#  Ripeline and Rmanual speed up
biological research and reporting

Alexey Shipunov  Minot State University, Minot ND, 58707

###### Abstract

The emergence of R, a freely available data analysis environment, brought to
the researcher in any science field a set of well-concerted instruments of
immense power and low cost. In botany and zoology, these instruments could be
used, for example, to speed up work in two distant but related fields:
analysis of DNA markers and preparation of natural history manuals. Both of
these tasks require a significant amount of monotonous work, which could be
automated with software.

I developed “Ripeline and “Rmanual,” two highly customizable R-based
applications, designed with a goal of simplicity, reproducibility, and
effectiveness. Ripeline is a pipeline that allows for a continuously updated
analysis of multiple DNA markers. Rmanual is a “living book” which allows the
creation and continuous update of manuals and checklists.

Comparing with more traditional ways of DNA marker analysis and manual
preparation, Ripeline and Rmanual allow for a significant reduction of time,
which is usually spent doing repetitive tasks. They also provide tools which
can be used in a broad spectrum of further applications.

Key words: R, sequences, pipeline, manuals

##  1 Introduction

DNA extraction, PCR amplification, and sequencing are now widely available for
researchers, and many computer software exists to generate phylogenetic trees
from sequences. However, the amount of work required is often overwhelming.
Even more complicated is the situation when the project requires an update,
and all steps have to be repeated. This is where software might be able to
help. It must link sequence generation and phylogeny estimation in the form of
a custom pipeline that works with multiple fragment types. Also, it should be
flexible (new analyses are easy to add), contemporary, updatable (allows to
rebuild the trees every time databases change), and use both local and
globally available data. It is also hard to find an existing portable software
which allows for proper sequence concatenation and gap coding. Ideally, this
pipeline should also allow to check, for example, which species within a group
are not yet successfully sequenced, or whether species names are congruent
between taxonomic and sequence parts of data.

Checklists and natural history manuals are cornerstones of biodiversity
research. However, much of biodiversity-related literature has spent years in
preparation. When taxonomic databases contain numerous name entries,
descriptions, habitat notes, and image links, the task to produce a checklist,
illustrated checklist, or manual is also overwhelmingly complex, especially in
the case when the product needs an update. The preparation process includes a
large amount of technical work, for example, adding images and descriptions to
species. I believe that a semi-automatic approach that combines processing of
relational text tables and typographic production will allow for a significant
increase in speed, and therefore will improve the production process of
manuals, checklists, pamphlets, and other similar literature.

Applications presented below help to improve both sequence analysis and manual
preparation. While they are devoted to different tasks, they have much in
common. Both are (1) based on “flat” text tables, (2) use R as a backbone for
databasing and organizing the input and output from the external software, and
(3) designed with the idea of continuous updates based on positive feedback.
The essential part of both applications are R scripts. Started as a
statistical environment (R Core Team, 2019), R is now rapidly gaining
popularity as the general data processing language for science, especially in
biology. This popularity is mostly due to extensive documentation, free
programming style, many high-level routines, advanced data manipulation and
plotting abilities. R is easy to embed, integrate, and call (both from it and
out of it). R scripts are one of the simplest ways to deploy and reproduce the
data processing workflow. As Ripeline and Rmanual are based on
reproducibility, portability, and integration, their R backbone is
indispensable.

##  2 Methods and Results

###  2.1 Ripeline (R-based sequence analysis pipeline)

Ripeline is an R-based pipeline application that allows for the wide variety
of sequence analyses (Fig. 1). In the core, it is a set of R and UNIX shell
script templates (i.e., simple text files), which are extensively commented
and easy to adapt for the needs of a particular project. I made the Ripeline
when I realized that I must run four independent phylogenetic projects, each
with plenty of marker data, and with only the restricted help from
undergraduate students. It saved a significant amount of my time, which
otherwise would have been spent on repetitive tasks. Now, thanks to Ripeline,
the first results of these projects are already published (Shipunov et al.,
2019).

Figure 1:  Ripeline: the main pipeline.

Ripeline includes both R-based and standalone methods of analysis. While many
of these are available separately, the actual workflows are diverse, not
standardized, and therefore not fully reproducible. Ripeline allows for the
fully standardized, reproducible sequence analyses. Installation of Ripeline
is also completely portable, as it is equal to downloading the single
directory.

Each Ripeline script (Table 1) is numbered (like in the UNIX “init” startup
system) and works independently (so if it fails for any reason, others will
continue to work). It also outputs a full report of what was done, thanks to
the “Rresults” utility from “shipunov” package (Shipunov, 2019).

Ripeline starts by checking the local DNA database (Table 1, Fig 1).

03_checks.r |  checks input tables
---|---
04_duplicated_ids_check.r |  checks duplicated IDs
13_make_wanted.r |  outputs names of non-sequenced species
20_make_sets.r |  combines DNA data into “sets”, one for each marker
30_align.r |  run the external multiple alignment tool
31_trim.r |  trims flanks of alignments
32_gapcode.r |  gap coding
40_concatenate_and_stat.r |  concatenates sequences and outputs statistics
51_make_r_raw_kmer_trees.r |  builds “raw” (based on all data) k-mer trees
52_make_r_semistrict_kmer_tree.r |  builds k-mer trees on semistrict super-matrix
53_make_r_nj_single_marker_trees.r |  builds NJ (neigbor joining) single marker trees
61_make_r_mp_semistrict_tree.r |  builds MP (maximum parsimony) trees
71_make_r_ml_modeltest.r |  tests maximal likelihood models
72_make_r_ml_trees.r |  estimates ML (maximal likelihood) trees using internal R functions
73_make_raxml_trees.r |  estimates ML trees with RAxML
81_make_mrbayes_semistrict_tree.r |  estimates Bayesian trees with MrBayes
Table 1:  Ripeline: basic R scripts

This database is the tab-delimited file with the following fields:

SOURCE

(where from). By default, Ripeline trusts locally obtained sequences more.

SEQUENCE.ID

Typically, something like “K-216” where “K” is a shortcut for the project.

FRAGMENT

Fragment name, like “abcd” or “efgh”.

SPECIES.NEW

Species name.

COMMENT

SELECT

(0 or 1). Which sequences to use.

SEQUENCE

The actual DNA string; allowed letters are “ATGCN”.

DNA-related data combined into “sets” (one per marker), aligned, trimmed, gap
coded, and then concatenated in super-matrices. Two types of concatenation are
employed:

Strict concatenation

Based on sequence IDs of locally obtained sequences.

Semi-strict concatenation

The next step. It uses “strict” dataset and adds sequences of any origin in
order to fill all gaps.

Ripeline uses some external tools like MUSCLE (Edgar, 2004), RAxML
(Stamatakis, 2014), and MrBayes (Ronquist & Huelsenbeck, 2003), but can also
work without them. Some proficiency in R is required to adopt these scripts
for the needs of a particular project. However, nothing beyond small
modifications (e.g., changing the size of the output image file) is necessary.
Only rudimentary shell scripting knowledge is required, on the level of
commenting or uncommenting particular lines. All in all, I believe that
Ripeline is suitable for users without excessive programming skills.

As all its components are cross-platform, Ripeline works on all major
platforms. Ripeline is fast. Template scripts are built around an artificial
example which involves two DNA markers from 12 species, multiple alignment,
trimming flanks, gap coding, concatenation, creation of “technical” k-mer and
NJ trees, a maximum parsimony example, maximum likelihood (ML) hypotheses
testing, two examples of ML, and Bayesian tree estimation. With parameters set
to minimum (like 100 bootstrap replicates), the whole pipeline takes about 2
minutes on an average laptop.

Ripeline is the new software, but there is already a series of use cases.
Results of the first project are recently published (Shipunov et al., 2019),
and two other projects are accessible in the author’s open repository
(http://ashipunov.info/shipunov/open/).

###  2.2 Rmanual (R-based natural history manual)

Software that assists in producing the natural history manual should be able
to extract data from databases, use images, and output typographically
formatted text. To my knowledge, R (R Core Team, 2019) and TeX (Knuth, 1984)
are the best candidates for making this kind of software.

Rmanual requires working TeX and R installations, plus one additional R
package. All of these are available on all major platforms. Rmanual includes
only one R script and also one shell script whose job is to use the former,
and then run the TeX engine. R code is short, fully commented, and is easy to
modify. There are only a few TeX definitions, which are easy to understand and
modify.

Before starting the Ripeline, I recommend utilities like TNRS (Boyle et al.,
2013) for the data cleaning and normalization. The basic idea of the Rmanual
is that after R outputs a text table (made from inter-combined input tables),
TeX uses it as a “body” (main, taxonomic part) of the book. In the end, each
page inside the PDF book consists of multiple table pieces (Fig. 2).

Figure 2:  Rmanual: the structure of the output manual page.

One of these pieces might look like:

    \FF{Subgenus \KK{Minores}}
     \SP{\KK{Kubrickus heus}}
     \DD{Photosynthetic, non-carnivorous, motile}
     \II{\I{kubrick_h.pdf}}

In the piece above, each field of the output table is formatted as an argument
of some TeX command: `\FF{}` for higher categories, `\SP{}` for species names,
`\DD{}` for species descriptions, and `\II{}` for figures.

Inter-combining of input tables uses the initial species list, “filters” and
“features”. Filters (like a table connecting synonyms and accepted names)
convert data, whereas features (like a table connecting genera and families)
add new information (Fig. 3).

Figure 3:  Rmanual: the main pipeline.

The header and footer of the book (the title page, first and last pages)
should be prepared manually using the supplied template. Apart from a PDF
book, Rmanual also outputs diagnostic data (e.g., which species do not
correspond with images).

The main strength of Rmanual is, therefore, the production of semi-automatic,
typographically ready output from the set of “flat” text tables and images.
Besides, Rmanual allows for the constant update of the output and therefore
produces books that are not only semi-automatic but also “living.”

Rmanual, similarly to Ripeline, requires some knowledge in R and TeX but does
not require extensive skills. If the researcher can modify R and TeX files
(note again that they are simple text files), it should be enough to run
Rmanual with their data and make simple PDF books. I tried the Rmanual
approach several times (Shipunov, 2013; Shipunov et al., 2015; Shipunov,
2018), and each time the process of adaptation did not take more than a few
hours.

Rmanual is very fast. When preparing this paper, I made a new illustrated
checklist, which included 122 plant species and their images within just two
hours. The actual book production (creation of a typographically ready PDF
file) takes only a few seconds on an average laptop. Even if an inexperienced
person will be ten times slower, 20 hours for a small book is a good result.
Therefore, I believe that when all data is ready, with a help from Rmanual or
with a Rmanual-like approach, the production of typographic results might span
only a few days.

The illustrated checklist of North Dakota plants (Butler, 2013; Shipunov,
2018) is one of the most advanced use cases of Rmanual. Two other working
examples of Rmanual, which use real data, are presented on Github:
https://github.com/ashipunov.

##  3 Conclusions

The biggest drawback of Ripeline and Rmanual is that the user must know some
basics of R and (in case of Rmanual) TeX. However, both of these systems are
free, and more importantly, extensively documented and community-supported.
Free manuals can be accessed both on CTAN (e.g., https://ctan.org/pkg/lshort-
english and https://ctan.org/pkg/beginlatex) and CRAN
(https://cran.r-project.org/other-docs.html.)

There are only a few tools comparable with Ripeline. PhyScraper (McTavish,
2019), for example, is the Python-based pipeline intended to do a similar job.
However, it is still in development (some parts are therefore not available),
and also uses a virtual environment which brings extra abstraction layers.
Phylogeny.fr (Dereeper et al., 2008) implements a similar idea but is Web-
oriented (i.e., depends on network services which can slow down the whole
process) and restricts the amount of allowable data. Other similar tools like
phyloGenerator (Pearse & Purvis, 2013) or ReproPhylo (Szitenberg et al., 2015)
are intended mostly for publicly available (GenBank) sequences.

The R package monographaR (Reginato, 2016) works on principles similar to
Rmanual, but its goal is taxonomic monographs and not checklists or manuals
(it is, therefore, more complicated than Rmanual), it also does not involve
TeX which consequently lowers the typographic quality of the result.

Ripeline, Rmanual, and two additional example manuals are freely available on
my GitHub: https://github.com/ashipunov. Ripeline and Rmanual are also
available as appendices to my “Visual Statistics. Use R!” book:
http://ashipunov.info/shipunov/software/r/r-en.htm.

##  4 Acknowledgements

The author is grateful to the Vitalij Arnold, Sergej Sukhov, Vera Shipunova,
and authors of R and TeX software.

##  5 Literature cited

Boyle, B., N. Hopkins, Z. Lu, J. A. R. Garay, D. Mozzherin, T. Rees, N.
Matasci, et al. 2013. The taxonomic name resolution service: an online tool
for automated standardization of plant names. BMC Bioinformatics. 14: 16.

Butler, J.L., 2013. Flora of North Dakota: Checklist by Alexey Shipunov.
Native Plants Journal, 14(3), pp.267-268.

Dereeper, A., V. Guignon, G. Blanc, S. Audic, S. Buffet, F. Chevenet, J. F.
Dufayard, et al. 2008. Phylogeny.fr: robust phylogenetic analysis for the non-
specialist. Nucleic Acids Research. 36: 465–469.

Edgar, R.C. 2004. MUSCLE: multiple sequence alignment with high accuracy and
high throughput. Nucleic Acids Research 32: 1792–1797.

Huelsenbeck, J.P., & Ronquist, R. 2001. MrBayes: Bayesian inference of
phylogenetic trees. Bioinformatics 17: 754–755.

Knuth, D.E. 1984. The TeXbook. Reading: Addison-Wesley.

McTavish, E.J. physcraper. URL:
https://github.com/McTavishLab/physcraper [last accessed 28 Nov. 2019]

Pearse, W.D. and Purvis, A. 2013. phyloGenerator: an automated phylogeny
generation tool for ecologists. Methods in Ecology and Evolution. 4: 692–698.

R Core Team. 2019. R: A language and environment for statistical computing. R
Foundation for Statistical Computing, Vienna, Austria. URL:
http://www.R-project.org/ [last accessed 19 Nov. 2019]

Reginato, M. 2016. monographaR: an R package to facilitate the production of
plant taxonomic monographs. Brittonia. 68: 212–216.

Shipunov, A. 2013. Atlas of plants of Tver region. 2013. URL:
http://ashipunov.info/shipunov/moldino/nauka/atlas.pdf [last accessed 19 Nov.
2019]

Shipunov, A. 2018. Flora of North Dakota: Illustrated checklist. Version 2.
Ed.: Kartesz, J., and Nishino, M. ISBN 978–1-4675–6379–6. URL:
http://ashipunov.info/shipunov/fnddb/index.htm [last accessed 19 Nov. 2019]

Shipunov, A. 2019. shipunov: Miscellaneous Functions from Alexey Shipunov. R
package version 1.2. URL: https://CRAN.R-project.org/package=shipunov [last
accessed 19 Nov. 2019]

Shipunov, A., Gladkova S., Timoshina P., Lee H. J., Choi J., Despiegelaere S.,
and Connolly B. 2019. Mysterious chokeberries: new data on the diversity and
phylogeny of Aronia Medik. (Rosaceae). European Journal of Taxonomy 570: 1–14.

Shipunov, A., L. Abramova, J. Beaudoin, J. H. Choi, D. Fry, R. Perry, V.
Shipunova, J. Schumaier, and J. Theodore. 2015. Flora of North Dakota project.
Phytoneuron 16: 1–10.

Stamatakis, A. 2014. RAxML version 8: a tool for phylogenetic analysis and
post-analysis of large phylogenies. Bioinformatics. 30:1312–1313.

Szitenberg, A., M. John, M. L. Blaxter, and D. H. Lunt. 2015. ReproPhylo: an
environment for reproducible phylogenomics. PLoS Computational Biology.
11:1004447.


Currently with ]ICFO - Instituto de Ciencias Fotonicas, The Barcelona
Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain

# Spectrally multimode squeezed states generation at telecom wavelengths

Victor Roman-Rodriguez1,2<EMAIL_ADDRESS>[  David Fainsin1 Guilherme L.
Zanin1, Nicolas Treps1, Eleni Diamanti2, and Valentina Parigi1 1Laboratoire
Kastler Brossel, Sorbonne Université, ENS-Université PSL, CNRS, Collège de
France, 4 place Jussieu, Paris F-75252, France  2Sorbonne Université, LIP6,
CNRS, 4 place Jussieu, 75005 Paris, France

###### Abstract

We report on the experimental demonstration of a source that generates
spectrally multimode squeezed states of light over the infrared C-Band. This
is achieved using a single-pass Spontaneous Parametric Down Conversion (SPDC)
process in a periodically-poled KTP waveguide that is pumped with the second
harmonic of a femtosecond laser. Our measurements show significant squeezing
in more than 21 frequency modes, with a maximum squeezing value over 2.5 dB.
Moreover, we demonstrate multiparty entanglement across 8 individual frequency
bands by measuring the covariance matrix of their quadratures. Finally, we use
reconfigurable mode-selective homodyne detection to mold the output into
cluster states of various shapes. This result paves the way for the
implementation of continuous variable quantum information protocols at
telecommunication wavelengths, with applications in multiparty, entanglement-
based quantum communication and computation.

††preprint: APS/123-QED

##  I Introduction

Continuous variable (CV) encoding of quantum information requires the
generation of multimode quantum states of light with tailored spectral,
spatial and temporal mode properties [1, 2]. In particular, measurement-based
protocols are based on the possibility of deterministically generating large
multimode entangled states, where entanglement is established between
amplitude and phase quadratures of different light modes [3, 4, 5, 6, 7]. This
in turn requires the generation of a large number of squeezed modes.

Second-order nonlinear waveguides are currently largely explored to generate
tailored squeezed modes: single-mode over a large bandwidth [8, 9, 10],
temporally multiplexed [11], and both spectrally and temporally multiplexed
[12].

Particular effort has been devoted to generate squeezed sources at
telecommunication wavelengths. Single-mode quantum states with significant
squeezing [13, 14], on-chip few-mode squeezed states [15, 16] and micro-comb
structures have been shown [17]. Recent results on single-mode squeezing
generation in nonlinear waveguides have been reported in a regime where
optical communication technologies can be exploited [8], where it is possible
to have quantum state transmission through fibers over long distances [18] and
sensing [19].

Here we demonstrate the generation of a vacuum squeezed field in the near
infrared C-Band that is intrinsically spectrally multimode, i.e., a system
that cannot be reduced to a single squeezer acting on a specific spectral
mode, but that involves many squeezers acting on different (orthogonal)
spectral modes. This implies that the many spectral modes can be directly
shaped, via linear optics transformations, into entangled networks of the same
number of nodes without mixing them with extra vacuum field states, thus not
degrading the squeezing and/or entanglement correlations. The networks built
in this way can be easily tailored in our setup via mode-selective homodyne
detection [4], and their quality can be characterized via the squeezing level
of their so-called nullifier operators. We can also independently check
entanglement correlations between different individual frequency bands, by
measuring the full quadrature covariance matrix for 8 different bands. A
Positive-Partial-Transpose (PPT) criterion can be used as an indication of
entanglement over all the possible band bipartitions.

Both intra-band entanglement and cluster structures can be used for frequency-
multiplexed QKD [20] and entangled-based multiparty quantum communication
protocols [21].

Moreover, the demonstrated source, being pumped with a pulsed laser in a
single-pass configuration, is compatible with simultaneous spectral and
temporal (pulse-based) multiplexing [22] as well as mode-selective non-
Gaussian operations [23]. This fact makes the source appealing for the
generation of scalable 3-dimensional entangled structures to be used in
measurement-based quantum computing [24].

The paper is structured as follows: in Section II we briefly summarize the
theoretical description of the generation of the multimode quantum states via
SPDC in type 0 nonlinear waveguides [25]. We also describe our scheme for the
experimental setup used in this work. In Section III, we show the experimental
results, covering multimode squeezing, covariance matrix in the basis of
equidistant frequency bands (that we call frexel basis, [26]), and squeezing
in the nullifiers of certain cluster states. We conclude this work with a
summary in Section IV.

##  II Experiment Description

###  II.1 Multimode SPDC

We proposed the generation and engineering of CV multipartite entangled states
of light in the frequency domain via SPDC, using nonlinear waveguides, in
[25]. Details about the theoretical considerations can be found there and in
the references within. In summary, we investigated the properties of the joint
spectral amplitude (JSA):

$J(\omega_{s},\omega_{i})=\sum_{k}\lambda_{k}h_{k}(\omega_{s})g_{k}(\omega_{i}),$
(1)

where $\omega_{s/i}$ is the signal/idler frequency, $\lambda_{k}$ are the
Schmidt coefficients and $h_{k}(\omega_{s})$ ($g_{k}(\omega_{i})$) are the
signal (idler) frequency modes composing the signal (idler) field after the
interaction. Given that the JSA contains all the frequency information of our
states, the output signal and idler fields can be described with the modes:

$\displaystyle\hat{A}_{k}^{\dagger}$
$\displaystyle=\int{\mathrm{d\omega_{s}}h_{k}(\omega_{s})\hat{a}^{\dagger}(\omega_{s})}$
(2) $\displaystyle\hat{B}_{k}^{\dagger}$
$\displaystyle=\int{\mathrm{d\omega_{i}}g_{k}(\omega_{i})\hat{b}^{\dagger}(\omega_{i})},$

which define the sometimes called supermode basis [27]. The Hamiltonian
describing the interaction in this basis takes the form:

$\hat{H}=\sum_{k}^{N}\lambda_{k}\hat{A}_{k}^{\dagger}\hat{B}_{k}^{\dagger}+\mathrm{h.c.}$
(3)

Evolution under this Hamiltonian is known to produce $N$ independent pairs of
completely entangled modes (EPR pairs). In [25], we focused on the EPR pair
multimode field as a starting point for the generation of the more general
cluster states in the context of type II SPDC. In the experiment described
here, for practical reasons, we work instead with the type 0 process in the
degenerate case, i.e., when the signal and idler modes are identical,
$\hat{A}_{k}=\hat{B}_{k}$, and hence there is a unique output field. In this
case the Hamiltonian of Eq. (3) reduces to:

$\hat{H}=\sum_{k}^{N}\lambda_{k}\left(\hat{A}_{k}^{\dagger}\right)^{2}+\mathrm{h.c.}$
(4)

Evolution under this Hamiltonian produces a multimode field with $N$
independent squeezed states. Therefore, we can take as the experimental
signature of the generation of these states the squeezing levels of our
multimode output field. Each individual squeezed mode can be addressed
experimentally via coherent (homodyne) detection. Indeed, since homodyne
detection is a projective measurement, interference with a properly spectrally
shaped Local Oscilator (LO) gives access to the noise properties of the
individual modes under study [26].

Once the squeezing levels are characterized, the generation of a particular
cluster state with up to $N$ nodes can be obtained by performing an adequate
unitary transformation [28, 29]. This can be done by adding to the
experimental setup a passive optical circuit performing the unitary operation,
or equivalently, changing appropriately the LO shape to access directly the
modes composing the cluster state with homodyne detection [1, 4].

Figure 1: Experimental scheme for the generation of the multimode states. A
telecom wideband (bandwidth around 55 nm) pulsed laser is up converted to its
second harmonic with a ppLN crystal and then coupled to a nonlinear ppKTP
waveguide. Type 0 SPDC interaction generates independent squeezing in a set of
frequency modes. Each mode is addressed individually with homodyne detection,
where the LO is shaped spectrally with the use of a pulse shaper. For more
details see text.

###  II.2 Experimental Setup

The experimental setup for generating the multimode squeezed state is depicted
schematically in Fig. 1. A broadband fiber femtosecond laser (characteristics:
bandwidth $\sim$ 55 nm, pulse width $\sim$ 57 fs, repetition rate 100 MHz,
power $\sim$ 500 mW, centered at 1560 nm), is partly directed to a
periodically poled Lithium Niobate (ppLN) crystal, engineered to produce the
second harmonic frequency. After the ppLN we obtain light with a bandwidth of
$\sim$ 2 nm, centered at 780 nm. This field is then coupled to a
single(spatial)-mode, rectangular, nonlinear, periodically poled Potassium
Titanyl Phosphate (ppKTP) waveguide111The waveguides were purchased from the
company AdvR. that down converts the second harmonic field to the C-band
telecom wavelengths and generates the multimode squeezed states.

On the other side, a big fraction of the original power from the laser is sent
to a pulse shaper in order to generate a spectrally configurable LO. The
signal field from the waveguide and the LO are mixed and directed to separate
photodiodes, whose electrical outputs are subtracted to obtain the homodyne
signal.

We constructed the pulse shaper by diffracting the wavelength components of
our input field using a grating, and directing them to a Spatial Light
Modulator (SLM), where the light is reflected back and recombined in a similar
grating (so called 4f configuration)[31]. In the SLM screen, each pixel, and
hence, each frequency component of the field, can be addressed individually,
resulting in a spectrally shaped pulse at the output. We used a home-made
interface in Python to control the different masks applied to the pulse
shaper.

The LO is then passed by another ppKTP waveguide (identical in dimensions)
before mixing it with the quantum signal. This is done to spatially match the
LO and the signal, effectively decreasing losses in the detection.

We perfomed numerical simulations in advance in order to predict the
properties of the independent squeezed modes following [25]. Details about
these numerical simulations and about the characterization of our waveguides
can be found in Appendix A.

##  III Experimental Results

###  III.1 Multimode Squeezing

Fig. 2 summarizes the squeezing values obtained in each of the measured modes,
where the shot noise stands as reference (0 dB). The quadrature noise was
measured as a function of the relative phase between the signal and the LO,
for different LO spectral shapes, via a spectrum analyzer. The relative phase
is modulated in time thanks to a piezoelectric mirror in the LO optical path.
We first performed the experiment by projecting the states into the family of
Hermite-Gauss (HG) modes. The particular family we used, derived from our
numerical simulation, is defined by a fundamental HG0 mode with 45 nm of FWHM
(in amplitude).

Figure 2: Multimode Squeezing Curves. Left : We summarize, for each Hermite-
Gauss mode (supermode) the quantity of squeezing and anti-squeezing measured
experimentally. The shot noise stands as reference (0 dB). For each mode, the
red area represents the part of the spectrum not accessible to our Local
Oscillator. Top Right : Measurement in the ‘flat’ mode basis (for details see
text). Down right: A zoom on the necessary spectral range for mode HG20.

The Sqz (squeezing) and ASqz (antisqueezing) values of Fig. 2 are the minimal
and maximal values of the quadrature noise measured via averaging on scans of
several phase-periods from the spectrum analyzer.

The asymmetry between squeezing and antisqueezing levels (typical values are
-1.0 dB for squeezing and 1.4 dB for antisqueezing), is attributed to
experimental optical losses.

The main loss source is the non-optimal spatial mode-matching between the
signal and LO optical modes. We characterize it by measuring the visibility of
fringes between the LO and a small fraction of telecom light coupled into the
waveguides. The associated quantum efficiency scales quadratically with the
measured visibility (other contributions to the global quantum efficiency in
the homodyne setup are defined in Appendix B).

In order to match the spatial modes of LO and signal, we inserted, in the LO
path, a waveguide identical to the one used for the generation of the signal.
The overall visibility in such a configuration, which is the one of the
measurements in Fig. 2, reaches the value of 77%.

We explain the non-ideal visibility by residual differences between the
spatial modes of the signal and the LO. This can be due to inhomogeneities in
the two waveguide structures that are known to appear when the waveguide is
long ($\sim$ cm scale).

Fig. 2 also shows the spectral shapes set in the LO pulse shaper to measure
the corresponding squeezing values. A detrimental effect in our setup is what
we call optical clipping, which arises due to the limited size of the
cylindrical mirror responsible for focusing the light beam onto the SLM for
pulse shaping. Since the beam comes from a refractive element (a grating),
some wavelengths at the extremes of the spectrum are therefore cut off. Due to
this effect, the LO spectral extremes cannot be used are not in homodyne
detection (the corresponding clipped regions fro the different modes are shown
in red in Fig. 2). We would then expect larger values for the measured
squeezing levels if the full spectrum was experimentally available.

It is also worth mentioning that due to the optical clipping, the cut HG modes
in which we project our state are technically not orthogonal anymore. The
dimension of the subspace spanned by the modes shown in Fig. 2 is nevertheless
close to 21 HG modes (details can be found in Appendix D, where we calculate
it to be around 18).

Furthermore, we projected the states into a basis of a form of orthogonal and
flattened HG modes that we call flat modes. For such modes we observed larger
squeezing values - with more than 2 dB of squeezing up to the 4th mode - than
for the HG modes. The flat modes and their squeezing level are shown in the
right side of Fig. 2, as well as in Appendix E. This result implies that the
HG basis shown on the left side of Fig. 2 is not the family of supermodes,
since those should be the most squeezed modes in the system.

Further evidence in this direction is given by the covariance matrix
measurements in the next section, showing that the spectral widths of the
theoretically predicted HG modes are probably underestimated.

We therefore expect that the measured squeezing values in Fig. 2 constitute a
lower bound for the potential squeezing that can be achieved with larger mode-
matching visibilities and without optical clipping. This is also witnessed by
the significant asymmetry in the squeezing and antisqueezing values measured
in the flat mode basis.

To summarize the above discussion, Fig. 2 demonstrates the experimental
realization of an optical multimode squeezed state composed of at least 21
frequency modes, with reasonable margin for improvement, and that can be used
for the quantum information protocols.

###  III.2 Covariance Matrix measurement

In order to check the presence of entanglement, i.e., quantum correlations, in
our multimode state, we measured the state in different basis from the one of
the supermodes [32]. In particular, we used the so-called frexel basis, which
is composed of a number of equally spaced frequency bands covering the total
spectrum of the LO. This is a suitable basis, not only because it is easily
accessible via the shaping capability of the LO, but also because the frexel
modes can be easily spatially separated via dispersive elements and sent to
different locations, which is necessary for multiparty quantum protocols. We
therefore measured the covariance matrix that characterizes our gaussian
state, in the frexel basis, using 8 equally spaced frequency bands.

The covariance matrix is shown in Fig. 3. Under the reasonable assumption of a
non-chirped pump, we expect no correlations between the different position,
$q_{i}$, and momentum, $p_{j}$, quadrature components and hence zero value for
the symmetrized expectation values of the form $q_{i}p_{j}$ [27, 33]. We then
measure only the subgroups of $q_{i}q_{j}$ or $p_{i}p_{j}$ quadratures. The
results are shown in the left of Fig. 3. The non-zero off-diagonal elements in
the data show the presence of correlations between the frequency bands
defining the 8 frexels in our state. This can be associated to the
entanglement between the frequency bands that can be tested via the positive
partial transpose (PPT) criterion [34] (see Appendix G for more details). We
obtained violation of the PPT criterion for all of the possible bipartitions
of our system.

Figure 3: Reconstructed covariance matrix. The covariance matrix on the left
is obtained by measuring the multimode state in the frexel basis, using 8
frequency bands. On the right, we show a numerical diagonalization of the
measured covariance matrix for recovering the squeezed supermodes. Figure 4:
Cluster state generation. The boxplots sum up the maximum squeezing value for
the first HG modes and the nullifiers of different cluster state size and
topologies. The values indicate the mean nullifiers’s squeezing value.

On the right of Fig. 3, we perform a numerical diagonalization of the measured
covariance matrix to recover the eigenmode basis, where no entanglement is
present. Thus we expect these eigenvectors to resemble discretized versions of
the supermodes, with eigenvalues related to a squeezing level value over the
frequency band composing the frexel. Overall, the numerical eigenvectors
obtained in the diagonalization are in good agreement with the theoretical
prediction of approximate Hermite-Gauss modes, except that their spectral
widths are larger than the theoretically predicted modes. This is consistent
with the measurement of larger squeezing values in the flat modes basis rather
than in the theoretically derived HG mode basis, as shown in the previous
section.

The eigenmodes and eigenvalues from the diagonalization of the covariance
matrix can be found in Appendix F.

###  III.3 Cluster State Generation

Finally, we used the experimental setup for the deterministic generation of
some few-node cluster states, as a proof of principle on the versatility of
the source.

We probe the generation of the cluster states by measuring squeezing in the
nullifiers that characterize a specific adjacency matrix, i.e., a particular
topology defining the graph. Changing from one topology to another can be
achieved by appropriately changing the mask on the pulse shaper [35]. The
nullifiers, $\\{\hat{\delta}_{i}\\}$, of a particular graph with quadratures
operators for the nodes denoted $\hat{q}_{i}$ and $\hat{p}_{i}$ are written
as:

$\hat{\delta}_{i}=\hat{p}_{i}-\sum_{j}V_{ij}\hat{q}_{j},$ (5)

where $V_{ij}$ are the elements of the adjacency matrix defining the cluster
state, which is known in advance[28]. Furthermore, the quality of the probed
cluster state can be qualitatively obtained by the amount of squeezing
measured in the nullifiers.

We measured 4-node cluster states with different topologies and whose
nullifier’s level of squeezing is summarized in Fig. 4 via statistical box
plots for each cluster topology. Additionally, we show that all nullifiers are
squeezed below the shot noise up to an 8-mode linear cluster. This proves the
generation of the cluster states. We leave further source optimization for
future studies and experiments.

##  IV Conclusion

In this work we report on a deterministic source of multimode quantum states
of light generated via type 0 SPDC in a nonlinear ppKTP rectangular waveguide.
We measured up to 21 squeezed optical spectral modes in the HG basis and over
2 dB of squeezing for four spectral flat modes. We measured and diagonalized
the covariance matrix in the frexel basis and demonstrated the realization of
different cluster states topologies, characterizing them with the squeezing in
their nullifier operators. The quality of the resource can be enhanced by
improving the signal-LO mode-matching and the optical configuration of the
pulse shaper, both within reach of current technology. The demonstrated source
can be used to implement entanglement based quantum communication protocols
and to build scalable resources for quantum computing at telecom wavelenghts.

###### Acknowledgements.

This work was supported by the European Research Council under the
Consolidator Grant COQCOoN (Grant No. 820079).

##  Appendix A Numerical simulation of the multimode state in our experimental
configuration

Numerical simulations were performed in order to predict the properties of the
independent squeezed modes at the output of the waveguide. Given the
experimental values measured from the second harmonic field (hence the pump to
the ppKTP waveguide), and the chosen waveguide dimensions (3 by 3 $\mu$m and
15 mm in length), the results are shown in Fig. 5. A number of about 34 modes
is expected with the Schmidt distribution, $\\{\lambda_{k}\\}$, shown in the
figure. The first three frequency modes, similar to Hermite-Gauss modes, are
also shown. For more details about the numerical simulation of the nonlinear
waveguides and how the experiment was designed, see [25].

Figure 5: Numerical simulation of the independent squeezed states. Left: JSA
function, Right: First three frequency modes, resembling Hermite-Gauss and the
distribution of the Schmidt coefficients for every supermode.

##  Appendix B Waveguide Characterization

The characterization of our waveguides was performed by coupling them to the
C-band wavelength (1560 nm). A measurement of the spectrum of the second
harmonic field produced by the C-band input gives an estimate on the
homogeneity of the waveguide along the propagation direction (a sinc-like
function is to be measured, which is the response to a square nonlinear
profile by the Fourier transform, expected from a homogeneuous structure). The
amount of second harmonic produced from the telecom input gives an estimate on
the nonlinear coefficient of the waveguide, that is fitted from data in the
next section. Finally, a measurement of the spatial profile of the output
telecom light can be compared with a numerical simulation (in our case with a
finite element method, [36]) to check the single-spatial-mode feature of the
waveguides, and the deviation from the expected fundamental mode propagating
in the actual structure.

The homodyne measurement was performed with a home-made detector, including
the two photodetectors and the transimpedance circuit outputting the homodyne
electrical signal. The total homodyne efficiency, $\eta_{\mathrm{h}}$, can be
decomposed in the following terms:

$\eta_{\mathrm{h}}=\eta_{\mathrm{PD}}\cdot\eta_{\mathrm{el}}\cdot\eta_{\mathrm{opt}}\cdot\eta_{\mathrm{mod}},$
(6)

where $\eta_{\mathrm{PD}}$ is the photodetector’s efficiency (around 85% in
our case), $\eta_{\mathrm{opt}}$ is related to the optical losses in the
homodyne circuit (near unity in our case), $\eta_{\mathrm{mod}}$ is the mode
matching efficiency, i.e., how close are the LO and the signal in terms of
polarization, spatial and temporal profile when interfering, and
$\eta_{\mathrm{el}}$ is the electronic efficiency.

The electronic efficiency can be written as
$\eta_{\mathrm{el}}=1-1/\mathrm{SNR}$ [37], with $\mathrm{SNR}$ the signal to
noise ratio, i.e., the ratio between the shot noise at a certain input
intensity and the value in the absence of any input signal (electronic noise).
We measured the best signal-to-noise ratio (also called clearance, if one
measures it in dB) at a demodulation frequency of 2 MHz, where the clearance
was about 20 dB at 2 mW of input power.

The mode-matching efficiency was limiting the amount of available squeezing in
our experiment and it is discussed in the main text.

##  Appendix C Phase Sensitive Amplification Results

Before measuring the multimode squeezing curves shown in the main text, we
built a degenerate Optical Parametric Amplifier, or OPA, by pumping the
waveguide with a relatively intense seed field at telecom wavelengths (taken
directly from the ultrafast laser) and a pump field (from our second harmonic
ppLN crystal at 780 nm).

Figure 6: Parametric gain vs pump power. Extrema in the modulation of the
amplitude for the degenerate OPA, fit to Eq. (7) gives the parametric
efficiency, related to the total squeezing level.

The phenomenon of parametric amplification can be observed as a modulation of
the seed amplitude at the output of the waveguide, depending on the relative
phase between the seed and the pump fields. This measurement allows us to show
the existence of parametric gain in our waveguides, which is a precondition of
squeezing generation, even though the levels of multimode squeezing cannot be
predicted in this way.

Figure 7: First 4 Hermite-Gauss modes measured (blue) and the corresponding
spectral flat modes obtained from them (orange). Figure 8: Eigenmodes of the
covariance matrix The amplitudes of the eight eigenmodes were plotted and
fitted with Hermite Gauss functions. The wavelength outside of the Local
Oscillator bandwidth was emphasized by marking it with a red hatched area.

The extrema of the parametric gain, $G_{\pm}$, can be approximated, for a
single-mode OPA, as [38, 39]:

$G_{\pm}\sim\exp(\pm 2\sqrt{\eta_{\mathrm{PSA}}P}),$ (7)

where $P$ is the pump power and $\eta_{\mathrm{PSA}}$ is the parametric
efficiency. Ideally, the minimum deamplification, $G_{-}$, should be symmetric
with respect to $G_{+}$, although a disparity between the two has been
reported when using pulsed lasers, and attributed to a distortion of the
spatial or temporal profile inside the nonlinear material. The disparity
appears at sufficiently high power density in the material.

Fig. 6 shows the parametric gain measured as a function of the pump power. The
data fits well Eq. (7) and gives two values for the parametric efficiency due
to their asymmetry. However, a measurement of the second harmonic efficiency
in the same nonlinear waveguide gives an efficiency of
$\eta_{\mathrm{SHG}}=0.33$ W-1, which is in good agreement with the extracted
parametric efficiency for the deamplification.

The conclusion of the phase sensitive experiment is that, at pump powers of
few mW, we can expect at least some dB of total squeezing in our multimode
states, which are the measured values showed in the main text.

##  Appendix D Independent Hermite-Gauss modes

As discussed in the main text, due to optical clipping, the 21 clipped
Hermite-Gauss modes implemented in the experiment were not exactly orthogonal,
and therefore the actual number of measured supermodes is expected to be lower
than 21. This number depends on the dimensionality of the vector space spanned
by the non-orthogonal modes. In order to account for that, we evaluate the
rank of the matrix composed by these modes performing a singular value
decomposition on the set of 21 vectors. The non-zero singular values count the
number of linearly independent modes and hence the dimension of the space we
are looking for. The singular values resulting from the decomposition is
summarized in Table 1.

Table 1: Singular values obtained from clipped Hermite-Gauss modes used in the experiment. Sing. Val. | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|---
| 1.42 | 1.40 | 1.31 | 1.28 | 1.21 | 1.19 | 1.18 | 1.12 | 1.09 | 1.08 |
Sing. Val. | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| 0.99 | 0.97 | 0.94 | 0.90 | 0.84 | 0.60 | 0.30 | 0.08 | 0.02 | 0.005 |

Since this is a numerical computation, we need a somewhat arbitrary criterion
to give a whole number indicating the dimension of our vector space from this
result. In our case, we obtain the dimension of the vector space by accounting
for singular values that are at least 10% of the highest one. This criterion
gives us 18 linearly independent modes. The value being close to the 21
Hermite-Gauss modes indicates that the optical clipping did not have a large
impact in reducing the dimensionality of the modes measured in the experiment.

##  Appendix E Flat mode basis

The spectral flat mode basis discussed in the main text is constructed to be
orthogonal, taking advantage of even and odd symmetries for the subsequent
mode functions. It was specifically designed to roughly resemble the first 4
Hermite-Gauss modes, so that the spectral width of each flat mode was obtained
by minimizing the $l^{2}$ norm distance to the corresponding Hermite-Gauss
mode counterpart. Fig. 7 shows the 4 flat modes constructed and implemented in
the experiment, together with the first 4 Hermite-Gauss modes from the
supermode basis.

##  Appendix F Eigenmodes from the diagonalization of the covariance matrix

The numerical diagonalization of the covariance matrix performed in the main
text gives back the eigenmode basis where there are no quantum correlations,
i.e., the supermode basis discussed in the text. For additional information
concerning the reconstruction of the covariance matrix see [12].

We therefore expect the numerical eigenmodes to resemble discretized versions
of the quasi-Hermite-Gauss supermodes found in our numerical simulations from
theory.

Fig. 8 shows the numerical eigenmodes obtained after the diagonalization. In
general, the supermode shapes are in very good agreement with the expected
theoretical Hermite Gauss modes. The bandwdith of every Hermite-Gauss is
systematically higher than the theoretical value, which together with the
optical clipping in the pulse shaper and the LO bandwidth could be causing the
degradation of the measured squeezed values described in the main text.

On a technical note, the 55 nm bandwidth of our LO was slightly smaller than
the bandwidth of our expected quantum signal. This limits the experimentally
accessible modes if the wavelengths outside the LO bandwidth are involved in
the spectral features of the supermode. This effect is noticeable for high-
order modes, which are the most broadband. In Fig. 8, the red area highlights
the wavelength range that was not accessible with our LO bandwidth.

Please also note that the eigenvalues of the covariance matrix are directly
related to the expected squeezing levels of each eigenmode. Fig. 9 shows the
eigenvalues obtained in the diagonalization, which are consistent with those
values directly measured in the supermode basis in the main text.

Figure 9: Squeezing values obtained from the numerical diagonalization of the
covariance matrix in the frequency basis. The modes have been arranged in
decreasing order of antisqueezed quadratures, and the first eigenmode is
squeezed in the $\hat{p}$ quadrature by convention.

##  Appendix G Peres–Horodecki (PPT) criterion

As explained in the main text, the Positive Partial Transposition (PPT)
criterion can be applied to determine whether quantum correlations are present
in our data of the covariance matrix.

The PPT criterion is based on the fact that given a density matrix defining a
quantum state, the partial transpose matrix of two possible bipartitions of
the density matrix has to be positive defined for the two bipartitions being
separable. The criterion applied to continuous variable systems can be found,
for example, in [34]. This theorem applies to the covariance matrix since it
completely defines any gaussian state.

In our case, given that we have an 8x8 matrix in our frexel basis, we have a
total of 127 possible bipartitions. For each bipartition, we can define the
minimum eigenvalue of the partial transpose matrix as the PPT value,
indicating quantum correlations if negative.

Fig. 10 shows the PPT value for the different bipartitions of our covariance
matrix, showing the violation of the PPT criterion, i.e., negativity in the
partial transpose matrix, in all of them.

More details about the PPT criterion applied to the covariance matrix can be
found in [40].

Figure 10: PPT values computed for all 127 possible bipartitions. The results
show that all of the values are negative, thereby indicating the non-
separability of the state.

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††thanks: e-mail<EMAIL_ADDRESS>
# Fusion reaction of a weakly-bound nucleus with a deformed target

Ki-Seok Choi and K. S. Kim  School of Liberal Arts and Science, Korea
Aerospace University, Koyang 412-791, Korea  Myung-Ki Cheoun  Department of
Physics, Soongsil University, Seoul 156-743, Korea  W. Y. So  Department of
Radiological Science, Kangwon National University at Dogye, Samcheok 245-905,
Korea  K. Hagino  Department of Physics, Kyoto University, Kyoto 606-8502,
Japan

###### Abstract

We discuss the role of deformation of the target nucleus in the fusion
reaction of the 15C + 232Th system at energies around the Coulomb barrier, for
which 15C is a well-known one-neutron halo nucleus. To this end, we construct
the potential between 15C and 232Th with the double folding procedure,
assuming that the projectile nucleus is composed of the core nucleus, 14C, and
a valence neutron. In addition, we also take into account the coupling to the
one-neutron transfer process to the 14C+233Th configuration. We show that such
calculation simultaneously reproduces the fusion cross sections for the 14C +
232Th and the 15C + 232Th systems, implying an important role of the transfer
coupling in fusion of neutron-rich nuclei.

###### pacs:

24.10.-i, 25.70.Jj

##  I Introduction

One of the most important discoveries in nuclei near the neutron drip line is
the halo phenomenon Tanihata1985 ; hansen1987neutron . It is characterized by
a spatially extended density distribution, originated from the weakly-bound
property of neutron-rich nuclei. Starting from 11Li Tanihata1985 , several
other halo nuclei have also been observed successively. For example, 6He,
14Be, and 17B are regarded as two-neutron halo nuclei, while 11Be, 15C and 19C
are categorized as one-neutron halo nuclei TANIHATA1985380 ; PhysRevC.57.2156
. Recently, heavier halo nuclei, such as 19CPhysRevLett.83.1112 , 22C
PhysRevLett.104.062701 , 31Ne PhysRevLett.103.262501 , and 37Mg
PhysRevLett.112.242501  have also been found at Radioactive Ion Beam Facility
(RIBF) in RIKEN.

Fusion reactions of halo nuclei have attracted lots of attention TS91 ; TKS93
; HPCD93 ; DV94 ; PhysRevC.61.037602 ; PhysRevC.65.024606 ; ITO200653 ;
Canto20061 ; Canto20151 ; CHOI2018455 ; choi2017estimating . It is generally
known that fusion cross sections at energies around the Coulomb barrier are
sensitive to the structure of colliding nuclei BT98 ; DHRS98 ; HT12 ; Back14 ;
MS17 , and it is thus likely that the halo structure significantly affects
fusion reactions, both in a static and a dynamical ways. With the development
of the radio-isotope technology, a large number of experimental data for
fusion of halo nuclei have been accumulated. For instance, fusion cross
sections for the 11Li + 208Pb PhysRevC.87.044603 , ${}^{\text{6}}$He + 238U
raabe2004no , 6He + 209Bi hassan2006investigation ; PhysRevLett.81.4580  ,
11Be + 209Bi SIGNORINI2004329 , and 15C + 232Th PhysRevLett.106.172701
systems have been reported.

Interestingly, it has been reported that fusion cross sections for the
${}^{\text{6}}$He + 238U system raabe2004no  do not show any significant
influence of the halo structure of 6He albeit that 6He is a well-known halo
nucleus. This is in contrast to fusion cross sections for the 11Li + 208Pb
PhysRevC.87.044603  system, which show an enhancement with respect to the
fusion cross sections for the 9Li + 208Pb system. The 6He + 209Bi system also
shows a similar trend as in the 11Li + 208Pb system PhysRevLett.81.4580 . In
the case one-neutron halo nuclei, cross sections for the 11Be+209Bi system are
reported to be similar to those for the 10Be+209Bi system SIGNORINI2004329 .
Origins for this apparent difference among these systems have not yet been
understood completely, even though the fissile nature of the 238U may play
some role.

In this regard, it is interesting to notice that 238U is a well deformed
nucleus while 208Pb and 209Bi are spherical nuclei. The aim of this paper is
to investigate the role of deformation of the target nucleus in fusion of a
halo nucleus. To this end, we shall discuss the fusion reaction of the
15C+232Th system. The 15C nucleus is a one-neutron halo nucleus
PhysRevC.57.2156 , and its structure is much simpler than the structure of the
two-neutron halo nuclei 11Li and 6He. The 15C+232Th system thus provides an
ideal opportunity to disentangle the deformation and the halo effects.
Moreover, 15C is heavier than 11Li and 6He, and more significant effects of
the target deformation can be expected for the 15C+232Th system as compared to
the 11Li, 6He+238U systems. Notice that the previous calculation for this
system used a very simple spectator model and did not take into account the
halo structure of 15C PhysRevLett.106.172701 . It has yet to be clarified how
much the measured fusion enhancement can be accounted for by taking into
account the halo structure of 15C.

The paper is organized as follows. In Sec. II, we first analyze the fusion of
the 14C + 232Th system by using a deformed Woods-Saxon potential. In Sec. III,
we analyze the fusion of the 15C + 232Th system and discuss the role of the
halo structure and the deformation effect. To this end, we construct the
potential between 15C and 232Th with the double folding formalism accounting
for the halo structure of the 15C nucleus. We finally summarize the paper in
Sec. IV

##  II Fusion reaction of the 14C+232Th system

Before we discuss the fusion cross sections for the 15C+232Th system, we first
analyze the 14C+232Th system. In order to take into account the deformation
effect of the target nucleus 232Th, we employ a deformed Woods-Saxon (WS)
potential for the relative motion between the target and the projectile nuclei
HT12 ; HRK99 ; PhysRevC.98.014607 :

$V_{{}^{14}{\rm
C}-T}(r,\theta)=-\frac{V_{0}}{1+\exp\left[\left(r-R_{0}-R_{T}\sum_{\lambda}\beta_{\lambda
T}Y_{\lambda 0}(\theta)\right)/a\right]},$ (1)

where $V_{0}$, $R_{0}$, and $a$ are the depth, the radius, and the diffuseness
parameters, respectively. $R_{T}$ and $\beta_{\lambda T}$ are the radius and
the deformation parameters of the target nucleus 232Th. The Coulomb potential
also has a deformed form given by HT12 ; HRK99

$\displaystyle V_{C}(r,\theta)$ $\displaystyle=$
$\displaystyle\frac{Z_{P}Z_{T}e^{2}}{r}$ (2)
$\displaystyle+\frac{3Z_{P}Z_{T}e^{2}}{5}\frac{R_{T}^{2}}{r^{3}}\left(\beta_{2T}+\frac{2}{7}\sqrt{\frac{5}{\pi}}\beta_{2T}^{2}\right)Y_{20}(\theta)$
$\displaystyle+\frac{3Z_{P}Z_{T}e^{2}}{9}\frac{R_{T}^{4}}{r^{5}}\left(\beta_{4T}+\frac{9}{7\sqrt{\pi}}\beta_{2T}^{2}\right)Y_{40}(\theta),$

with the second order in the quadrupole deformation parameter, $\beta_{2T}$,
and the first order in the hexadecapole deformation parameter, $\beta_{4T}$.
$Z_{P}$ and $Z_{T}$ are the atomic number of the projectile and the target
nuclei, respectively. Fusion is simulated with the incoming wave boundary
condition HT12 ; HRK99 . For simplicity, in this paper we assume that 14C is
inert.

The fusion cross sections for the 14C+232Th system so obtained are presented
in Fig. 1. The actual values for the parameters in the WS potential employed
in this calculation are given in Table 1. We use the same values for the
radius and the diffuseness parameters as those of the global type of Akÿz-
Winther (AW) potential akyuz1981proceedings , while we adjust the depth
parameter, $V_{0}$ to fit the measured fusion cross sections. For the
deformation parameters, we employ $\beta_{2T}=0.233$ and $\beta_{4T}=0.0946$
zumbro1986 . In the figure, the blue dashed line shows the cross sections in
the absence of the deformation effect, while the black solid line shows the
cross sections with the deformation effect. One can clearly see that the
enhancement of the fusion cross sections below the Coulomb barrier region can
be well accounted for by taking into account the deformation of 232Th. It is
apparent that the deformation plays an important role in this system.

Table 1: The depth, $V_{0}$, the radius, $r_{0}$, and the diffuseness, $a$, parameters for the deformed Woods-Saxon potential for the 14C + 232Th reaction. Here, the radius parameter $r_{0}$ is defined as $R_{0}=r_{0}(A_{P}^{1/3}+A_{T}^{1/3}$), where $A_{P}$ and $A_{T}$ are the mass numbers of the projectile and the target nuclei, respectively. The resultant barrier height, $V_{b}$, the barrier position, $R_{b}$, and the barrier curvature, $\hbar\Omega$ are also shown.  $V_{0}$ (MeV) |  $r_{0}$ (fm) |  $a$ (fm) |  $V_{b}$ (MeV) |  $R_{b}$ (fm) |  $\hbar\Omega$ (MeV)
---|---|---|---|---|---
68.174 | 1.231 | 0.548 | 60.66 | 12.22 | 4.96
Figure 1: Fusion cross sections for the 14C + 232Th system. The dashed line
denotes cross sections in the absence of the deformation effect of the target
nucleus 232Th while the solid line is obtained by taking into account the
deformation effect with a deformed Woods-Saxon potential. The experimental
data are taken from Ref. PhysRevLett.106.172701 .

##  III Fusion reaction of the 15C+232Th system

###  III.1 The internuclear potential

Let us now discuss the fusion reaction of the 15C + 232Th system. We first
construct the potential between 15C and 232Th taking into account the
deformation effect of the target nucleus as well as the halo structure of the
projectile. To this end, we employ the double folding approach and construct
the potential as

$\displaystyle V_{{}^{15}{\rm
C}-T}(\mbox{\boldmath$r$};\mbox{\boldmath$r$}_{\rm d})$ $\displaystyle=$
$\displaystyle\int d\mbox{\boldmath$r$}_{p}\int
d\mbox{\boldmath$r$}_{T}\rho_{p}(r_{p})\rho_{T}(\mbox{\boldmath$r$}_{T};\mbox{\boldmath$r$}_{\rm
d})$ (3) $\displaystyle\times
V_{NN}(\mbox{\boldmath$r$}-\mbox{\boldmath$r$}_{p}+\mbox{\boldmath$r$}_{T}),$

where $V_{NN}$ is an effective nucleon-nucleon interaction, while
$\rho_{p}(r_{p})$ and
$\rho_{T}(\mbox{\boldmath$r$}_{T};\mbox{\boldmath$r$}_{\rm d})$ are the
density profiles for the projectile and the target nuclei, respectively. Here,
the density of the deformed target is defined with respect to the orientation
angle, $\mbox{\boldmath$r$}_{\rm d}$, in the space fixed frame. In this paper,
we employ the deformed Woods-Saxon density given by

$\rho_{T}(\mbox{\boldmath$r$};\mbox{\boldmath$r$}_{\rm
d})=\frac{\rho_{0}}{1+\exp\left[\left(\frac{r-c(1+\sum_{\lambda}\beta_{\lambda
T}Y_{\lambda 0}(\theta_{rd}))}{z}\right)\right]},$ (4)

where $\theta_{rd}$ is the angle between $r$ and $\mbox{\boldmath$r$}_{\rm
d}$, and $\beta_{\lambda T}$ are the deformation parameters. We take $c$=6.851
fm, $z$=0.518 fm and $\rho_{\text{0}}$=0.162 fm-3 for the 232Th nucleus
DEVRIES1987495 . For the nucleon-nucleon interaction, $V_{NN}$, we use the M3Y
interaction BERTSCH1977399  given by

$V_{NN}(r)=-2134\frac{e^{-2.5r}}{2.5r}+7999\frac{e^{-4r}}{4r}-275.81\delta(r),$
(5)

where the energy and the length are given in units of MeV and fm,
respectively. Notice that this interaction also includes the knock-on exchange
effect in the zero-range approximation.

To evaluate the double folding potential, the target density (4) is expanded
as

$\displaystyle\rho_{T}(\mbox{\boldmath$r$};\mbox{\boldmath$r$}_{\rm d})$
$\displaystyle=$ $\displaystyle\sum_{\lambda}\rho_{T\lambda}(r)Y_{\lambda
0}(\theta_{rd})$ (6) $\displaystyle=$
$\displaystyle\sum_{\lambda,\mu}\rho_{T\lambda}(r)\sqrt{\frac{4\pi}{2\lambda+1}}Y_{\lambda\mu}(\hat{\mbox{\boldmath$r$}})Y^{*}_{\lambda\mu}(\hat{\mbox{\boldmath$r$}}_{\rm
d}).$ (7)

Substituting this into Eq. (3), one obtains the potential in a form of

$V_{{}^{15}{\rm C}-T}(\mbox{\boldmath$r$};\mbox{\boldmath$r$}_{\rm
def})=\sum_{\lambda,\mu}V_{\lambda}(r)\sqrt{\frac{4\pi}{2\lambda+1}}Y_{\lambda\mu}(\hat{\mbox{\boldmath$r$}})Y^{*}_{\lambda\mu}(\hat{\mbox{\boldmath$r$}}_{\rm
d}),$ (8)

with

$V_{\lambda}(r)=\int d\textbf{r}_{p}\int
d\textbf{r}_{T}\rho_{n}(r_{p})\rho_{T\lambda}(r_{T})V_{NN}(\mbox{\boldmath$r$}-\mbox{\boldmath$r$}_{p}+\mbox{\boldmath$r$}_{T}).$
(9)

In the isocentrifugal approximation, one then sets
$\hat{\mbox{\boldmath$r$}}_{\rm d}=0$ and finally obtains HT12

$V_{{}^{15}{\rm C}-T}(r,\theta)=\sum_{\lambda}V_{\lambda}(r)Y_{\lambda
0}(\theta).$ (10)

We assume that the projectile nucleus 15C takes the two-body structure, with
the spherical core nucleus 14C and a valence neutron. The density of the
projectile is then given as

$\rho_{p}(\mbox{\boldmath$r$})=\rho_{c}(r)+\rho_{n}(r),$ (11)

where $\rho_{c}(r)$ and $\rho_{n}(r)$ are the density for the core nucleus and
the valence neutron, respectively. If one uses this density, the folding
potential of Eq. (3) is also separated into two parts:

$V_{{}^{15}{\rm C}-T}(r,\theta)=V_{{}^{14}{\rm
C}-T}(r,\theta)+V_{n-T}(r,\theta).$ (12)

For simplicity, we replace the interaction between the core and the target
nuclei, $V_{{}^{14}{\rm C}-T}(r,\theta)$, by the deformed Woods-Saxon
potential determined in the previous section.

For the density for the valence neutron, we construct it using a $2s_{1/2}$
neutron wave function in a Woods-Saxon potential as

$\rho_{n}(r)=\frac{1}{4\pi}\,\left[R_{2s_{1/2}}(r)\right]^{2},$ (13)

where $R_{2s_{1/2}}(r)$ is the radial part of the wave function. To this end,
we use the Woods-Saxon potential with set C in Ref. HS07 , which reproduces
the empirical neutron separation energy for this state,
$\epsilon_{2s_{1/2}}=-1.21$ MeV. Figure 2 shows the projectile density thus
obtained. The blue dashed line shows the density for the core nucleus, 14C,
while the red dot-dashed line denotes the valence neutron density. For the
description of the core density, we use the modified harmonic-oscillator
model, whose parameters can be found in Ref. DEVRIES1987495 . One can see that
the valence neutron density has a long tail, reflecting the halo structure of
the 15C nucleus.

Figure 2: The density distribution of the 15C nucleus (the solid line). The
dashed and the dot-dashed lines denote the contribution of the core nucleus
and the valence neutron, respectively.
---
Figure 3: The potential between the valence neutron in 15C and the target
nucleus 232Th for the 15C + 232Th reaction. The top, the middle, and the
bottom panels are for $V_{0}(r)$, $V_{2}(r)$, and $V_{4}(r)$ in Eq. (9),
respectively. The black solid lines show the results of the double folding
potential, while the red dashed lines shows fits with the Woods-Saxon
function.

The solid lines in Fig. 3 shows the neutron-target potential obtained with the
double folding procedure. The top, the middle, and the bottom panels show the
monopole, the quadrupole, and the hexadecapole components, respectively. In
order to discuss properties of these potentials, we fit them with a Woods-
Saxon function and its first derivative. That is,

$\displaystyle V_{\lambda}(r)$ $\displaystyle=$
$\displaystyle\frac{-V_{0}}{1+\exp\left[\left(r-R_{0}\right)/a\right]}~{}~{}~{}(\lambda=0),$
$\displaystyle=$
$\displaystyle\frac{-V_{0}\exp\left[\left(r-R_{0}\right)/a\right]}{\left(1+\exp\left[\left(r-R_{0}\right)/a\right]\right)^{2}}~{}~{}~{}(\lambda=2,4),$

The results of the fitting are shown in Fig. 3 by the dashed lines (see Table
2 for the parameters). Since the region around the position of the Coulomb
barrier is most important for fusion cross sections, the fitting are performed
mainly in the surface region, $r>$ 8 fm. In Fig. 3, one can see that the
folding potential can be well fitted with the Woods-Saxon function. The
hexadecapole component, $V_{4}(r)$, has some deviation from the Woods-Saxon
function, but its contribution to the total potential is much smaller than the
monopole and the quadrupole components. We also find that the contribution of
$\lambda=6$, that is, $V_{6}(r)$, is negligible, with a small depth size of
about $-$0.1 MeV. Note that the Coulomb barrier parameters, obtained by
setting the deformation parameters $\beta_{T\lambda}$ to be zero in the
folding procedure, are $V_{b}$=58.81 MeV, $R_{b}$=12.33 fm, and the
$\hbar\Omega$=4.10 MeV, which can be compared to those for the 14C+232Th
system listed in Table 1. The Coulomb barrier height is lowered by 1.53 MeV
owing to the weakly bound valence neutron in 15C.

Table 2: Parameters for the neutron-target part of the nuclear potential for the 15C+232Th system. Those are obtained by fitting the double folding potential to the Woods-Saxon function and its derivative. The depth, $V_{0}$, the radius, $R_{0}$, and the diffuseness, $a$, parameters are shown for each multipole component.  $V_{\lambda}$ |  $V_{0}$ (MeV) |  $R_{0}$ (fm) |  $a$ (fm)
---|---|---|---
$V_{0}(r)$ | 105.923 | 7.981 | 1.572
$V_{2}(r)$ | 28.756 | 7.080 | 1.890
$V_{4}(r)$ | 4.398 | 8.131 | 1.944

Figure 4 shows the total potential (that is, the sum of the nuclear and the
Coulomb potentials). The red dashed and the green dot-dashed lines show the
potential for the 14C+232Th reaction for $\theta=0^{o}$ and at
$\theta=90^{o}$, respectively. Note that the case of $\theta=0^{o}$ is
referred to as a tip collision while that of $\theta=90^{o}$ is referred to as
a side collision. The black solid and the blue dotted lines show the
corresponding potentials for the 15C+232Th reaction, obtained with the Woods-
Saxon fit to the double folding potential between the valence neutron and the
core nucleus. For both the cases, the Coulomb barrier is significantly lowered
due to the addition of the valence neutron in 15C. It is well known that
lowering of the Coulomb barrier leads to an increase of the penetration
probability for fusion cross sections.

Figure 4: The total potentials (the sum of the nuclear and the Coulomb
potentials) for the 14C + 232Th and the 15C + 232Th systems for $\theta=0^{o}$
and $\theta=90^{o}$. The solid and the dotted lines are for the 15C + 232Th
system, for which the solid line corresponds to $\theta=0^{o}$ and the dashed
line corresponds to $\theta=90^{o}$. The dashed and the dot-dashed lines are
the same as the solid and the dotted lines, but for the 14C + 232Th system.

###  III.2 Fusion cross sections

Let us now calculate fusion cross sections for the 15C + 232Th system using
the potential constructed in the previous subsection. For simplicity, we
include up to $\lambda=4$ in the multipole expansion of the double folding
potential between the valence neutron in 15C and the target nucleus.

Figure 5: Fusion cross sections for the 15C + 232Th system. The dot-dashed
line shows fusion cross sections obtained by scaling the potential for the 14C
+ 232Th system with the mass number of the colliding nuclei. The dotted line
shows fusion cross sections with the double folding potential, which takes
into account the halo structure of the 15C nucleus, while the solid line shows
the result of the coupled-channels calculation with the transfer coupling. For
comparison, the figure also shows the theoretical fusion cross sections for
the 14C + 232Th system by the dashed line. The experimental data are taken
from Ref. PhysRevLett.106.172701  .

Figure 5 presents a comparison between the calculated fusion cross sections
and the experimental data. The red circles and the blue stars show the
experimental data for the 15C + 232Th and 14C + 232Th systems, respectively.
The violet dashed curve shows the results for the 14C + 232Th system, which is
the same as the solid line in Fig.1. The coral dot-dashed curve shows the
results for the 15C + 232Th system, obtained using the potential which is
simply scaled from that for the 14C + 232Th system. That is, the radius
parameter in the Woods-Saxon potential for the 14C + 232Th system is changed
from $r_{0}\left(14^{1/3}+232^{1/3}\right)$ to
$r_{0}\left(15^{1/3}+232^{1/3}\right)$. This calculation does not take into
account the weakly bound nature of the 15C projectile, and corresponds to the
calculation presented in Ref. PhysRevLett.106.172701 . In fact, the dashed and
the dot-dashed lines show similar fusion cross sections to each other, as has
been argued in Ref. PhysRevLett.106.172701 . Even though this calculation
reproduces the experimental data at energies above the Coulomb barrier,
$E_{\rm c.m.}$ $\geqq$ 60 MeV, it considerably underestimates fusion cross
sections in the energy region below the Coulomb barrier.

Fusion cross sections evaluated with the halo nature of the 15C nucleus are
shown by the brown dotted line. This result clearly shows an enhancement of
fusion cross sections with respect to the dot-dashed line, and reproduce the
experimental data at the two lowest energies. However, the fusion cross
sections in the region of 60 MeV $\lesssim E_{\rm c.m.}\lesssim$ 65 MeV are
clearly overestimated.

In order to investigate a possible origin for the discrepancy, we follow Ref.
CHOI2018455  and consider a transfer coupling to the 14C+233Th channel. That
is, we consider a transfer to a single effective channel Rowley2001  and solve
the coupled-channels equations HT12 ; HRK99  of

$\displaystyle\left(\begin{array}[]{cc}K+V_{1}(r,\theta)&F_{1\rightarrow
2}(r)\\\ F_{1\rightarrow 2}(r)&K+V_{2}(r,\theta)-Q\\\
\end{array}\right)\left(\begin{array}[]{c}\psi_{1}(r)\\\ \psi_{2}(r)\\\
\end{array}\right)$ (20) $\displaystyle\hskip
128.0374pt=E\left(\begin{array}[]{c}\psi_{1}(r)\\\ \psi_{2}(r)\\\
\end{array}\right).$ (23)

Here, the channels 1 and 2 denote the 15C+232Th and the 14C+233Th systems,
respectively. $K$ is the kinetic energy (with the centrifugal potential) and
$V_{i}(r,\theta)$ ($i$=1,2) is the inter-nucleus potential for each partition.
$Q$ is the effective $Q$-value for the one-neutron transfer process, while
$F_{1\rightarrow 2}(r)$ is the coupling form factor. Notice that the effective
transfer channel may mock up also the breakup channel to some extent.

Table 3: The parameters for the transfer coupling. Here, those in the coupling form factor, Eq. (24), are determined by fitting the results of the coupled-channels calculations to the experimental fusion cross section for the 15C+232Th system with $Q=0$ for the transfer $Q$-value.  $Q$ (MeV) |  $F_{t}$ (MeV fm) |  $R_{\rm coup}$ (fm) |  $a_{\rm coup}$ (fm)
---|---|---|---
0 | 27.5 | 14.638 | 0.69

In the calculation, we assume that the potential $V_{2}(r,\theta)$ for the 14C
+ 233Th channel is the same as the potential for the 14C + 232Th presented in
Table 1. For the coupling form factor, $F_{1\rightarrow 2}(r)$, we employ the
derivative form of the Woods-Saxon potential DV86 ; dasso1985macroscopic
given by

$F_{1\rightarrow 2}(r)=F_{t}\,\frac{d}{dr}\left(\frac{1}{1+\exp((r-R_{\rm
coup})/a_{\rm coup})}\right).$ (24)

The parameters are determined by fitting to the experimental data for fusion
cross sections (see Table III). To this end, we take the transfer $Q$-value to
be zero, $Q=0$, rather than the ground-state-to-ground-state $Q$-value,
$Q_{\rm gg}=+3.568$ MeV, taking into account the $Q$-value matching condition
Rowley2001 . The black solid line in Fig. 5 shows the fusion cross sections so
obtained. One can see that the experimental data are well reproduced in the
whole energy region shown in the figure, indicating an importance of the
dynamical effect on fusion of the neutron-rich nucleus, 15C.

##  IV Summary

We have calculated fusion cross sections for the 15C+232Th system, for which
15C is a well-known one-neutron halo nucleus while 232Th is a well deformed
nucleus. To this end, we have evaluated the cross sections within the double
folding formalism taking into account the halo structure of 15C and the
deformation of the target nucleus. In addition, we have also taken into
account the coupling to the one-neutron transfer channel to the 14C+233Th
system with the coupled-channels formalism. We have shown that such
calculation reproduces simultaneously well the experimental data for the
15C+232Th and 14C+232Th systems. This clearly indicates that all of the halo
structure of 15C, the deformation of 232Th, and the dynamical effect such as
transfer and breakup play an important role in fusion of the 15C+232Th system.
In this regard, it would be useful to investigate systematically fusion of a
halo nucleus with deformed target nuclei in order to gain a deeper insight
into the role of deformation of the target in fusion of neutron-rich nuclei. A
comparison of the 15C + 232Th system to the 15C + 238U and 15C+208Pb systems
might also provide useful information.

## Acknowledgment

This work was supported by the National Research Foundation of Korea (Grant
No. NRF-2016R1C1B1012874, 2018R1D1A1B07045915,
NRF-2017R1E1A1A01074023,NRF-2019R1H1A1102164, NRF-2020R1A2C3006177 and
NRF-2013M7A1A1075764) and by JSPS KAKENHI (Grant No. JP19K03861). K. S. Kim’s
work was supported by MSIT (No.2018R1A5A1025563).

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