terms of the determinant, respectively. For example, in the two-states case,
the determinant is equal to $\textrm{Im}[f_{1\rightarrow
1}]\textrm{Im}[f_{2\rightarrow 2}]-(\textrm{Im}[f_{1\rightarrow 2}])^{2}$, and
we recover expression (22). For the three states case, for example, the matrix
$\bm{\Sigma}$ is then defined as:

$\bm{\Sigma}=\frac{4\pi}{k}\begin{pmatrix}\textrm{Im}[f_{1\rightarrow
1}]&\textrm{Im}[f_{1\rightarrow 2}]&\textrm{Im}[f_{1\rightarrow 3}]\\\
\textrm{Im}[f_{1\rightarrow 2}]&\textrm{Im}[f_{2\rightarrow
2}]&\textrm{Im}[f_{2\rightarrow 3}]\\\ \textrm{Im}[f_{1\rightarrow
3}]&\textrm{Im}[f_{2\rightarrow 3}]&\textrm{Im}[f_{3\rightarrow
3}]\end{pmatrix}.$ (26)

The determinant of this matrix is given by:

$\begin{split}&\textrm{det}(\bm{\Sigma})=\frac{4\pi}{k}\Bigg{(}\textrm{Im}[f_{1\rightarrow
1}]\Big{(}\textrm{Im}[f_{2\rightarrow 2}]\textrm{Im}[f_{3\rightarrow
3}]-\textrm{Im}[f_{2\rightarrow 3}]^{2}\Big{)}\\\
-&\textrm{Im}[f_{1\rightarrow 2}]\Big{(}\textrm{Im}[f_{1\rightarrow
2}]\textrm{Im}[f_{3\rightarrow 3}]-\textrm{Im}[f_{2\rightarrow
3}]\textrm{Im}[f_{1\rightarrow 3}]\Big{)}\\\ +&\textrm{Im}[f_{1\rightarrow
3}]\Big{(}\textrm{Im}[f_{1\rightarrow 2}]\textrm{Im}[f_{2\rightarrow
3}]-\textrm{Im}[f_{2\rightarrow 2}]\textrm{Im}[f_{1\rightarrow
3}]\Big{)}\Bigg{)}.\end{split}$ (27)

Using the definition (25), the control index takes the form:

$R_{c}=\sqrt[3]{\frac{\textrm{det}(\bm{\Sigma})_{-}}{\textrm{det}(\bm{\Sigma})_{+}}},$
(28)

where:

$\begin{split}\textrm{det}(\bm{\Sigma})_{-}=&\textrm{Im}[f_{1\rightarrow
1}](\textrm{Im}[f_{2\rightarrow 3}])^{2}+\textrm{Im}[f_{2\rightarrow
2}](\textrm{Im}[f_{1\rightarrow 3}])^{2}\\\ +&\textrm{Im}[f_{3\rightarrow
3}](\textrm{Im}[f_{1\rightarrow 2}])^{2},\end{split}$ (29)

and

$\begin{split}\textrm{det}(\bm{\Sigma})_{+}=&\textrm{Im}[f_{1\rightarrow
1}]\textrm{Im}[f_{2\rightarrow 2}]\textrm{Im}[f_{3\rightarrow 3}]\\\
+&2\textrm{Im}[f_{1\rightarrow 2}]\textrm{Im}[f_{2\rightarrow
3}]\textrm{Im}[f_{1\rightarrow 3}],\end{split}$ (30)

when $\textrm{Im}[f_{1\rightarrow 2}]\textrm{Im}[f_{2\rightarrow
3}]\textrm{Im}[f_{1\rightarrow 3}]>0$. Otherwise the term
$2\textrm{Im}[f_{1\rightarrow 2}]\textrm{Im}[f_{2\rightarrow
3}]\textrm{Im}[f_{1\rightarrow 3}]$ must be transferred to
$\textrm{det}(\bm{\Sigma})_{-}$. As stated previously, the value of $R_{c}$
can be used to interpret systems and coherent superpositions that give large
control of the total ICS, as is done in the next section. Specifically,
$R_{c}$ close to one indicates the most efficient coherent control.

##  V Systems with extensive control of the total cross section

The theory above provides a foundation for coherent control of the total ICS.
In the next section, we consider the coherent control of realistic atomic and
molecular collisions at ultralow temperatures. In particular, we demonstrate
the possibility of extensive coherent control of total ICS for O2+O2 and Rb+Rb
collisions.

###  V.1 O2+O2 scattering

Figure 2: Coherent control of the total cross section for the ultracold
85Rb+85Rb collisions in their lower hyperfine states $F=2$ at 50 $\mu$K from
the initial superpositions (a) $\Psi_{E}$ and (b) $\Psi_{NE}$. The relative
phases $\beta,\gamma$ are varied, while the relative populations
$\eta=11\pi/32$, $\epsilon=3\pi/8$ are fixed for achieving the best control.

The first system considered is ultracold scattering of two oxygen molecules in
their rovibrational ground states at 10 $\mu$K. This system has been realized
experimentally in a magnetic trap at 50 mK [40], with further evaporative or
sympathetic cooling projected to achieve the $\mu$K regime. Hence, it is an
advantageous system for study. The oxygen molecules have spin 1, and a spin-
exchange processes can occur during the scattering. The initial internal
states considered here is an entangled superposition between the symmetrized
states $\ket{S_{A}=1,M_{S_{A}}=-1,S_{B}=1,M_{S_{B}}=1}$ and
$\ket{S_{A}=1,M_{S_{A}}=0,S_{B}=1,M_{S_{B}}=0}$:

$\Psi_{E}(\vec{r},\xi)=\textrm{e}^{ikz}\,\left(\cos\eta\ket{1,-1,1,1}+\,\sin\eta\textrm{e}^{i\beta}\ket{1,0,1,0}\right),$
(31)

where $M_{S_{A/B}}$ is the projection of the electronic spin $S_{A/B}$ along
the space-fixed $z$ axis. The symmetrized initial states of identical bosons
are defined as:

$\begin{split}\ket{S_{1},m_{1},S_{2},m_{2}}=\frac{1}{\sqrt{2(1+\delta_{1,2})}}\Big{[}&\ket{S_{1},m_{1}}_{A}\ket{S_{2},m_{2}}_{B}\\\
+&\ket{S_{2},m_{2}}_{A}\ket{S_{1},m_{1}}_{B}\Big{]},\end{split}$ (32)

where $m_{2}\geq m_{1}$, and where the subscript on, for example,
$\ket{S_{1},m_{1}}_{A}$ denotes the states of particle A.

The scattering results (see Appendix D for computational details) show that
the control index is close to 1, i.e $R_{c}=0.97$
($\textrm{Im}[f_{1\rightarrow 1}]$=158.96 a.u., $\textrm{Im}[f_{2\rightarrow
2}]$=162.39 a.u. and $\textrm{Im}[f_{1\rightarrow 2}]$=-156.00 a.u.). Figure 1
(a) shows the total ICS as a function of the relative population $\eta$ and
phase $\beta$ of the initial superposition (31). The minimum value of the
total ICS is seen to be 68 times smaller than the maximum value (11745 and
795427 $\AA^{2}$, respectively), confirming the analysis in terms of the
control index. This control can also be analyzed by decomposing the total
cross section into the different state-to-state cross sections. Two
explanations of the large extent of control can be identified. First, in
accord with the Wigner threshold law [41], only two final channels
($\ket{1,-1,1,1}$ and $\ket{1,0,1,0}$) substantially contribute to the total
cross section. Second, there is complete control of the state-to-state cross
sections to these two channels, as illustrated in our earlier work [25].

###  V.2 Rb+Rb scattering

A second system of interest is the ultracold scattering of 85Rb atoms in their
lower hyperfine states $F=2$ at 50 $\mu$K. This system can be readily realized
experimentally in either an optical dipole trap [42, 43] or an optical tweezer
[44], allowing for precise control over internal and external atomic states.
The scattering state is prepared in an entangled superposition of three
symmetrized states $\ket{F_{1}=2,M_{F_{1}}=-2,F_{2}=2,M_{F_{2}}=2}$,
$\ket{F_{1}=2,M_{F_{1}}=-1,F_{2}=2,M_{F_{2}}=1}$ and
$\ket{F_{1}=2,M_{F_{1}}=0,F_{2}=2,M_{F_{2}}=0}$:

$\begin{split}\Psi_{E}(\vec{r},\xi)=\textrm{e}^{ikz}\Big{(}&\cos\eta\,\sin\epsilon\ket{2,-2,2,2}\\\
+&\sin\eta\,\sin\epsilon\,\textrm{e}^{i\beta}\ket{2,-1,2,1}\\\ +&\cos\epsilon\
\textrm{e}^{i\gamma}\ket{2,0,2,0}\Big{)}.\end{split}$ (33)

Here, $M_{F}$ denotes the projection of the total angular momentum
$\vec{F}=\vec{I}+\vec{S}$ along the z axis. The angles
$\eta,$$\epsilon\in[0,\pi/2]$ determine the relative populations, and
$\beta,\gamma\in[0,2\pi]$ the relative phases of the superposition.
In this case, the value of the control index (Eq. (28)) is $0.83$, showing the
robustness of control. We find that the minimum value of the total cross
section (see Fig.2 (a)) is 11 times smaller than the maximal value (24,669 and
263,065 $\AA^{2}$ respectively). Here, the control is less effective than for
the case of the scattering of oxygen molecules, but the advantage of the
rubidium case is in experimental implementation. That is, ultracold rubidium
atoms are widely available in optical dipole traps [42, 43] or, more recently,
in optical tweezer setups [44]. Furthermore, initial steps toward the creation
of entangled superposition similar to Eq.(33) between two 85Rb atoms have been
demonstrated in optical tweezers[44].

###  V.3 Non-entangled superpositions

The superpositions (31) and (33) are entangled and cannot be obtained via
independent preparation of each of the collision partners. That poses an
experimental challenge, since independent preparation is the easiest way to
experimentally realize coherent control. To examine the effect of initial
state entanglement, we consider scattering of unentangled independently
prepared molecules.

For the case of O2+O2 collisions, the two colliding molecules are prepared in
two different superpositions:

$\displaystyle\ket{\psi_{A}}$
$\displaystyle=N\left(\sqrt{\cos\eta}\ket{1,-1}+\sqrt{\sin\eta}\,\textrm{e}^{i\frac{\beta}{2}}\ket{1,0}\right)$
(34) $\displaystyle\ket{\psi_{B}}$
$\displaystyle=N\left(\sqrt{\sin\eta}\,\textrm{e}^{i\frac{\beta}{2}}\ket{1,0}+\sqrt{\cos\eta}\ket{1,1}\right),$
(35)

where $N=(\sin\eta+\cos\eta)^{-1/2}$ is a normalization factor. Here, for
simplicity, we consider states of A and B with population and relative phase
in both being determined by the same parameter $\eta$ and $\beta$. Different
choices of control parameters in A and B are also possible.
The initial superposition is then obtained by symmetrizing the product
$\ket{\psi_{A}}\ket{\psi_{B}}$ to give:

$\Psi_{NE}(\vec{r},\xi)=N^{2}\textrm{e}^{ikz}\Big{[}\cos\eta\ket{{1,-1,1,1}}+\sin\eta\,\textrm{e}^{i\beta}\ket{{1,0,1,0}}\\\
+\sqrt{\cos\eta\,\sin\eta}\,\textrm{e}^{i\frac{\beta}{2}}(\ket{{1,-1,1,0}}+\ket{{1,0,1,1}})\Big{]}.$
(36)

Note that Eq. (36) is unentangled as compared to Eq. (31). Preparation at the
level of the individual molecules implies that the superposition contains a
range of states with different values of $M_{int}$. Here, there are two states
with $M_{int}$=0, one state with $M_{int}$=1 and one state with $M_{int}$=-1.
Since these supplementary states do not interfere, the extent of the control
is reduced. Also, the normalization factor, which depends on $\eta$, has an
impact on the control. However, with $\eta=\pi/4$ fixed (see Fig. 1(d)),
coherent control by varying the relative phase is still extensive, with a
factor of 41 between the maximal and minimal values. Therefore, even with a
non-entangled superposition, it is possible to substantially control the total
cross section for O2+O2 scattering.

As an example in the Rb+Rb case, the two colliding atoms are prepared in two
different three-state superpositions:

$\begin{split}\ket{\psi_{A}}=N_{3}\Big{(}&\sqrt{\sin\eta\,\sin\epsilon}\ket{2,-2}\\\
+&\sqrt{\cos\eta\,\sin\epsilon}\,\textrm{e}^{i\frac{\beta}{2}}\ket{2,-1}\\\
+&\sqrt{\cos\epsilon}\,\textrm{e}^{i\frac{\gamma}{2}}\ket{2,0}\Big{)},\end{split}$
(37)
$\begin{split}\ket{\psi_{B}}=N_{3}\Big{(}&\sqrt{\sin\eta\,\sin\epsilon}\ket{2,2}\\\
+&\sqrt{\cos\eta\,\sin\epsilon}\,\textrm{e}^{i\frac{\beta}{2}}\ket{2,1}\\\
+&\sqrt{\cos\epsilon}\,\textrm{e}^{i\frac{\gamma}{2}}\ket{2,0}\Big{)},\end{split}$
(38)

where
$N_{3}=(\sin\eta\,\sin\epsilon+\cos\eta\,\sin\epsilon+\cos\epsilon)^{-1/2}$.
As in the oxygen case, the initial non-entangled superposition is obtained by
symmetrizing the product $\ket{\psi_{A}}\ket{\psi_{B}}$:

$\Psi_{NE}(\vec{r},\xi)=N_{3}^{2}\,\textrm{e}^{ikz}\Big{[}\cos\eta\,\sin\epsilon\ket{2,-2,2,2}\\\
+\sin\eta\,\sin\epsilon\,\textrm{e}^{i\beta}\ket{2,-1,2,1}+\cos\epsilon\,\textrm{e}^{i\gamma}\ket{2,0,2,0}\\\
+\sqrt{\sin\eta\,\cos\eta}\,\sin\epsilon\,\textrm{e}^{i\frac{\beta}{2}}(\ket{2,-2,2,1}+\ket{2,-1,2,2})\\\
+\sqrt{\sin\eta\,\sin\epsilon\,\cos\epsilon}\,\textrm{e}^{i\frac{\gamma}{2}}(\ket{2,-2,2,0}+\ket{2,0,2,2})\\\
+\sqrt{\cos\eta\,\sin\epsilon\,\cos\epsilon}\,\textrm{e}^{i\frac{\beta+\gamma}{2}}(\ket{2,-1,2,0}+\ket{2,0,2,1})\Big{]}.$
(39)

In this case, there are four groups of states: one group of three states with
$M_{int}=0$, two groups of two states with $M_{int}=\pm 1$ and two groups of
one state with $M_{int}=\pm 2$. The total cross section is the sum of the ICS
correspondingly to each group. Coherent control by varying the relative phases
is shown in Figure 2 (b). Here, the ratio max/min is significantly
reduced,relative to the entangled case, to a factor of two (163,911 and 74,512
$\AA^{2}$ respectively). With the non-entangled superposition, the quasi-
vanishing of the total cross section is lost. This difference between the
oxygen and rubidium cases arises from the number of uncontrollable "satellite"
terms [23], prevalent in Eq. (39). Nevertheless, if the cross section can be
accurately measured, the predicted control is sufficiently large to be
experimentally measurable.

##  VI Conclusion

We derived the generalized multichannel optical theorem for the scattering of
initial coherent superpositions, a fundamental contribution to scattering
theory, introducing new interference contributions. As an example, it was then
used to address three issues. The first was to determine how the generalized
optical theorem reflects contributions from an initial superposition of
internal states. As in the standard form of the optical theorem, the forward
scattering plays an essential role. However, the generalized theorem shows
that inelastic scattering between the states involved in the initial
superpositions is crucial in the optical theorem and here in applications to
the coherent control of the total integral cross section.

The second issue concerned optical theorem insights into the coherent control
of the total scattering cross section. The maximal and the minimal values of
the total cross sections were found to be directly related to the elastic and
inelastic forward scattering amplitudes. If these quantities fulfill the
Schwartz equality, the minimal value vanishes and complete control is
possible. Furthermore, the Schwartz equality allowed us to define a control
index through which the extent of control can be understood. These statements
were generalized to include the initial superpositions of $N_{sup}$ internal
states of the collision partners.

Finally, the third issue concerned identifying cold atomic and molecular
collision systems with a large extent coherent control to motivate
experimental demonstrations of coherent control of scattering. Two interesting
cases were examined: O2+O2 scattering in their rovibrational ground state, and
the scattering of rubidium atoms 85Rb. To examine experimental requirements on
the initial state preparation, we considered both entangled and non-entangled
superpositions, where the latter could be created by preparation of the
individual molecules. Although the entangled states display greater control,
the non-entangled initial superpositions show sufficient control to be both
significant and experimentally observable. Ultracold O2+O2 scattering was
found to allow better control, while ultracold 85Rb+85Rb collisions could be
easier to probe experimentally, given widespread availability of ultracold Rb
atoms in either optical dipole traps [42, 43] or optical tweezers [44]. These
two cases demonstrate that a large degree of control can be obtained for
multi-channel scattering, and motivate the first experimental demonstration of
the quantum interference based control of the total cross section in ultracold
atomic and molecular collisions.

## Acknowledgements

This work was supported by the U.S. Air Force Office of Scientific Research
(AFOSR) under Contract No. FA9550-19-1-0312. SciNet computational facilities
are gratefully acknowledged.

## Appendix A: Calculation of the incoming flux

The incoming current density for the superposition state takes the form:

$\begin{split}\vec{j}_{in}(\vec{r})=\Bigg{[}\int\textrm{d}\xi\frac{\hbar}{2i\mu}\left(\sum_{i=1}^{N_{sup}}a_{i}\ket{i}\textrm{e}^{ikz}\right)^{*}\\\
\vec{\nabla}\left(\sum_{i^{\prime}=1}^{N_{sup}}a_{i^{\prime}}\ket{i^{\prime}}\textrm{e}^{ikz}\right)\Bigg{]}+\textrm{c.c}.\end{split}$
(40)

Using the orthogonality of the channel basis states
$\braket{i}{i^{\prime}}=\delta_{i,i^{\prime}}$ and
$\sum_{i=1}^{N_{sup}}|a_{i}|^{2}=1$, one obtains:

$\vec{j}_{in}(\vec{r})=\frac{\hbar k}{\mu}\hat{e}_{z},$ (41)

where $\hat{e}_{z}$ is the unit vector along $\vec{z}$.
The integral on the closed surface gives:

$I_{in}=\lim_{r\rightarrow\infty}r^{2}\frac{\hbar
k}{\mu}\int\hat{e}_{z}\cdot\hat{e}_{r}\textrm{d}\Omega,$ (42)
$I_{in}=\lim_{r\rightarrow\infty}r^{2}\frac{\hbar
k}{\mu}2\pi\int_{0}^{\pi}\,\sin\theta\,\cos\theta\textrm{d}\theta.$ (43)

Since $\int_{0}^{\pi}\,\sin\theta\,\cos\theta\textrm{d}\theta=0$, the incoming
contribution vanishes:

$I_{in}=0.$ (44)

## Appendix B: Calculation of the outgoing flux

The outgoing current density is:

$\begin{split}\vec{j}_{out}(\vec{r})=\Bigg{[}\int\textrm{d}\xi\frac{\hbar}{2i\mu}\left(\sum_{j}f_{sup\rightarrow
j}(\theta,\phi)\frac{\textrm{e}^{ik_{j}r}}{r}\ket{j}\right)^{*}\\\
\vec{\nabla}\left(\sum_{j^{\prime}}f_{sup\rightarrow
j^{\prime}}(\theta,\phi)\frac{\textrm{e}^{ik_{j^{\prime}}r}}{r}\ket{j^{\prime}}\right)\Bigg{]}+\textrm{c.c}.\end{split}$
(45)

Using the orthogonality relation, equation (45) becomes:

$\vec{j}_{out}(\vec{r})=\frac{\hbar}{\mu}\sum_{j}\frac{k_{j}}{r^{2}}|f_{sup\rightarrow
j}(\theta,\phi)|^{2}\hat{e}_{r}+O\left(\frac{1}{r^{3}}\right).$ (46)

Note that the angular components of the gradient operator (included in
$O\left(\frac{1}{r^{3}}\right)$) have been neglected. Indeed, when the limit
of $r$ to infinity is taken, those angular components become negligible in
comparison to the radial component.

The integrated quantity takes the form:

$I_{out}=\frac{\hbar}{\mu}\sum_{j}k_{j}\int|f_{sup\rightarrow
j}(\theta,\phi)|^{2}\textrm{d}\Omega.$ (47)

Using the definitions of the differential cross section ,
$\frac{d\sigma_{sup}}{d\Omega}=\frac{k_{j}}{k}|f_{sup\rightarrow
j^{\prime}}(\theta,\phi)|^{2}$, and of the total ICS,
$\sigma_{sup}^{tot}=\sum_{j}\int\frac{d\sigma_{sup}}{d\Omega}d\Omega$,
$I_{out}$ becomes:

$I_{out}=\frac{\hbar k}{\mu}\sigma_{sup}^{tot}.$ (48)

## Appendix C: Calculation of the interfering flux

The interference current density takes the form:

$\begin{split}\vec{j}_{int}(\vec{r})=\Bigg{[}&\int\textrm{d}\xi\frac{\hbar}{2i\mu}\left(\sum_{i=1}^{N_{sup}}a_{i}\ket{i}\textrm{e}^{ikz}\right)^{*}\\\
&\vec{\nabla}\left(\sum_{j}f_{sup\rightarrow
j}(\theta,\phi)\frac{\textrm{e}^{ik_{j}r}}{r}\ket{j}\right)\\\
+&\frac{\hbar}{2i\mu}\left(\sum_{j}f_{sup\rightarrow
j}(\theta,\phi)\frac{\textrm{e}^{ik_{j}r}}{r}\ket{j}\right)^{*}\\\
&\vec{\nabla}\left(\sum_{i=1}^{N_{sup}}a_{i}\ket{i}\textrm{e}^{ikz}\right)\Bigg{]}+\textrm{c.c}.\end{split}$
(49)

After the applications of the gradient and the orthogonality relation, the
interference current density becomes:

$\begin{split}\vec{j}_{int}(\vec{r})=\Bigg{[}\frac{\hbar}{2\mu}\sum_{i}^{N_{sup}}a_{i}^{*}f_{sup\rightarrow
i}(\theta,\phi)\textrm{e}^{ikr(1-\cos\theta)}\\\
\left[\frac{k}{r}(1+\cos\theta)+\frac{i}{r^{2}}\right]\hat{e}_{r}\Bigg{]}+\textrm{c.c}.\end{split}$
(50)

As for the outgoing contribution, only the radial component of the gradient is
considered. The integral on the closed surface is equal to:

$\begin{split}I_{int}=\Bigg{[}\frac{\hbar}{2\mu}\sum_{i}^{N_{sup}}a_{i}^{*}\lim_{r\rightarrow\infty}\int_{0}^{2\pi}d\phi\int_{0}^{\pi}\textrm{d}\theta\,\sin\theta
f_{sup\rightarrow i}(\theta,\phi)\\\
\textrm{e}^{ikr(1-\cos\theta)}\left[rk(1+\cos\theta)+i\right]\Bigg{]}+\textrm{c.c}.\end{split}$
(51)

Due to the uniform convergence of the limit, the order of the integral on
$\phi$ and the limit on $r$ can be interchanged. We focus in the $\theta$
integral first. Then, two limits must be calculated:

$\lim_{r\rightarrow\infty}rk\int_{0}^{\pi}\textrm{d}\theta\,\sin\theta
f_{sup\rightarrow
i}(\theta,\phi)\textrm{e}^{ikr(1-\cos\theta)}(1+\cos\theta),$ (52)
$\lim_{r\rightarrow\infty}i\int_{0}^{\pi}\textrm{d}\theta\,\sin\theta
f_{sup\rightarrow i}(\theta,\phi)\textrm{e}^{ikr(1-\cos\theta)}.$ (53)

We consider the second limit(53) and we make the changes of variables
$x=1-\cos\theta$ and $\kappa=kr$. We also, for notational clarity, suppress
the $\phi$ dependence of $f_{sup\rightarrow i}(\theta,\phi)$ until Eq.(62).
Eq. (53) then becomes:

$\begin{split}\lim_{r\rightarrow\infty}\int_{0}^{\pi}\textrm{d}\theta\,\sin\theta
f(\theta)\textrm{e}^{ikr(1-\cos\theta)}=\\\
\lim_{\kappa\rightarrow\infty}\int_{0}^{2}\textrm{d}xf(x)\textrm{e}^{i\kappa
x}.\end{split}$ (54)

By the Riemann-Lebesgue lemma, this limit vanishes:

$\lim_{\kappa\rightarrow\infty}\int_{0}^{2}\textrm{d}xf(x)\textrm{e}^{i\kappa
x}=0.$ (55)

Now, we focus on the limit (52) and make the same change of variables:

$\begin{split}\lim_{r\rightarrow\infty}kr\int_{0}^{\pi}\textrm{d}\theta\,\sin\theta
f(\theta)\textrm{e}^{ikr(1-\cos\theta)}(1+\cos\theta)=\\\
\lim_{\kappa\rightarrow\infty}\kappa\int_{0}^{2}\textrm{d}xf(x)\textrm{e}^{i\kappa
x}(2-x).\end{split}$ (56)

After integration by parts, we obtain:

$\begin{split}\lim_{\kappa\rightarrow\infty}\kappa\int_{0}^{2}\textrm{d}xf(x)\textrm{e}^{i\kappa
x}(2-x)=\\\
\frac{1}{i}\Bigg{(}\lim_{\kappa\rightarrow\infty}[f(x)(2-x)\textrm{e}^{i\kappa
x}]_{0}^{2}\\\
-\lim_{\kappa\rightarrow\infty}\int_{0}^{2}(f(x)(2-x))^{\prime}\textrm{e}^{i\kappa
x}\textrm{d}x\Bigg{)}.\end{split}$ (57)

Using the Riemann-Lebesgue lemma, the second term evaluates to zero:

$\begin{split}\lim_{\kappa\rightarrow\infty}\kappa\int_{0}^{2}\textrm{d}xf(x)\textrm{e}^{i\kappa
x}(2-x)=\\\
\frac{1}{i}\left(\lim_{\kappa\rightarrow\infty}[f(x)(2-x)\textrm{e}^{i\kappa
x}]_{0}^{2}\right).\end{split}$ (58)

The first term gives:

$\lim_{\kappa\rightarrow\infty}\kappa\int_{0}^{2}\textrm{d}xf(x)\textrm{e}^{i\kappa
x}(2-x)=\frac{-2f(x=0)}{i},$ (59)
$\lim_{\kappa\rightarrow\infty}\kappa\int_{0}^{2}\textrm{d}xf(x)\textrm{e}^{i\kappa
x}(2-x)=2if(x=0).$ (60)

$x=0$ corresponds to $\theta=0$. Therefore, one obtains:

$\begin{split}&\lim_{r\rightarrow\infty}kr\int_{0}^{\pi}\textrm{d}\theta\,\sin\theta
f(\theta)\textrm{e}^{ikr(1-\cos\theta)}(1+\cos\theta)=\\\
&2if(\theta=0).\end{split}$ (61)

Then, $I_{int}$ becomes, where we restore the $\phi$ dependence in $f$ and
consider the $\phi$ integral:

$I_{int}=\frac{i\hbar}{\mu}\sum_{i}^{N_{sup}}a_{i}^{*}\int_{0}^{2\pi}\textrm{d}\phi
f_{sup\rightarrow i}(\theta=0,\phi)+\textrm{c.c}.$ (62)

The forward scattering amplitudes from the superposition $f_{sup\rightarrow
i}(\theta=0,\phi)$ is then expanded in state-to-state scattering amplitudes
(see Eq. (3)):

$I_{int}=\frac{i\hbar}{\mu}\sum_{i,i^{\prime}=1}^{N_{sup}}a_{i}^{*}a_{i^{\prime}}\int_{0}^{2\pi}\textrm{d}\phi
f_{i^{\prime}\rightarrow i}(\theta=0,\phi)+\textrm{c.c}.$ (63)

The forward scattering amplitude does not depend on $\phi$, so that the
integral on $\phi$ simply gives:

$I_{int}=\frac{2\pi
i\hbar}{\mu}\sum_{i,i^{\prime}=1}^{N_{sup}}a_{i}^{*}a_{i^{\prime}}f_{i^{\prime}\rightarrow
i}(\theta=0)+\textrm{c.c},$ (64)
$I_{int}=\frac{-4\pi\hbar}{\mu}\sum_{i,i^{\prime}=1}^{N_{sup}}\textrm{Im}[a_{i}^{*}a_{i^{\prime}}f_{i^{\prime}\rightarrow
i}(\theta=0)].$ (65)

## Appendix D: Details of the numerical calculations

### O2+O2 scattering

We perform quantum scattering calculations on O2 \+ O2 collisions using a
coupled-channel (CC) methodology [45] based on the expansion of the scattering
wavefunction in an uncoupled symmetrized space-fixed basis set composed of
direct products of molecular rotational and spin basis functions and the
orbital angular momentum eigenstates. Most of the computational details are
essentially the same as reported in the previous work of one of the authors
[45]. The CC basis set was composed of three rotational states ($N=0{-}4$) and
6 partial waves ($\ell=0{-}10$) at 10 $\mu$K. The hyperfine structure of 17O2
was neglected to make the calculations computationally feasible. The CC
equations were integrated on the radial grid from $R_{\text{min}}=4.0\,a_{0}$
to $R_{\text{max}}=150\,a_{0}$ with a grid step of $0.04\,a_{0}$. The T-matrix
elements are obtained from this CC results and are used to calculate the
forward scattering amplitudes:

$f_{i\rightarrow
j}(\theta=0)=\frac{1}{2k}\sum_{\ell}\sum_{\ell^{\prime}}i^{\ell-\ell^{\prime}+1}\sqrt{(2\ell+1)(2\ell^{\prime}+1)}T_{i\ell
0\rightarrow j\ell^{\prime}0}.$ (66)

Finally, the total cross section is calculated using the generalized optical
theorem (Eq. 12).

### Rb+Rb scattering

Quantum scattering calculations of ultracold 85Rb+ 87Rb collisions were
performed following the same methodology as in the previous works [41, 46,
47]. The CC equations were integrated on the radial grid from
$R_{\text{min}}=2.0\,a_{0}$ to $R_{\text{max}}=300\,a_{0}$ with a grid step of
$0.005\,a_{0}$. The calculation of the forward scattering amplitude and of the
total cross section is the same as that for O2-O2 scattering.

## References

  * Taylor [1972] J. R. Taylor,  _Scattering Theory: The Quantum Theory of Nonrelativistic Collisions_ (John Wiley & Sons, 1972).
  * Friedrich [2013] H. Friedrich,  _Scattering Theory_ (Springer, 2013).
  * Sellmeier [1871] E. Sellmeier, Zur erklärung der abnormen farbenfolge in spectrum einiger substanzen, Ann. Phys. Chem. 143, 272 (1871).
  * Rayleigh [1871] L. Rayleigh, On the light from the sky, its polarization and colour, Philos. Mag. 47, 375 (1871).
  * Feenberg [1932] E. Feenberg, The scattering of slow electrons by neutral atoms, Phys. Rev. 40, 40 (1932).
  * Bohr  _et al._ [1939] N. Bohr, R. Peierls, and G. Placzek, Nuclear reactions in the continuous energy region, Nature 144, 200 (1939).
  * Joshipura and Patel [1996] K. N. Joshipura and P. M. Patel, Total electron scattering cross sections for NO, CO, NO2, N2O, CO2 and NH3. (Ei > 50 eV), J. Phys. B: A. Mol. Opt. Phys. 29, 3925 (1996).
  * Jiang  _et al._ [1995] Y. Jiang, J. Sun, and L. Wan, Total cross sections for electron scattering by polyatomic molecules at 10-1000 eV: C2H2, C2H4, C2H6, C3H6, C3H8 and C4H8, Z. Phys. D 34, 29 (1995).
  * Scholz  _et al._ [1990] T. Scholz, H. R. J. Walters, and P. G. Burke, The total cross section for electron scattering by atomic hydrogen, J. Phys. B: A. Mol. Opt. Phys. 23, L467 (1990).
  * Kim  _et al._ [1997] Y. E. Kim, Y. J. Kim, and A. L. Zubarev, Optical theorem formulation of low-energy nuclear reactions, Phys. Rev. C 55, 801 (1997).
  * Lipperheide [1987] R. Lipperheide, Heavy-ion reaction cross sections from elastic scattering data, Nuclear Physics , 190 (1987).
  * Udagawa and Tamura [1980] T. Udagawa and T. Tamura, Derivation of breakup-fusion cross sections from the optical theorem, Phys. Rev. C 24, 1348 (1980).
  * Pancheri and Srivastava [2016] G. Pancheri and Y. N. Srivastava, Introduction to the physics of the total cross section at LHC, Eur. Phys. J. C. 77, 150 (2016).
  * R. V. Krems and Friedrich [2009] W. C. S. R. V. Krems and B. Friedrich,  _Cold Molecules_ (CRC Press, 2009).
  * Karra  _et al._ [2016] M. Karra, K. Sharma, B. Friedrich, S. Kais, and D. Herschbach, Prospects for quantum computing with an array of ultracold polar paramagnetic molecules, J. Chem. Phys. 144, 094301 (2016).
  * Park  _et al._ [2017] J. W. Park, Z. Z. Yan, H. Loh, S. A. Will, and M. W. Zwierlein, Second-scale nuclear spin coherence time of ultracold 23Na40K molecules, Science 357, 372 (2017).
  * Yu  _et al._ [2019] P. Yu, L. W. Cheuk, I. Kozyryev, and J. M. Doyle, A scalable quantum computing platform using symmetric-top molecules, New J. Phys. 21, 093049 (2019).
  * Albert  _et al._ [2020] V. V. Albert, J. P. Covey, and J. Preskill, Robust encoding of a qubit in a molecule, Phys. Rev. X 10, 031050 (2020).
  * Carr  _et al._ [2009] L. D. Carr, D. DeMille, R. V. Krems, and J. Ye, Cold and ultracold molecules: science, technology and applications, New J. Phys. 11, 055049 (2009).
  * Heazlewood and Softley [2021] B. Heazlewood and T. P. Softley, Towards chemistry at absolute zero, Nat. Rev. Chem. 5, 125–140 (2021).
  * Chin  _et al._ [2010] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Rev. Mod. Phys. 82, 1225 (2010).
  * Krems [2019] R. V. Krems,  _Molecules in Electromagnetic Fields_ (Wiley-VCH, 2019).
  * Shapiro and Brumer [2012] M. Shapiro and P. Brumer,  _Quantum Control of Molecular Processes_ (Wiley-VCH, 2012).
  * Zhou  _et al._ [2021] H. Zhou, W. E. Perreault, N. Mukherjee, and R. N. Zare, Quantum mechanical double slit for molecular scattering, Science 374, 960 (2021).
  * Devolder  _et al._ [2021] A. Devolder, P. Brumer, and T. V. Tscherbul, Complete quantum coherent control of ultracold molecular collisions, Phys. Rev. Lett. 126, 153403 (2021).
  * Zelevinski  _et al._ [2008] T. Zelevinski, S. Kotochigova, and J. Ye, Precision test of mass-ratio variations with lattice-confined ultracold molecules, Phys. Rev. Lett. 100, 043201 (2008).
  * Kotochigova  _et al._ [2009] S. Kotochigova, T. Zelevinski, and J. Ye, Prospects for application of ultracold Sr2 molecules in precision measurements, Phys. Rev. A 79, 012504 (2009).
  * Zhu  _et al._ [1995] L. Zhu, V. Kleiman, X. Li, S. P. Lu, K. Trenteiman, and R. J. Gordon, Coherent laser control of the product distribution obtained in the photoexcitation of HI, Science 270, 77 (1995).
  * Shapiro and Brumer [1996] M. Shapiro and P. Brumer, Coherent control of collisional events: Bimolecular reactive scattering, Phys. Rev. Lett. 77, 2574 (1996).
  * Abrashkevich  _et al._ [2001] A. Abrashkevich, M. Shapiro, and P. Brumer, Coherent control of atom-diatom reactive scattering: isotopic variants of H+H2 in three dimensions, Chem. Phys. 267, 81 (2001).
  * Gong  _et al._ [2003] J. Gong, M. Shapiro, and P. Brumer, Entanglement-assisted coherent control in nonreactive diatom–diatom scattering, J. Chem. Phys. 118, 2626 (2003).
  * Zeman  _et al._ [2004] V. Zeman, M. Shapiro, and P. Brumer, Coherent control of resonance-mediated reactions: F + HD, Phys. Rev. Lett. 92, 133204 (2004).
  * Brumer  _et al._ [1999] P. Brumer, A. Abrashkevich, and M. Shapiro, Laboratory conditions in the coherent control of reactive scattering, Faraday Discuss. 113, 291 (1999).
  * Perreault  _et al._ [2017] W. E. Perreault, N. Mukherjee, and R. N. Zare, Quantum control of molecular collisions at 1 Kelvin, Science 358, 356 (2017).
  * Perreault  _et al._ [2018] W. E. Perreault, N. Mukherjee, and R. N. Zare, Cold quantum-controlled rotationally inelastic scattering of HD with H2 and D2 reveals collisional partner reorientation, Nat. Chem. 10, 561 (2018).
  * Mukherjee  _et al._ [2014] N. Mukherjee, W. Dong, and R. Zare, Coherent superposition of m-states in a single rovibrational level of H2 by Stark-induced adiabatic raman passage, J. Chem. Phys. 140, 074201 (2014).
  * Liu  _et al._ [2021] Y. Liu, M.-G. Hu, M. A. Nichols, D. Yang, D. Xie, H. Guo, and K.-K. Ni, Precision test of statistical dynamics with state-to-state ultracold chemistry, Nature 593, 379 (2021).
  * Margulis  _et al._ [2020] B. Margulis, J. Narevicius, and E. Narevicius, Direct observation of a Feshbach resonance by coincidence detection of ions and electrons in Penning ionization collisions, Nature Communications 11, 3553 (2020).
  * Frishman  _et al._ [1999] E. Frishman, M. Shapiro, and P. Brumer, Coherent enhancement and suppression of reactive scattering and tunneling, J. Chem. Phys. 110, 9 (1999).
  * Segev  _et al._ [2019] Y. Segev, M. Pitzer, M. Karpov, N. Akerman, J. Narevicius, and E. Narevicius, Collisions between cold molecules in a superconducting magnetic trap, Nature 572, 189 (2019).
  * Hermsmeier  _et al._ [2021] R. Hermsmeier, A. Devolder, P. Brumer, and T. V. Tscherbul, Near-threshold scaling of resonant inelastic collisions at ultralow temperatures, arXiv: https://arxiv.org/abs/2105.11995 (2021).
  * Grimm  _et al._ [2000] R. Grimm, M. Weidemüller, and Y. B. Ovchinnikov, Optical dipole traps for neutral atoms, Adv. At. Mol. Opt. Phys. 42, 95 (2000).
  * Courteille  _et al._ [1998] P. Courteille, R. S. Freeland, D. J. Heinzen, F. A. van Abeelen, and B. J. Verhaar, Observation of a Feshbach resonance in cold atom scattering, Phys. Rev. Lett. 81, 69 (1998).
  * Sompet  _et al._ [2019] P. Sompet, S. S. Szigeti, E. Schwartz, A. S. Bradley, and M. F. Andersen, Thermally robust spin correlations between two 85Rb atoms in an optical microtrap, Nature Communications 10, 1889 (2019).
  * Tscherbul  _et al._ [2009] T. V. Tscherbul, Y. V. Suleimanov, V. Aquilanti, and R. V. Krems, Magnetic field modification of ultracold molecule–molecule collisions, New J. Phys. 11, 055021 (2009).
  * Li and Krems [2007] Z. Li and R. V. Krems, Electric-field-induced Feshbach resonances in ultracold alkali-metal mixtures, Phys. Rev. A 75, 032709 (2007).
  * Li  _et al._ [2008] Z. Li, S. Singh, T. V. Tscherbul, and K. W. Madison, Feshbach resonances in ultracold ${}^{85}\mathrm{Rb}\text{$-$}^{87}\mathrm{Rb}$ and ${}^{6}\mathrm{Li}\text{$-$}^{87}\mathrm{Rb}$ mixtures, Phys. Rev. A 78, 022710 (2008).