# Exclusive Top Threshold Matching at Lepton Colliders

DESY, Hamburg, Germany
E-mail  Bijan Chokoufé Nejad
DESY, Hamburg, Germany
E-mail<EMAIL_ADDRESS>Andre Hoang
University of Vienna, Vienna, Austria
E-mail<EMAIL_ADDRESS>Wolfgang Kilian
University of Siegen, Germany
E-mail<EMAIL_ADDRESS>Maximilian Stahlhofen
Johannes-Gutenberg-University, Mainz, Germany
E-mail<EMAIL_ADDRESS>Thomas Teubner
University of Liverpool, United Kingdom
E-mail<EMAIL_ADDRESS>Christian Weiss
DESY, Hamburg, Germany
E-mail<EMAIL_ADDRESS>
###### Abstract:

The threshold scan at future lepton colliders is the most precise known method
to determine the top quark mass (well below 100 MeV), a fundamental parameter
of the Standard Model that co-determines the stability properties of the
electroweak vacuum. We present a new method to match the continuum next-to-
leading order QCD corrections with the next-to-leading logarithmic resummation
of the Coulomb singularities of the quasi-toponium bound state at threshold
where fixed-order perturbation theory is invalid. This matching is performed
at the level of the fully exclusive WbWb final state. It allows to study all
kinds of differential distributions at or close to threshold. The top mass
dependence of these distributions opens up new possibilities for the top mass
determination that might be competitive with the inclusive threshold scan.

##  1 Introduction

Future high-energy lepton colliders will measure the top quark properties like
its mass, width and couplings with an unprecedented accuracy and use the top
quark as a means to search for new physics beyond the Standard Model. Here, we
discuss the matching of fixed-order QCD next-to-leading order (NLO-QCD
calculations for exclusive $e^{+}e^{-}\to W^{+}bW^{-}\bar{b}$ final states in
the continuum, based on [1], with a resummed calculation in the threshold
region, where fixed-order perturbation theory in the strong coupling
$\alpha_{s}$ is not a good approximation anymore, but the top velocity $v$ is
an additional expansion parameter and Coulomb-singular terms
$\sim(\alpha_{s}/v)^{n}$ and (ideally also) large logarithms
$\sim(\alpha_{s}\log v)^{n}$ have to be resummed. In Sec. 2, after reviewing
our calculational framework and the details of the continuum calculation for
completeness, we discuss, based on [2], a previously known non-relativistic
QCD (NRQCD) effective field theory setup to compute a form factor accounting
for the resummation of the threshold-singular terms at NLL accuracy,
implemented it in the fixed-order calculation and matched the result to the
QCD-NLO cross section in the transition region between threshold and
continuum. We thus obtained a fully-differential cross section, which gives
reliable predictions for all center-of-mass energies. Depending on how
inclusive the process is, we achieve LL + QCD-NLO (for very exclusive
processes) or NLL + QCD-NLO precision (for inclusive processes) in the
threshold region. Finally, we conclude in Sec. 3.

##  2 QCD-NLO (fixed-order) & Threshold Matching

In the continuum, i.e. away from the threshold, QCD corrections are properly
described by fixed-order relativistic QCD-NLO perturbation theory for the off-
shell top pair production. For that purpose, we study either the process
$e^{+}e^{-}\to W^{+}bW^{-}\bar{b}$ or
$e^{+}e^{-}\to\ell^{+}e^{-}\bar{\nu}_{e}\mu^{+}\nu_{\mu}b\bar{b}$ including
leptonic $W$ decays. Within the full four- or six-particle final state, there
are double-resonant diagrams included (involving a top and an anti-top
propagator), single-resonant diagrams and non-resonant irreducible background
processes. To calculate total and fully differential QCD-NLO corrections for
the top production processes, we take the WHIZARD framework for (QCD-)NLO
processes. WHIZARD [3] is a multi-purpose event generator with its own matrix-
element generator for tree-level amplitudes, O’Mega [4, 5] with support for a
plethora of models like e.g. supersymmetry [6]. Users can use external models
by the interface to FeynRules [7]. WHIZARD uses the color-flow formalism [8],
and it comes with its own parton shower implementation [9]. QCD-NLO
applications within WHIZARD started with a hard-coded implementation for the
production of $b$ jets at LHC [10, 11], while matching between resummed terms
and fixed-order calculations have been tackled by combining fixed-order
electroweak corrections to chargino production at the ILC with an all-order
QED initial-state structure function [12, 13]. WHIZARD is also able to do
automatic POWHEG matching for $e^{+}e^{-}$ processes [14].

WHIZARD uses FKS subtraction [15] and generates the automatically generates
the phase space for all singular emission regions. Virtual matrix elements,
color-correlated and spin-correlated matrix elements for the collinear and
soft splittings are taken from the one-loop provider (OLP) program OpenLoops
[16]. The complex mass scheme is used, leading to a complex weak mixing
angles. The input values are as follows: $m_{W}=80.385$ GeV, $m_{Z}=91.1876$
GeV, $m_{t}=173.2$ GeV, $m_{H}=125$ GeV. We use massive $b$-quarks of mass
$m_{b}=4.2$ GeV. Widths need to be calculated at the same order and in the
same scheme than the scattering process in order to guarantee properly
normalized branching ratios: $\Gamma_{Z}^{\text{LO}}=2.4409$ GeV,
$\Gamma_{Z}^{\text{NLO}}=2.5060$ GeV, $\Gamma_{W}^{\text{LO}}=2.0454$ GeV,
$\Gamma_{W}^{\text{NLO}}=2.0978$ GeV, $\Gamma_{t\to Wb}^{\text{LO}}=1.4986$
GeV, $\Gamma_{t\to Wb}^{\text{LO}}=1.3681$ GeV. As the matrix elements for the
full off-shell processes contain narrow resonances, particularly the $H\to bb$
resonance, we use a resonance-aware version of the FKS subtraction formalism
to make sure that cancellations between real emissions and subtraction terms
do cancel though the real emission could shift the kinematics on or off the
resonance compared to Born kinematics. This resonance-aware treatment is
automatically done in WHIZARD. As we are using massive $b$-quarks, no cuts are
necessary for the process $e^{+}e^{-}\to W^{+}W^{-}b\bar{b}$. The integrations
for the full QCD-NLO are very stable. We did two independent own integrations
with the serial and the non-blocking MPI-parallelizable version [17] of VAMP
[18] inside WHIZARD.

For the QCD-NLO corrections, we take the top mass as renormalization scale.
The scale variations for the process $e^{+}e^{-}\to W^{+}bW^{-}\bar{b}$ is
very small, at the level of two per cent. After one has replaced the top width
in the matrix elements by a running top width $\Gamma_{t}(\mu_{R})$ , the
scale variations for the on-shell process $e^{+}e^{-}\to t\bar{t}$ behave the
same way as for the off-shell process. The WHIZARD infrastructure immediately
enables QCD-NLO calculations/simulations for polarized beams, to include QED
initial-state photon radiation as well as collider-specific beamspectra.

A kinematic fit to the shape of the rising of the cross section at the top
threshold is believed to be the most precise method to measure the top quark
mass with an ultimate precision of 30-80 MeV. For this the systematic
uncertainties of the experimental measurement – especially the details of the
beam spectrum – as well as the theoretical uncertainties have to be well under
control. As shown above, close to the kinematical threshold for the on-shell
production of a $t\bar{t}$

Figure 1: Matched NRQCD-NLL + QCD-NLO calculation without (left) and with
(right) QED ISR. The dashed vertical line is the value of twice
$M^{1\text{S}}$. Blue is the fixed QCD-NLO calculation, red is the fully
matched calculation. The matched calculation has a full envelope over
(symmetrized) scale uncertainties as well as variations over switch-off
functions.

pair, fixed-order perturbation theory is not a good approximation. Very close
to threshold, the effective field theory of (v/p)NRQCD separates the hard
scale $m_{t}$, the soft scale given by the top momentum of the non-
relativistic top quark with velocity $v$, $m_{t}v$ and the ultrasoft scale,
given by the kinetic energy of the top quark, $m_{t}v^{2}$ and allows to resum
large logarithms of $v$ with $\alpha_{s}\sim v\sim 0.1$ close to threshold.
”Fixed-order” calculations resumming only Coulomb singularities, but no
velocity logarithms, for the totally inclusive $t\bar{t}$ production have been
carried out in NRQCD to NNNLO [19]. The large velocity logarithms have been
resummed to next-to-next-leading logarithmic (NNLL) [22] order (cf. also [20,
21] for predictions not containing the full set of NNLL ultrasoft logarithms).
These NRQCD calculations, based on the optical theorem, hold only for the
total inclusive cross section in a narrow window around the $t\bar{t}$
threshold. Here, we combine and match the NLL NRQCD-resummed process close to
the top threshold with the fixed-order (relativistic) QCD-NLO process in the
continuum. By a carefully performed matching procedure, our approach smoothly
interpolates between threshold region and continuum, and allows to study all
kinds of differential distributions.

The matching is embedded into the WHIZARD-OpenLoops QCD-NLO fixed-order
framework discussed above. The NLL resummed NRQCD contributions are included
in terms of (S-/P-wave) form factors to the (vector/axial vector)
$\gamma/Z-t-\bar{t}$ vertex. These form factors are obtained from the
numerical solution of Schrödinger-type equations for the NLL Green functions
computed by the Toppik [23, 24, 25] code, which is included in WHIZARD, for
technical details cf. [2]. In order to avoid double-counting between the
fixed-order QCD-NLO part and the resummed NLL-NRQCD part, one has to expand
the form factors to first order in $\alpha_{s}$ and subtract those pieces. As
the NRQCD resummed calculations are only available for the top-vector and
axial-vector currents, this removal of double-counting has to be done in a
factorized approach within a double-pole approximation. In order to maintain
gauge-invariance of the factorized amplitudes, an on-shell projection of the
exclusive final states to the top mass shell is performed, for details cf.
[2]. The implementation inside WHIZARD has been

Figure 2: Left panel: Matched NRQCD-NLL + QCD-NLO total cross section as on
the left of Fig. 1, but for a single choice of switch off-function. $h$ and
$f$ are renormalization scale parameters as defined in [2, 22]. The grey bands
display the corresponding scale variations with and without symmetrization.
Right panel: $Wb$ invariant mass distribution at threshold ($\sqrt{s}=344$
GeV) as obtained with WHIZARD. The red line represents the full NRQCD-NLL +
QCD-NLO matched, and the blue line the pure QCD-NLO result. The associated
bands are generated by the same scale variations as in the left panel, here
without symmetrization.

validated with analytical calculations for different invariant mass cuts on
the reconstructed top quarks from Ref. [26].

For larger top velocity ($v\gtrsim 0.4$) only the relativistic QCD-NLO result
is valid. We define a switch-off function that smoothly interpolates between
the two regions. The possibility to vary this arbitrary function and its
parameters adds another theory uncertainty to the different scale variations
[2]. The results of our matching procedure are displayed in Fig. 1. These
plots show the total inclusive cross section for the process $e^{+}e^{-}\to
W^{+}bW^{-}\bar{b}$, in the left panel without and in the right panel with QED
initial-state radiation (ISR). The dashed vertical line gives the value for
$2M^{1\text{S}}$. The 1S mass $M^{1\text{S}}$ is defined as half of the
perturbative mass of a would-be 1S toponium state and represents a renormalon
free short-distance mass, which we treat as an input parameter in WHIZARD. The
blue line shows the QCD-NLO cross section including scale variations in the
blue shaded areas. The red curve shows the NRQCD-NLL + QCD-NLO result, while
the shaded band contains all (symmetrized) scale variations of the hard, soft
and ultrasoft factorization/renormalization scales according to [22] as well
as variations of the switch-off function to a reasonable extent [2]. The
dotted black line shows the matched results without applying a switch-off
function to the factorized NRQCD terms which deviates above threshold from the
relativistic QCD-NLO result. In Fig. 2, left panel, we see the matched result
in the threshold region for a single choice of switch-off parameters, but
scale variations over the full two-dimensional renormalization parameter range
defined in [22]. This shows that the scale variation bands for the resummed
NLL result in the threshold region are highly asymmetric with respect to the
central value which motivates to apply a symmetrization of the error bands
around the central value. This symmetrization is also shown in Fig. 1. In the
right panel of Fig. 2 we show as an example for a differential distribution
the invariant mass of the $W-b$ jet system. Blue is the fixed-order QCD-NLO
distribution, while red is the fully matched distribution including scale
variations, here un-symmetrized. The ratio plot in the bottom does not show a
K factor, but the ratio of the matched result to the QCD-NLO fixed order
result. It shows an enhancement in the top mass peak due to threshold
resummation by a factor of 10-12.

##  3 Conclusions

In order to be able to study experimental event selections as well as
differential distributions, we presented a matched threshold calculation that
smoothly interpolates the threshold region described by non-relativistic QCD
to the relativistic QCD-NLO calculation. It constitutes the highest precision
available at the level of the completely exclusive final state. Any of the
presented differential distributions depending on the top mass may serve as a
different means to determine the top mass. We were not accomplishing this task
here, but rather showed a framework as a proof-of-principle of the matching
procedure between threshold and continuum. For the proper matching to the
continuum the fixed-order QCD-NLO calculations for top-quark pair production
including top and (leptonic) $W$ decays have been done. All of this has been
done in the QCD-NLO framework of the WHIZARD event generator which allows to
include all important physics of a lepton collider like polarization, QED ISR
radiation and non-trivial beam spectra.

## References

  * [1] B. Chokoufé Nejad, W. Kilian, J. M. Lindert, S. Pozzorini, J. Reuter and C. Weiss, JHEP 1612, 075 (2016) [arXiv:1609.03390 [hep-ph]].
  * [2] F. Bach, B. C. Nejad, A. Hoang, W. Kilian, J. Reuter, M. Stahlhofen, T. Teubner and C. Weiss, JHEP 1803, 184 (2018) [arXiv:1712.02220 [hep-ph]].
  * [3] W. Kilian, T. Ohl and J. Reuter, Eur. Phys. J. C 71, 1742 (2011) [arXiv:0708.4233 [hep-ph]].
  * [4] M. Moretti, T. Ohl and J. Reuter, hep-ph/0102195.
  * [5] B. Chokoufe Nejad, T. Ohl and J. Reuter, Comput. Phys. Commun. 196, 58 (2015) [arXiv:1411.3834 [physics.comp-ph]].
  * [6] T. Ohl and J. Reuter, Eur. Phys. J. C 30, 525 (2003) [hep-th/0212224].
  * [7] N. D. Christensen, C. Duhr, B. Fuks, J. Reuter and C. Speckner, Eur. Phys. J. C 72, 1990 (2012) [arXiv:1010.3251 [hep-ph]].
  * [8] W. Kilian, T. Ohl, J. Reuter and C. Speckner, JHEP 1210, 022 (2012) [arXiv:1206.3700 [hep-ph]].
  * [9] W. Kilian, J. Reuter, S. Schmidt and D. Wiesler, JHEP 1204, 013 (2012) [arXiv:1112.1039 [hep-ph]].
  * [10] T. Binoth, N. Greiner, A. Guffanti, J. Reuter, J.-P. Guillet and T. Reiter, Phys. Lett. B 685, 293 (2010) [arXiv:0910.4379 [hep-ph]].
  * [11] N. Greiner, A. Guffanti, T. Reiter and J. Reuter, Phys. Rev. Lett. 107, 102002 (2011) [arXiv:1105.3624 [hep-ph]].
  * [12] W. Kilian, J. Reuter and T. Robens, Eur. Phys. J. C 48, 389 (2006) [hep-ph/0607127].
  * [13] T. Robens, J. Kalinowski, K. Rolbiecki, W. Kilian and J. Reuter, Acta Phys. Polon. B 39, 1705 (2008) [arXiv:0803.4161 [hep-ph]].
  * [14] J. Reuter, B. Chokoufe, A. Hoang, W. Kilian, M. Stahlhofen, T. Teubner and C. Weiss, J. Phys. Conf. Ser. 762, no. 1, 012059 (2016) [arXiv:1602.06270 [hep-ph]].
  * [15] S. Frixione, Z. Kunszt and A. Signer, Nucl. Phys. B 467, 399 (1996) [hep-ph/9512328].
  * [16] F. Cascioli, P. Maierhöfer and S. Pozzorini, Phys. Rev. Lett. 108, 111601 (2012) [arXiv:1111.5206 [hep-ph]]; F. Buccioni, S. Pozzorini and M. Zoller, Eur. Phys. J. C 78, no. 1, 70 (2018) [arXiv:1710.11452 [hep-ph]].
  * [17] S. Braß, W. Kilian, J. Reuter, DESY 18-198, SI-HEP-2018-32, in preparation
  * [18] T. Ohl, Comput. Phys. Commun. 120, 13 (1999) [hep-ph/9806432].
  * [19] M. Beneke, Y. Kiyo, P. Marquard, A. Penin, J. Piclum and M. Steinhauser, Phys. Rev. Lett. 115, no. 19, 192001 (2015) [arXiv:1506.06864 [hep-ph]].
  * [20] A. H. Hoang, A. V. Manohar, I. W. Stewart and T. Teubner, Phys. Rev. D 65, 014014 (2002) [hep-ph/0107144].
  * [21] A. Pineda and A. Signer, Nucl. Phys. B 762, 67 (2007) [hep-ph/0607239].
  * [22] A. H. Hoang and M. Stahlhofen, JHEP 1405, 121 (2014) [arXiv:1309.6323 [hep-ph]].
  * [23] M. Jezabek, J. H. Kuhn and T. Teubner, Z. Phys. C 56, 653 (1992).
  * [24] R. Harlander, M. Jezabek, J. H. Kuhn and T. Teubner, Phys. Lett. B 346, 137 (1995) [hep-ph/9411395].
  * [25] A. H. Hoang and T. Teubner, Phys. Rev. D 60, 114027 (1999) [hep-ph/9904468].
  * [26] A. H. Hoang, C. J. Reisser and P. Ruiz-Femenia, Phys. Rev. D 82, 014005 (2010) [arXiv:1002.3223 [hep-ph]].


# Gerald Edelman’s Steps Toward a Conscious Artifact

Jeffrey L. Krichmar
Department of Cognitive Sciences
Department of Computer Science
University of California, Irvine
Irvine, CA 92697-5100
<EMAIL_ADDRESS>

###### Abstract

In 2006, during a meeting of a working group of scientists in La Jolla,
California at The Neurosciences Institute (NSI), Gerald Edelman described a
roadmap towards the creation of a Conscious Artifact. As far as I know, this
roadmap was not published. However, it did shape my thinking and that of many
others in the years since that meeting. This short paper, which is based on my
notes taken during the meeting, describes the key steps in this roadmap. I
believe it is as groundbreaking today as it was more than 15 years ago.

_K_ eywords Artificial Intelligence $\cdot$ Consciousness $\cdot$ Machine
Consciousness $\cdot$ Robotics

##  1 Introduction

In February of 2020, I participated in the "On Consciousness" podcast with
Bernie Baars and David Edelman. We talked about my work at The Neurosciences
Institute (NSI) in La Jolla, California on the Darwin series of Brain-Based
Devices, as well as my current research in neurorobotics. Unsurprisingly, the
conversation turned to consciousness. I happened to mention that a page from
my old lab notebook, which is pinned to a bulletin board in my office at UC
Irvine, outlines a roadmap towards the creation of a Conscious Artifact. The
key steps in this roadmap were laid out by Gerald Edelman, who was the
director of the NSI at the time I was a research fellow there.

Several months later, alert listener Grant Castillou contacted me about the
podcast. In his email he wrote, "I have read many of Dr. Edelman’s works, but
never come across anything like this before". At that time, my university was
in lockdown due to the COVID-19 pandemic and I did not have access to my
office. Recently, though, I was in my office and found the roadmap (see Figure
1). This brief paper describes the roadmap and my recollections of the event
and the thinking behind it more than 15 years ago.

Figure 1: Copy of the roadmap for a Conscious Artifact from my lab notebook in
2006.

##  2 Background and History

It is necessary to provide some background on what led to this event. Of
course, it was a long time ago, so I may not have all the facts and dates
exactly right. It is well known that memory is labile; time and the
accumulation of experience can lead to false memories. Keeping that caveat in
mind, what follows is my best recollection of the event and what was proposed.

Although the date in the upper right corner of my notebook looks like January
3rd, I think this is either a misprint or my bad handwriting. I believe the
event was shortly after the Society for the Study of Artificial Intelligence
and Simulation of Behaviour (AISB) meeting held in Bristol UK in April of
2006. I participated in the GC5: Architecture of Brain and Mind symposium and
also attended the symposium on Integrative Approaches to Machine
Consciousness.

When I returned to the NSI, I was asked to give a briefing on the meeting at
the Institute’s Fellows’ Symposium. I described what I saw and, in particular,
the current state of machine consciousness and artificial intelligence.

Later that day, we had our weekly "Build A Brain" meeting. The "Build A Brain"
group comprised an eclectic mix of theoreticians, engineers, and
neuroscientists. Every week we would discuss the progress on our brain-based
devices and how we could test theories of neuroscience using those
simulations. Although Gerald Edelman had given much thought to the topic of
consciousness, up until that point he had not wanted to construct a conscious
machine. In fact, he either did not think it was possible or thought it was
not a worthwhile endeavor.

However, at this particular meeting, I think my report on AISB’06 compelled
him to state, "I think the time has come to build a conscious artifact." My
notes from that meeting are shown in Figure 1. After the meeting, Eugene
Izhikevich, who was also present, told me to save this page because it might
be important. So, I made an extra copy of the page for myself, and it
eventually followed me to UC Irvine, where it now occupies a corner of my
bulletin board.

The rest of this paper describes, to the best of my recollection, my
interpretation of this roadmap toward a Conscious Artifact.

##  3 Roadmap to a Conscious Artifact

Table 1 is re-typed from my notes in Figure 1. Anyone who had the good fortune
to be in a meeting with Gerald Edelman would know that thoughts peppered
listeners in rapid fire. Or, as he would say, a "drink from a fire hose."
Without notes or other written prompts, Edelman rattled off 10 key steps
toward a Conscious Artifact. I tried as best as I could to keep up. The steps
were presented roughly in order of how one would proceed to construct a
Conscious Artifact. The first four steps were areas in which the team had made
some progress. The remaining steps were areas that needed further
investigation. I would say that, to date, none of the steps listed have been
adequately addressed. However, the purpose of this paper is not to review the
current state of the art, but rather to describe these steps as best I can.

Table 1: Conscious Artifact Step | Description
---|---
1) | Reentrant Architecture
2) | Thalamo-Cortical System
3) | Value Systems
4) | Phenotype
5) | Motor Control
6) | Generalization in Learning and Memory
7) | Communication
8) | Thought
9) | Language
10) | Kindergarten
A) | Computer Power
B) | Minimum Neurons Necessary
Note: | Any Sensor will Do But Need Haptics

###  3.1 Reentrant Architecture

A major component of Edelman’s Neural Darwinism and Theory of Neuronal Group
Selection (TNGS) was reentrant signaling [1]. Reentrant, because it was
different than feedback signals. Neuronal groups were bi-directionally linked
with synaptic connections. Different groups with specific features could share
information with these connections. Experience-dependent plasticity selected
associations among these groups. These global mappings (see margin of Figure
1) ultimately gave rise to perceptual categories and action plans.

The theoretical neuroscientists working at the NSI had created a number of
models to test these ideas, both in simulation and instantiated on physical
robotic platforms [2, 3]. With these models, they were able to show feature
mapping, binding through synchrony, memory recall, and other brain properties.
These were colorfully termed, "noodling," as can be read in my notes.

###  3.2 Thalamo-Cortical System

In a series of books, Edelman described his theory of consciousness, which was
based on the TNGS [4, 5]. A key to this theory, as enumerated in some of his
later publications, was the concept of a Dynamic Core [6]. The Dynamic Core
was essentially reentrant signalling between the thalamus and the neocortex.
The dynamics of the Dynamic Core were necessary to produce conscious thought
or higher order consciousness.

At the time of the meeting, researchers at the NSI were developing extremely
detailed computational models of the thalamus and neocortex. These models
showed sleep wake cycles, as well as other other brain rhythms observed during
conscious thought [7, 8]. In one such model, Eugene Izhikevich and Gerald
Edelman were able to show the formation of neuronal groups due to plasticity
and dynamic neuronal activity [9].

###  3.3 Value Systems

Value systems are neural structures that are necessary for an organism to
modify its behavior based on the salience or value of an environmental cue.
The value system in a brain-based device is analogous to neuromodulatory
systems in that its units show phasic responses when activated and its output
acts diffusely across multiple pathways to promote synaptic change.

Value systems were first explored by Karl Friston in theoretical work
conducted at the NSI when the Institute was located on the campus of The
Rockefeller University in New York City [10]. Every Brain-Based Device was
equipped with a value system for shaping behavior and building associations
between predicted value and observed value [11]. Edelman noted that value
could signal hunger, fear, and reward, among other signals salient to the
behaving agent.

###  3.4 Phenotype

In the case of the robotic platforms created at the NSI, the phenotype
consisted of the physical design of a given device. In particular, it was the
shape or morphology of the robot, as well as the layout and type of on-board
sensors. The Darwin series of automata had a basic shape; notably, all were
wheeled robots. But despite their relatively simple external design, they had
a wide range of sensors for hearing, vision, taste, and touch. Later versions
of the Darwin automata included omni-wheels, as well as articulated arms and
legs.

Interestingly, Edelman noted that the phenotype needed to be compliant and
must necessarily include proprioception. Compliance through soft materials and
elasticity is an important property of biological systems. Proprioception
would, Edelman believed, lead to a notion of self and body awareness. These
phenotypes were not a part of our design at the time and needed to be further
investigated through incorporation into additional robotic platforms. Also
notable is the fact that Edelman came back to the importance of touch toward
the end of the meeting. See the last line of Figure 1 and Table 1, where I
quoted him saying "any sensor will do, but need haptics."

###  3.5 Motor Control

Edelman’s theory of consciousness was very much tied to behavior and intended
actions. Therefore, in his mind (and presumably those who attended the
meeting) motor control was an important step towards creation of a Conscious
Artifact. In particular, Edelman mentioned efference copy and body sense.
After every action, a copy of the motor command is fed back to the nervous
system. This is called a "motor efference copy". The brain uses the motor
efference copy to check if the action generated yields the expected sensory
stimuli and expected body position. In this way, the agent might produce a
body sense.

It is also notable that Edelman singled out the Basal Ganglia (BG in my notes)
as an important aspect of motor control and key functional anatomical feature
of consciousness. Therefore a key step towards achieving a conscious artifact
is incorporating Basal Ganglia function. The Basal Ganglia selects actions and
generates putative motor sequences. In my notes, I wrote "proto syntax".
Edelman thought that action selection and the sequence of actions was a first
step towards language. He was a strong proponent of the idea that language was
rooted in action and motor control.

###  3.6 Generalization in Learning and Memory

Around this time, researchers at the NSI were constructing sophisticated
models of hippocampus and long-term memory [12, 13]. However, these models
were brittle. They suffered from an inability to transfer information from one
task to another, as well an incapacity for generalization. I find it
interesting that, to this day, transfer learning and generalization continue
to limit AI systems. Moreover, the ability to learn over a lifetime of
experience remains beyond the reach of current artificial systems.

###  3.7 Communication

Critically important to demonstrating a Conscious Artifact would be some form
of accurate report. Before language, a report could be realized by
communication between Brain-Based Devices (BBD <–> BBD, in Figure 1). By
reporting its intentions and state to another agent, the agent is showing a
degree of self-awareness. It is also interesting to speculate that
consciousness, in particular self-awareness (or higher-order consciousness),
might not be observable in a single agent. Rather, it might require social
interaction.

###  3.8 Thought

Unfortunately, I don’t have much to add to this step. I can only guess that
here, Edelman was alluding to mental simulation and imagination. This is a
prediction of the dynamic core theory, which was briefly described in the
Thalamo-Cortical section and in more detail in Edelman’s writings.

###  3.9 Language

Again, report is brought up as an important step toward creating a Conscious
Artifact. However, Edelman makes it clear that language is distinctively more
sophisticated than communication, as described in the Communication step.
Language is nuanced, suffused as it is with emotion, thought, intention, and
action. It is safe to say that Edelman was thinking that language, as
instantiated in a Conscious Artifact, would be something far beyond Natural
Language Processing or simply passing a Turing test. An accurate report via
language would need to demonstrate that the agent had a form of higher order
consciousness closely tied to body sense and a self.

###  3.10 Kindergarten

Similar to Turing’s theory and the field of developmental robotics, Edelman
proposed that to achieve all of the above, the Conscious Artifact would need
to be subjected to a curriculum of sorts. It was too much to load these
characteristics upon initialization of a given simulation. And more
importantly, consciousness is tied to the individual’s experience. This is
where learning and memory from experience becomes critically important.
Furthermore, communication and language are necessary in order to interact
with a teacher or caretaker, not to mention, report one’s intentions and
state.

###  3.11 Other Notes

Towards the end of the meeting, some pragmatic issues were brought up. One
limiting factor at the time was computational power, which remains a
constraint today. I am unsure what is meant by the "minimum neurons
necessary", which I quoted in my notebook. It could be taken to be tantamount
to computing power. Or it could be related to sparseness and energy. The
metabolic constraints imposed by biology was a topic that Edelman often
revisited during internal meetings at the NSI.

##  4 Hidden Gems

Anyone who had the experience of interacting with Gerald Edelman will tell you
that conversations were generously sprinkled with anecdotes and a seemingly
infinite supply of jokes. He was a consummate storyteller. Given any context,
he would suddenly and unexpectedly tell a joke that not only made a point with
great clarity, but was also delivered like a seasoned standup comedian. This
meeting was no different in content or character.

During such meetings, I would make notes in the margins to remind me of the
particular story or joke.

The first note is "7th position violin" and "Yasha Haifitz fiddles muscle".
Edelman was an accomplished violinist. It was said that he could have had a
career as a concert violinist. The 7th position is the most advanced position
of the hand; necessary for generating the highest notes on a violin. According
to members of Edelman’s family, his mother wanted him to be a doctor. As the
story goes, when he expressed the desire to become a concert violinist, his
mother said, "I have two words for you: Jascha Heifetz". Heifetz was one of
the greatest violinists of all time. This quickly ended the conversation.
Edelman realized he could never be on a par with Heifetz and decided to go
into medicine instead. This, of course, led to an accomplished research career
and a Nobel Prize in Physiology or Medicine for deciphering the protein
structure of the antibody (i.e., gamma gobulin or IgG). I believe the
discussion of seventh position during the meeting had to do with motor control
sequences and converting those sequences into muscle memory. In any case, it
nicely follows from: 5) Motor Control to 6) Generalization in Learning and
Memory.

I can only guess what "global mapping" meant. It certainly fits with the idea
of reentrant architecture and the dynamic core.

Lastly, "Mayef the Meshuggenah" is an element of one of Edelman’s classic
jokes. Though the phrase certainly alludes to Albert Mayev, Edelman’s violin
teacher, who was good friends with a number of the great violinists of the
mid-20th Century, including Heifetz and Nathan Milstein. Sadly, I cannot
remember the details. It is not something that you can Google to understand
the precise context. Perhaps one of my colleagues can recall the details of
this joke or story.

##  5 Final Thoughts

I strongly believe that the steps laid out on that day in 2006, comprise the
correct roadmap toward creation of machine consciousness. With this in mind, I
believe it is important to share these steps with others. I have tried to
follow them in my own research. But it is important to temper expectations
here; each of these steps is a career in itself. For example, I decided to
focus on "3) Value Systems" when I transitioned from the NSI to UC Irvine in
2008 [14]. I am still researching this fascinating topic [15, 16]. Along the
way, I have turned my attention to other steps on the list, but certainly have
not addressed all the steps listed in Figure 1 and Table 1.

After I left the NSI, Edelman, along with Yanqing Chen, Jason Fleischer, Joe
Gally, Jeff McKinstry, and others, began working in earnest on a Conscious
Artifact project. Among other things, they made notable progress on sequence
learning and mental imagery [17, 18].

Nevertheless, I think I can safely say that no one has put all these steps
together within one system. I am hopeful that someone, either in my generation
or the next, will one day be able to achieve this lofty goal.

## Acknowledgments

I would like to thank Bernie Baars and David Edelman for the discussion that
brought this roadmap to mind, as well as for many interesting conversations.
David was extremely helpful with editing and filling in details. A big shout
out goes to Grant Castillou for bringing this to my attention and leading me
to put this in writing so that it could be shared with others. Lastly, I would
like to thank the "Build A Brain" group at The Neurosciences Institute for
generating these ideas, for the progress toward a Conscious Artifact, and for
many fond memories.

## References

  * [1] G. M. Edelman.  Neural Darwinism: The Theory Of Neuronal Group Selection.  Basic Books, 1987.
  * [2] A. K. Seth, J. L. McKinstry, G. M. Edelman, and J. L. Krichmar.  Visual binding through reentrant connectivity and dynamic synchronization in a brain-based device.  Cerebral Cortex, 14(11):1185–1199, 2004.
  * [3] J. Wray and G. M. Edelman.  A model of color vision based on cortical reentry.  Cereb Cortex, 6(5):701–16, 1996.
  * [4] G. M. Edelman.  The Remembered Present: A Biological Theory of Consciousness.  Basic Books, 1990.
  * [5] G. M. Edelman.  Bright Air, Brilliant Fire: On the Matter of Mind.  Basic Books, 1993.
  * [6] G. M. Edelman and G. Tononi.  A Universe Of Consciousness: How Matter Becomes Imagination.  Basic Books, 2000.
  * [7] E. D. Lumer, G. M. Edelman, and G. Tononi.  Neural dynamics in a model of the thalamocortical system. i. layers, loops and the emergence of fast synchronous rhythms.  Cereb Cortex, 7(3):207–27, 1997.
  * [8] E. D. Lumer, G. M. Edelman, and G. Tononi.  Neural dynamics in a model of the thalamocortical system. ii. the role of neural synchrony tested through perturbations of spike timing.  Cereb Cortex, 7(3):228–36, 1997.
  * [9] E. M. Izhikevich and G. M. Edelman.  Large-scale model of mammalian thalamocortical systems.  Proc Natl Acad Sci U S A, 105(9):3593–8, 2008.
  * [10] K. J. Friston, G. Tononi, Jr. Reeke, G. N., O. Sporns, and G. M. Edelman.  Value-dependent selection in the brain: simulation in a synthetic neural model.  Neuroscience, 59(2):229–43, 1994.
  * [11] J. L. Krichmar and G. M. Edelman.  Brain-based devices for the study of nervous systems and the development of intelligent machines.  Artificial Life, 11(1-2):63–77, 2005.
  * [12] J. G. Fleischer, J. A. Gally, G. M. Edelman, and J. L. Krichmar.  Retrospective and prospective responses arising in a modeled hippocampus during maze navigation by a brain-based device.  Proceedings of the National Academy of Sciences of the United States of America, 104(9):3556–3561, 2007.
  * [13] J. L. Krichmar, D. A. Nitz, J. A. Gally, and G. M. Edelman.  Characterizing functional hippocampal pathways in a brain-based device as it solves a spatial memory task.  Proceedings of the National Academy of Sciences of the United States of America, 102(6):2111–2116, 2005.
  * [14] J. L. Krichmar.  The neuromodulatory system: A framework for survival and adaptive behavior in a challenging world.  Adaptive Behavior, 16(6):385–399, 2008.
  * [15] M. C. Avery and J. L. Krichmar.  Neuromodulatory systems and their interactions: A review of models, theories, and experiments.  Frontiers in Neural Circuits, 11, 2017.
  * [16] J. L. Krichmar and F. Rohrbein.  Value and reward based learning in neurorobots.  Frontiers in Neurorobotics, 7, 2013.
  * [17] J. L. McKinstry and G. M. Edelman.  Temporal sequence learning in winner-take-all networks of spiking neurons demonstrated in a brain-based device.  Front Neurorobot, 7:10, 2013.
  * [18] J. L. McKinstry, J. G. Fleischer, Y. Chen, W. E. Gall, and G. M. Edelman.  Imagery may arise from associations formed through sensory experience: A network of spiking neurons controlling a robot learns visual sequences in order to perform a mental rotation task.  PLoS One, 11(9):e0162155, 2016.


# All-in-fiber dynamic orbital angular momentum mode sorting

Alvaro Alarcón  Linköpings Universitet, Institutionen för Systemteknik,
Linköpings Universitet, Linköping 581 83, Sweden.  Santiago Gómez  Universidad
de Concepción, Departamento de Física, Universidad de Concepción, Casilla
160-C, Concepción, Chile.  Universidad de Concepción, Millennium Institute for
Research in Optics, Universidad de Concepción, Casilla 160-C, Concepción,
Chile.  Departamento de Física, Universidad del Bío-Bío, Collao 1202, Casilla
5C, Concepción, Chile.  Daniel Spegel-Lexne  Linköpings Universitet,
Institutionen för Systemteknik, Linköpings Universitet, Linköping 581 83,
Sweden.  Joakim Argillander  Linköpings Universitet, Institutionen för
Systemteknik, Linköpings Universitet, Linköping 581 83, Sweden.  Jaime Cariñe
Departamento de Ingeniería Eléctrica, Universidad Católica de la Santísima
Concepción, Concepción, Chile.  Gustavo Cañas  Departamento de Física,
Universidad del Bío-Bío, Collao 1202, Casilla 5C, Concepción, Chile.  Gustavo
Lima  Universidad de Concepción, Departamento de Física, Universidad de
Concepción, Casilla 160-C, Concepción, Chile.  Universidad de Concepción,
Millennium Institute for Research in Optics, Universidad de Concepción,
Casilla 160-C, Concepción, Chile.  Guilherme B. Xavier  Linköpings
Universitet, Institutionen för Systemteknik, Linköpings Universitet, Linköping
581 83, Sweden<EMAIL_ADDRESS>
###### Abstract

The orbital angular momentum (OAM) spatial degree of freedom of light has been
widely explored in many applications, including telecommunications, quantum
information and light-based micro-manipulation. The ability to separate and
distinguish between the different transverse spatial modes is called mode
sorting or mode demultiplexing, and it is essential to recover the encoded
information in such applications. An ideal $d$ mode sorter should be able to
faithfully distinguish between the different $d$ spatial modes, with minimal
losses, have $d$ outputs, and have fast response times. All previous mode
sorters rely on bulk optical elements such as spatial light modulators, which
cannot be quickly tuned and have additional losses if they are to be
integrated with optical fiber systems. Here we propose and experimentally
demonstrate, to the best of our knowledge, the first all-in-fiber method for
OAM mode sorting with ultra-fast dynamic reconfigurability. Our scheme first
decomposes the OAM mode in fiber-optical linearly polarized (LP) modes, and
then interferometrically recombines them to determine the topological charge,
thus correctly sorting the OAM mode. In addition, our setup can also be used
to perform ultra-fast routing of the OAM modes. These results show a novel and
fiber integrated form of optical spatial mode sorting that can be readily used
for many new applications in classical and quantum information processing.

##  I Introduction

The orbital angular momentum modes (OAM) of light [1, 2] have captured the
interest of the scientific community due to their many possible applications,
such as optical manipulation [3, 4], information transfer [5], data
multiplexing [6], quantum key distribution [7] and high-dimensional quantum
information processing [8]. OAM modes are commonly represented using a
Laguerre-Gaussian (LG) basis set, which has a ring-like amplitude profile and
azimuthal phase dependence $\exp(i\ell\phi)$. LG modes are characterized by
$\ell$ azimuthal and $p$ radial integer numbers [1]. In contrast, the modes
that only carry orbital angular momentum (OAM) have only an azimuthal phase
structure and are described only by the index $\ell$. The fact that OAM±ℓ
modes are mutually orthogonal and can generate an infinite state space makes
them a suitable platform for applications requiring $d$-spatial modes (or
$d$-dimensions), such as classical and quantum communications. For many
applications it is crucial to develop OAM-supporting devices and systems that
are compatible with optical fibers, can be integrated into other systems and
are scalable. Direct compatibility with optical fibers was not an issue in the
past, but since the advent of space-division multiplexing (SDM) optical
fibers, which can support the propagation of OAM modes [9], the use of bulk
optical elements can be a major disadvantage.

Although some efforts have been made to generate OAM modes using fiber systems
[10, 11, 12, 13, 14, 15, 16], the efficient detection of OAM modes using all-
in-fiber platforms remains an ongoing challenge. An OAM mode sorter is a
device capable of multiplexing different incoming electromagnetic fields that
carry OAM. Its internal operation depends on the spatial information carried
by the incoming mode [17, 18]. Fig. 1 schematically shows the behavior of a
general OAM mode sorter, where several OAM modes with different azimuthal
indexes can be sorted at the output of the system.

Figure 1: General ideal OAM mode sorter. The purpose of an OAM mode sorter is
to sort incoming OAM modes to a predetermined output, which depends on both
the topological charge of the incoming beam and any operations carried out by
the sorter.

OAM mode sorters have been implemented using several techniques, such as with
the use of spatial light modulators [19, 17, 20, 21], q-plates [22, 23],
geometrical transformations [24], log-polar coordinate transformations [25,
26], spiral transformations [27], or multiplane converters [28, 29, 30]. There
are also free-space interferometric-based schemes that perform OAM mode
sorting by applying relative phases related to the topological charge $\ell$
of the incoming modes using dove prisms [31, 32, 33]. All these systems are
based on performing operations on OAM modes using bulk optics, which makes
them difficult to integrate into current telecommunications networks.
Furthermore, when it comes to scaling up the number of spatial modes to be
distinguished, this type of scheme becomes complex. A fiber-based technique
recently presented is a sorter using optical fiber delays [34], taking
advantage of the fact that the different modes propagating in a fiber will
separate due to modal dispersion. This method has a limited bandwidth due to
the long delays needed. Finally, a static multiplexer/demultiplexer based on a
photonic lantern has been presented [35], however as it has no reconfigurable
capabilities, its applications in quantum information processing or in dynamic
optical networks are impractical.

In this paper, we present a novel all-in-fiber platform, with ultra-fast
reconfigurability, for interferometric sorting of arbitrary incoming OAM±ℓ
modes. Our approach is based on the observation that any OAM±ℓ mode can be
decomposed into a combination of an odd and an even linearly polarized (LP)
mode, with a relative phase difference of $\pm\pi/2$ [36, 37]. The central
components of our platform consist of an optical fiber that can support
multiple transverse spatial propagation modes and a photonic lantern (PL) [38,
39]. We can immediately obtain complete information about the parameter
$\pm\ell$ of the incoming OAM mode by detecting which output of our system the
light beam exits through. Our scheme can be scaled up in dimensionality by
employing components that support more spatial modes. In addition, we can
operate our system from a passive mode, where the modes are sorted, to an
active mode where we show that the OAM mode information can be arbitrarily
redirected to another route. To experimentally demonstrate the feasibility of
the scheme, we successfully implement an all-in-fiber reconfigurable mode
sorter for the OAM+1 and OAM-1 modes. As we have used off-the-shelf
components, our system can be easily integrated into fiber-optical
telecommunication networks. Our sorter can operate at high speeds, being able
to perform routing operations with a switching time as low as $7$ ns, limited
by the driving electronics. To the best of our knowledge, this represents a
record in speed response for OAM mode sorters, making it an excellent
candidate for active (or passive) demultiplexing systems in current optical
networks and for applications in quantum information processing.

##  II All-in-fiber OAM mode sorter

In Figure 2a we present the scheme of our proposal for sorting OAM states. The
inputs to our scheme are any state between $\text{OAM}_{+\ell}$ or
$\text{OAM}_{-\ell}$, where $\ell$ can only take integer values. The mode
sorter consists of a few-mode fiber [40] connected to a photonic lantern [41].
The physical principle behind our experiment is the fact that any OAM±ℓ mode
within a few-mode fiber can be decomposed into the following LP modal
superposition [14, 42, 43, 37]:

$\text{OAM}_{\pm\ell}=\text{LP}_{\ell 1(\text{even})}\pm i\cdot\text{LP}_{\ell
1(\text{odd})}$ (1)

The photonic lantern maps the LP modes that follow the above decomposition
from a specific OAM±ℓ mode coming from the few-mode fiber with a one-to-one
correspondence into a pair of single-mode fiber outputs only supporting the
fundamental LP01 mode (Gaussian). The lantern also preserves the phase
relation between the LP mode pair. The single-mode fibers are then recombined
interferometrically with a multi-port beam splitter (MBS) [44]. Phase
modulators (PMs) are placed in each arm of this high-dimensional
interferometer to adjust the internal relative phases. It is therefore
possible to identify uniquely the state’s topological charge $\pm\ell$ by
detection at the output of the interferometer. This method is completely in-
fiber, and this makes it perfect for integration to OAM sources and systems
which are already done fully in-fiber, which is becoming more common [9, 45,
13, 46]. Observe that if the appropriate phases are set in all the phase
modulators [44] it is possible to route the mode information to any output.
Furthermore, this reconfigurable setup allows one to implement any unitary
transformation onto a high-dimensional OAM encoded quantum state, which is
highly sought after in quantum information. In this work we perform an
experimental demonstration of the scheme to sort the OAM+1 and OAM-1 modes
(Fig. 2b), based on 3-mode few-mode fibers and photonic lanterns.

Figure 2: The proposed OAM mode sorter. a) In this general high-dimensional
case, there is a source emitting OAM modes with different azimuthal indices.
Each OAM mode is mapped onto different output paths from the photonic lantern
(PL), which are then superposed by a multiport beam splitter (MBS). Each
output path is connected to a detector to determine the topological charge of
each OAM mode. b) The experiment realized on the two-dimensional case allowing
the sorting of the OAM+1 OAM-1 modes, with detectors placed at the two
outputs. D: Detector; FMF: Few-mode fiber; PL: Photonic lantern; PM: Phase
modulator.

##  III Experimental setup

We now describe in detail the experimental setup for the two-dimensional
version of our scheme (Fig. 3). We choose to create the OAM±1 modes through
the well-known technique of generating a computer-generated hologram on a
spatial light modulator (SLM) to mimic the desired optical element capable of
converting a Gaussian laser beam into a helical mode [47]. Although this is
done in free-space for convenience, our mode-sorting system is completely in-
fiber, and it would work equally well if the OAMs were generated with an in-
fiber method [14]. A continuous wave (CW) laser operating at $1546.92$ nm is
spatially filtered through a single-mode fiber (SMF) and collimated with a 10x
objective lens to reach a waist of $\omega_{p}\approx 1250$ $\mu m$. A
polarizing beam splitter (PBS), is used to set the laser’s polarization state
to horizontal for the optimal operation of the SLM. A manual polarization
controller placed before the objective is used to optimize the optical power
following the objective. Note that $\omega_{p}$ is sufficient to illuminate
all the $\ell$-forked holograms, which consists of a helical phase profile
superposed with a linear phase ramp to spatially isolate the encoded field
from the Gaussian mode, resulting in a diffraction grating that produces the
OAMℓ mode in the first diffraction order [48]. We use a Holoeye Pluto-
Telco-013 phase-only SLM with a resolution of $1920$ x $1080$ pixels and $8\mu
m$ pixel pitch to implement the holograms. Each pixel acts as a programmable
phase shifter between $0$ and $2\pi$, which is controlled by the grayscale
value of the corresponding pixels of the hologram.

Figure 3: Experimental setup. The setup for demonstrating the sorting of OAM±1
modes can be divided into two main parts: the OAM±1 mode source and the OAM±1
mode sorter. The OAM source can switch between OAM+1 and OAM-1 by changing the
forked diffraction grating on the spatial light modulator (SLM). The mode
sorter is an all-fiber interferometer based on space-division-multiplexing
fiber technology. The computer-generated holograms used to create the OAM±1
are a superposition of a helical phase and a diffraction grating. The inset
shows the generated spatial intensity and phase profiles (interference
patterns) of the OAM modes. The phase profiles are obtained from the
interference pattern between the corresponding OAM mode and a Gaussian beam to
distinguish the topological charge $\ell=\pm 1$. These spatial patterns are
detected by an InGaAs CCD camera. A control laser at a different wavelength is
injected onto the FMF through the beamsplitter (BS) before the CCD camera and
is used as feedback signal for a microcontroller (MC) running a control
algorithm to stabilize the phase drift noise inside the mode-sorter
interferometer. A phase modulator (PMc) placed in one of the arms is used as
the actuator for this control system.

The SLM reflects the beam onto a $4$f system composed of two lenses, L1 and
L2, each with a focal length of $150$ mm. The Fourier transform of the
reflected field is formed in the focal plane of L1, where there is a pinhole.
This spatial filter aims to pick out the diffracted first order, corresponding
to the Fourier spectrum of the encoded field. The second lens (L2) performs an
inverse Fourier transform, resulting in the OAM±1 modes. A beam splitter (BS)
is placed at the end of the 4f system in order to split the beam and allow us
to image the generated mode during the experiment. The transmitted mode is
imaged by an InGaAs CCD camera placed in the image plane of L2 (see Fig. 3 and
inset). Both the amplitude and phase profiles of the OAM modes are imaged on
the camera. For the phase profiles, we create a reference mode by splitting
the laser’s beam with a 50:50 fiber coupler before the input 10x objective
(not shown in the figure for the sake of simplicity). This reference beam is
connected to the fiber launcher named ”Control input” in Fig. 3, which
consists of another 10x objective, where it is superposed with the OAM beam on
the beamsplitter. Therefore, the phase profiles are imaged as the interference
pattern between these two beams, and shown in the inset of Fig. 3. This
control laser input is otherwise used as a launch device to a second diode
laser as part of an active phase stabilization system for the mode sorter
interferometer (further explained below).

On the reflected output of the BS there is an $20$x objective used to couple
the OAM±1 mode into a FMF. The FMF used in this experiment is a commercially
available graded index telecom fiber (OFS 80730) that can support three modes:
the fundamental Gaussian LP01 and the higher order LP11a, and LP11b modes. The
linearly polarized modes are particular modes that propagate in a weakly
guided medium. Therefore, when the $20$x objective couples the OAM±1 mode into
the FMF, they are decomposed into LP modes propagating in the FMF. A manual
fiber polarization controller, where the FMF is wound, is used to correctly
adjust the modal demultiplexing in the photonic lantern. This fiber carrying
the incoming OAM±1 modes is then connected to a commercial PL (Phoenix
Photonics 3PLS-GI-15) consisting of one FMF at its input and three single-mode
fibers at its output, corresponding to each of the supported LP modes. The
internal structure of the PL is made of an adiabatic region that provides a
low transition from the input FMF to the output of the three SMFs in such a
way that each LP mode is mapped to the appropriate output port [38, 39]. This
mapping is produced by a matching process between the effective indices of the
tapered region and the incoming spatial modes [37, 49, 50]. Since the incoming
OAM mode only decomposes into the LP11a and LP11b modes, we will use only the
two outputs of the PL corresponding to these modes leaving out the optical
port that decomposes the LP01 mode. After the OAM±1 modes are decomposed, the
following mapping is done: LP11a→ $E_{1}$; LP11b→ $E_{2}$, where $E_{1}$ and
$E_{2}$ represent both electrical fields propagating through the two different
paths after PL.

A lithium niobate pigtailed telecom phase modulator (PM)(Thorlabs LN65S-FC) is
placed on the upper arm. The PM is driven by an electrical signal from a
function generator, so any arbitrary relative phase $\phi$ can be added to the
upper arm between 0 and $2\pi$. A variable optical attenuator (not shown for
simplicity) is located in the lower arm to equalize the optical power from
both arms of the interferometer before recombining on the $50:50$ fiber
coupler (FC). We also have manual polarization controllers before the FC to
ensure path indistinguishability. D+1 and D-1 are amplified p-i-n photodiodes
to measure the optical intensity at the two outputs, with the results recorded
by a digital oscilloscope.

Due to the natural instability of the interferometric configuration, we inject
a laser at a different wavelength ($1546.12$ nm) through the free port of the
beam splitter before the InGaAs CCD camera (Fig. 3). This laser beam co-
propagates with the OAM modes through the sorter, and is used as a feedback
signal for a phase stabilization system. This reference beam is split from the
OAM information before D+1 with a dense wavelength division multiplexer
(DWDM), and is detected by a third amplified p-i-n photodiode Dr, whose
electrical output is read by a microcontroller running a perturb-and-observe
algorithm [51] to stabilise the environmentally-induced phase drift in the
mode sorter interferometer. The phase controller employs a second phase
modulator (PMc) placed in the opposing arm to compensate the phase drift. The
control system operates continuously, but is briefly switched off when there
is a need to operate the sorter in routing mode. The optical loss from the PL
input to the detectors is 8.3 dB, which are mainly given by the PM and the
internal losses of the PL ($\sim$ 5 dB). These losses can be greatly reduced
by new lantern designs that have much lower losses (up to 0.2 dB) [52],
showing a path to having a mode sorter with very low losses.

##  IV Results

The first goal is to demonstrate the successful sorting between the OAM±1
modes with our scheme. Given that the mode sorter Mach-Zehnder interferometer
(MZI) is actively stabilized by the controller, the power detected in D+1 and
D-1 will depend on the additional relative phase $\phi$ applied by the PM and
also of the topological charge $\ell=\pm 1$ of the incoming OAM beam. It is
well known that in an MZI the optical intensity registered in $D_{+1}$
($D_{-1}$) is proportional to ${|E_{1}+e^{i\phi}E_{2}|}^{2}$
(${|E_{1}-e^{i\phi}E_{2}|}^{2}$). From Eq. 1, we notice that the incoming
OAM±1 also carries a relative phase between both LP modes of $\pm\pi/2$. By
adjusting the controller to maintain a relative phase between both arms of
$\pi/2$, then if the incoming beam is an OAM+1 (OAM-1) then we obtain maximum
optical intensity detected at $D_{+1}$ ($D_{-1}$). We then continuously switch
between OAM+1 and OAM-1 every second by changing the corresponding forked
diffraction gratings in the SLM. Figure 4 shows the electrical signal
corresponding to the optical intensity measured at D+1, when fixing
$\phi=\pi/2$ in the interferometer, with the periodic switching of the OAM
modes. From this result we show the deterministic identification of the
direction of rotation of the OAM mode, thus its topological charge.

Figure 4: OAM mode sorting. Here we show how our setup can sort the OAM modes
when the source is constantly changing between OAM+1 and OAM-1 with a period
of 1 s. A relative phase of $\pi/2$ is being applied inside the interferometer
by PM, such that OAM+1 is sent to D+1 and OAM-1 is sent to D-1. The intensity
curve above as a function of time is measured at D+1. The OAM mode labels
indicate when each particular OAM mode was being sent.

To demonstrate the stability of the sorting process over longer time periods,
we have repeated this process continuously for one hour. We have calculated
the optical visibility using the consecutive optical intensities between the
maxima and minima at the output of D+1 as the input OAM mode is continuously
switched. For any two consecutive maxima and minima, we calculate the
visibility $V$ as $V=(I_{+}-I_{-})/(I_{+}+I_{-})$, where $I_{+}$ ($I_{-}$) is
the detected optical intensity when the OAM+1 (OAM-1) mode is sent. We plot
the visibilities calculated for every mode transition as a function of time in
Fig. 5, while also displaying in the inset, the histogram of all the measured
visibilities giving an average of $92.59\pm 1.87\%$.

Figure 5: Long-term performance of the OAM mode sorter. We plot the
visibilities during a period of one hour calculated from the constant
switching between the OAM+1 and OAM-1 modes from the source, through the SLM.
The inset shows a histogram comes from these results, showing that the average
visibility is 92.59$\pm 1.87$%.

Our system can also operate as an active router for OAM modes by applying fast
relative phase changes through the PM. For this demonstration we fix the OAM+1
in the source while electrically driving a train of ten pulses, each being 100
$\mu$s long, to the PM. During the period the pulse train is being sent the
phase stabilization algorithm is suspended. The amplitude and offset of these
driving pulses was previously calibrated in order to have a relative phase
applied between the paths equivalent to $\phi=\pi/2$ and $\phi=-\pi/$2
respectively. We once again observe the output of the system by recording the
optical intensity present at D+1. By applying the sequential pulses to the PM,
the OAM+1 mode can be routed to either D+1 or D-1 depending on the value of
the driving relative phase, with these results shown in Fig. 6(a), where we
can clearly see the sequential routing of the OAM+1 mode between the two
outputs. The system works equally well if we now observe the D-1 output as
shown in Fig.6(b), with the routing operation complementary as expected. The
average visibility for both cases is $92.75\pm 1.27\%$, with the main
limitation to the optical visibility originating from the modal crosstalk in
the photonic lantern.

Figure 6: OAM mode routing with the source transmitting the OAM+1 state. a)
Optical intensity at the D+1 output when the internal PM in the mode sorter
changes $\phi$ from $\pi/2$ to $-\pi/2$ ten times consecutively in a period of
1.1 ms, generated from a field programmable gate array (FPGA) electronic
module. During the switching time, the phase stabilization algorithm is
suspended, and resumed after the last switching pulse is applied. In this way,
it is possible to route the OAM+1 mode to any detector arbitrarily. b) The
same measurement, but observed in detector D-1, where the complementary nature
of the interferometer can be observed, as expected. The routing speed in
measurements (a) and (b) are limited by our p-i-n photodetectors. In (c) we
demonstrate the speed response of our mode sorter by changing the p-i-n
photodiode at D+1 for an InGaAs single-photon detector, which has a timing
resolution of 200 ps. For this measurement we have attenuated the output power
of the source to a few photons per detector gate window of 2.5 ns. We measure
a rise/fall time of approximately 7 ns, limited by our employed driving
electronics.

The ultra-fast reconfigurability is a major advantage of our configuration
stemming from the use of telecom electro-optical fiber pig-tailed phase
modulators. For this demonstration, we replace the p-i-n photodiode, which has
a limited bandwidth of 10 MHz, with an InGaAs single-photon counting module
(IdQuantique id210), which has a timing resolution of 200 ps. The detector is
running with 10% detection efficiency and a 2.5 ns wide gate window. We then
modulate the internal PM in the mode sorter with a series of 20 ns wide
transitions between $\pi/2$ and $-\pi/2$ and synchronize the detector’s gate
window with these transitions. We then vary the electrical delay to the
trigger signal fed to the InGaAs detector in steps of 1 ns, allowing us to
scan the optical phase pulse which switches between the OAM modes, and the
result is shown in Fig. 6(c), where we show a rise/fall time of $7$ ns, which
is limited by our home-made electrical driver amplifying the electrical signal
feeding the phase modulator. This value can be further improved with
appropriate electronics, as the response bandwidth of the phase modulator is
within the 10 GHz range. For this measurement the OAM laser beam is attenuated
such that only a few photons per detection window are present and thus, the
detector is not saturated.

##  V Conclusions

Appropriate tools for manipulating OAM states are highly sought in many areas
of photonics, opening up new possibilities based on this degree-of-freedom of
light. Sorting of OAM modes has been one particular area of considerable
intense research in the last years, using many different techniques. Common to
most of these previous methods is the use of bulk optical elements to a
greater or lesser degree. The downside of these schemes is the difficulty of
integration with optical fiber systems, due to the additional losses and
complexity needed to couple the light to and from a fiber. This is specially
acute in cases where long-distance transmission is desired, such as in optical
and quantum communications. In the few cases where optical-fiber-based
techniques were used, no reconfigurability or high speed responses were
possible, severely limiting those to applications where dynamic operation is
essential, such as in quantum information processing. Here we solve this issue
by combining photonic lanterns with interferometric analysis of the components
of an OAM mode. Our sorter is designed to receive an OAM mode coming from a
few-mode fiber, and decomposes it into its linearly polarized mode components
with a photonic lantern. Then by recombining these modes interferometrically
onto a beamsplitter, we are able to sort the incoming modes. Our scheme is
generalizable to higher dimensionalities by employing few-mode fibers,
photonic lanterns and multi-port beamsplitters that support more transverse
spatial modes. For instance, MBSs with 7 ports [44] or alternatively arrays of
2 port beamsplitters on a photonics integrated circuit can function as a
general $d$-port beamsplitter [53] can be used as well as lanterns with
15[54], 19 [55] and 61 [56] ports have been demonstrated, showing the
potential of our scheme. Furthermore very recent results have demonstrated
photonic lanterns with very low loss ($<$ 0.2 dB) [52], further showing the
feasibility of our proposal for practical applications.

Finally, due to the interferometric nature of our configuration, it can also
be used in an active/reconfigurable mode, where the input mode can be routed
to a different output on demand, or a unitary transformation can be applied by
the phase modulator, allowing the measurement in different mutually unbiased
bases in quantum information [44]. In this active mode, we have demonstrated,
to be the best of our knowledge, the fastest response in an OAM router, with
the potential to reach sub-ns response times due to the use of fast telecom
phase modulators. Our results show how the use of space-division multiplexing
technology (i.e. few-mode fibers and photonic lanterns) can be used to
implement a fast and reconfigurable OAM mode sorter that is fully compatible
with optical fiber networks, further paving the way for this popular degree-
of-freedom to be used as a reliable transport medium for information in
classical and quantum networks.

### Acknowledgements

The authors acknowledge support from CENIIT Linköping University, the Swedish
Research Council (VR 2017-04470), the Knut and Alice Wallenberg Foundation
through the Wallenberg Center for Quantum Technology (WACQT) and by the
QuantERA grant SECRET (VR 2019-00392). Also, this work was supported by the
Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) (Grant Nos.,
1230796, 11201348 and 1200859) – Millennium Science Initiative Program –
ICN17-012.

## References

  * Allen  _et al._ [1992] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes, Phys. Rev. A 45, 8185 (1992).
  * Shen  _et al._ [2019] Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, Optical vortices 30 years on: Oam manipulation from topological charge to multiple singularities, Light: Science & Applications 8, 90 (2019).
  * Grier [2003] D. G. Grier, A revolution in optical manipulation, Nature 424, 810 (2003).
  * Padgett and Bowman [2011] M. Padgett and R. Bowman, Tweezers with a twist, Nat. Photon. 5, 343 (2011).
  * Gibson  _et al._ [2004] G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, Free-space information transfer using light beams carrying orbital angular momentum, Opt. Express 12, 5448 (2004).
  * Wang  _et al._ [2012] J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, Terabit free-space data transmission employing orbital angular momentum multiplexing, Nat. Photon. 6, 488 (2012).
  * Mirhosseini  _et al._ [2015] M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, High-dimensional quantum cryptography with twisted light, New Journal of Phys. 17, 033033 (2015).
  * Erhard  _et al._ [2018] M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, Twisted photons: new quantum perspectives in high dimensions, Light: Science & Applications 7, 17146 (2018).
  * Xavier and Lima [2020] G. B. Xavier and G. Lima, Quantum information processing with space-division multiplexing optical fibres, Commun. Phys. 3, 9 (2020).
  * Li  _et al._ [2015] S. Li, Q. Mo, X. Hu, C. Du, and J. Wang, Controllable all-fiber orbital angular momentum mode converter, Opt. Lett. 40, 4376 (2015).
  * Chang  _et al._ [2022] W. Chang, M. Feng, B. Mao, P. Wang, Z. Wang, and Y.-G. Liu, All-fiber fourth-order oam mode generation employing a long period fiber grating written by preset twist, J. Lightwave Technol. 40, 4804 (2022).
  * Wu  _et al._ [2022] G. Wu, X. Wu, S. Gao, J. Tu, J. Zhou, Q. Sui, W. Liu, and Z. Li, Multi-channel higher-order oam generation and switching based on a mode selective interferometer, Opt. Express 30, 25093 (2022).
  * Lian  _et al._ [2022] Y. Lian, X. Qi, Y. Wang, Z. Bai, Y. Wang, and Z. Lu, Oam beam generation in space and its applications: A review, Opt. Lasers Eng. 151, 106923 (2022).
  * Alarcón  _et al._ [2023] A. Alarcón, J. Argillander, D. Spegel-Lexne, and G. B. Xavier, Dynamic generation of photonic spatial quantum states with an all-fiber platform, Opt. Express 31, 10673 (2023).
  * Lu  _et al._ [2023] Y. Lu, Z. Jiang, Z. Chen, M. Jiang, J. Yang, Q. Zhou, J. Zhang, D. Zhang, J. Chai, H. Yang, and W. Liu, High-power orbital angular momentum beam generation using adaptive control system based on mode selective photonic lantern, J. Lightwave Tech. , 1 (2023).
  * Heng  _et al._ [2018] X. Heng, J. Gan, Z. Zhang, J. Li, M. Li, H. Zhao, Q. Qian, S. Xu, and Z. Yang, All-fiber stable orbital angular momentum beam generation and propagation, Opt. Express 26, 17429 (2018).
  * Fontaine  _et al._ [2019] N. K. Fontaine, H. C. R. Ryf, D. T. Neilson, K. Kim, and J. Carpenter, Laguerre-gaussian mode sorter, Nat. Commun. 96, 7 (2019).
  * Ariyawansa  _et al._ [2021] A. Ariyawansa, E. J. Figueroa, and T. G. Brown, Amplitude and phase sorting of orbital angular momentum states at low light levels, Optica 8, 147 (2021).
  * Jia  _et al._ [2022] Q. Jia, R. Feng, B. Shi, Y. Zhang, H. Li, X. Li, F. Sun, Y. Cao, H. Shi, J. Wang, and W. Ding, Universal translation operator for laguerre–gaussian mode sorting, Appl. Phys. Lett. 121, 191104 (2022).
  * Fickler  _et al._ [2020] R. Fickler, F. Bouchard, E. Giese, V. Grillo, G. Leuchs, and E. Karimi, Full-field mode sorter using two optimized phase transformations for high-dimensional quantum cryptography, J. Opt. 22, 024001 (2020).
  * Lavery  _et al._ [2012] M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, Refractive elements for the measurement of the orbital angular momentum of a single photon, Opt. Express 20, 2110 (2012).
  * Jia  _et al._ [2019] J. Jia, Z. Chang, H. Yang, Q. Liu, F. Wang, H. Gao, F. Li, and P. Zhang, Mode sorter designed for (de)multiplexing vector vortex modes, Appl. Opt. 58, 7094 (2019).
  * Karimi  _et al._ [2009] E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates, Appl. Phys. Lett. 94, 231124 (2009).
  * Huang  _et al._ [2015] H. Huang, G. Milione, M. P. J. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. A. Nguyen, D. A. Nolan, M.-J. Li, M. Tur, R. R. Alfano, and A. E. Willner, Mode division multiplexing using an orbital angular momentum mode sorter and mimo-dsp over a graded-index few-mode optical fibre, Sci. Rep. 5, 14931 (2015).
  * Berkhout  _et al._ [2010] G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Efficient sorting of orbital angular momentum states of light, Phys. Rev. Lett. 105, 153601 (2010).
  * Dudley  _et al._ [2013] A. Dudley, T. Mhlanga, M. Lavery, A. McDonald, F. S. Roux, M. Padgett, and A. Forbes, Efficient sorting of bessel beams, Opt. Express 21, 165 (2013).
  * Wen  _et al._ [2018] Y. Wen, I. Chremmos, Y. Chen, J. Zhu, Y. Zhang, and S. Yu, Spiral transformation for high-resolution and efficient sorting of optical vortex modes, Phys. Rev. Lett. 120, 193904 (2018).
  * Kupianskyi  _et al._ [2023] H. Kupianskyi, S. A. R. Horsley, and D. B. Phillips, High-dimensional spatial mode sorting and optical circuit design using multi-plane light conversion, APL Photonics 8, 026101 (2023).
  * Zhang  _et al._ [2020] Y. Zhang, H. Wen, A. Fardoost, S. Fan, N. K. Fontaine, H. Chen, P. LiKamWa, and G. Li, Simultaneous sorting of wavelengths and spatial modes using multi-plane light conversion, arXiv:2010.04859 10.48550/arXiv.2010.04859 (2020).
  * Morizur  _et al._ [2010] J.-F. Morizur, L. Nicholls, P. Jian, S. Armstrong, N. Treps, B. Hage, M. Hsu, W. Bowen, J. Janousek, and H.-A. Bachor, Programmable unitary spatial mode manipulation, J. Opt. Soc. Am. A 27, 2524 (2010).
  * Leach  _et al._ [2002] J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Measuring the orbital angular momentum of a single photon, Phys. Rev. Lett. 88, 257901 (2002).
  * Leach  _et al._ [2004] J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon, Phys. Rev. Lett. 92, 013601 (2004).
  * Cozzolino  _et al._ [2019] D. Cozzolino, D. Bacco, B. Da Lio, K. Ingerslev, Y. Ding, K. Dalgaard, P. Kristensen, M. Galili, K. Rottwitt, S. Ramachandran, and L. K. Oxenløwe, Orbital angular momentum states enabling fiber-based high-dimensional quantum communication, Phys. Rev. Appl. 11, 064058 (2019).
  * Sulimany and Bromberg [2022] K. Sulimany and Y. Bromberg, All-fiber source and sorter for multimode correlated photons, npj Quantum Information 8, 4 (2022).
  * Eznaveh  _et al._ [2018] Z. S. Eznaveh, J. C. A. Zacarias, J. E. A. Lopez, K. Shi, G. Milione, Y. Jung, B. C. Thomsen, D. J. Richardson, N. Fontaine, S. G. Leon-Saval, and R. A. Correa, Photonic lantern broadband orbital angular momentum mode multiplexer, Opt. Express 26, 30042 (2018).
  * Ma  _et al._ [2012] P. Ma, P. Zhou, Y. Ma, X. Wang, R. Su, and Z. Liu, Generation of azimuthally and radially polarized beams by coherent polarization beam combination, Opt. Lett. 37, 2658 (2012).
  * Zeng  _et al._ [2018] X. Zeng, Y. Li, L. Feng, S. Wu, C. Yang, W. Li, W. Tong, and J. Wu, All-fiber orbital angular momentum mode multiplexer based on a mode-selective photonic lantern and a mode polarization controller, Opt. Lett. 43, 4779 (2018).
  * Leon-Saval  _et al._ [2010] S. G. Leon-Saval, A. Argyros, and J. Bland-Hawthorn, Photonic lanterns: a study of light propagation in multimode to single-mode converters, Opt. Express 18, 8430 (2010).
  * Birks  _et al._ [2015] T. A. Birks, I. Gris-Sánchez, S. Yerolatsitis, S. G. Leon-Saval, and R. R. Thomson, The photonic lantern, Adv. Opt. Photon. 7, 107 (2015).
  * Sillard  _et al._ [2014] P. Sillard, M. Bigot-Astruc, and D. Molin, Few-mode fibers for mode-division-multiplexed systems, J. Lightwave Tech. 32, 2824 (2014).
  * Velázquez-Benítez  _et al._ [2018] A. M. Velázquez-Benítez, J. E. Antonio-López, J. C. Alvarado-Zacarías, N. K. Fontaine, R. Ryf, H. Chen, J. Hernández-Cordero, P. Sillard, C. Okonkwo, S. G. Leon-Saval, and R. Amezcua-Correa, Scaling photonic lanterns for space-division multiplexing, Sci. Rep. 8, 8897 (2018).
  * Zhang  _et al._ [2019] H. Zhang, B. Mao, Y. Han, Z. Wang, Y. Yue, and Y. Liu, Generation of orbital angular momentum modes using fiber systems, Appl. Sci. 9, 1033 (2019).
  * Lu  _et al._ [2018] J. Lu, L. Meng, F. Shi, X. Liu, Z. Luo, P. Yan, L. Huang, F. Pang, T. Wang, X. Zeng, and P. Zhou, Dynamic mode-switchable optical vortex beams using acousto-optic mode converter, Opt. Lett. 43, 5841 (2018).
  * Cariñe  _et al._ [2020] J. Cariñe, G. Cañas, P. Skrzypczyk, I. Šupić, N. Guerrero, T. Garcia, L. Pereira, M. A. S. Prosser, G. B. Xavier, A. Delgado, S. P. Walborn, D. Cavalcanti, and G. Lima, Multi-core fiber integrated multi-port beam splitters for quantum information processing, Optica 7, 542 (2020).
  * Heng  _et al._ [2017] X. Heng, J. Gan, Z. Zhang, Q. Qian, S. Xu, and Z. Yang, All-fiber orbital angular momentum mode generation and transmission system, Opt. Commun. 403, 180 (2017).
  * Alarcón and Xavier [2020] A. Alarcón and G. B. Xavier, A few-mode fiber mach-zehnder interferometer for quantum communication applications, in  _Frontiers in Optics / Laser Science_ (Optica Publishing Group, 2020) p. LM1F.6.
  * Arrizón  _et al._ [2007] V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, Pixelated phase computer holograms for the accurate encoding of scalar complex fields, J. Opt. Soc. Am. A 24, 3500 (2007).
  * Bekshaev  _et al._ [2010] A. Bekshaev, O. Orlinska, and M. Vasnetsov, Optical vortex generation with a “fork” hologram under conditions of high-angle diffraction, Opt. Commun. 283, 2006 (2010).
  * Li  _et al._ [2019] Y. Li, Y. Li, L. Feng, C. Yang, W. Li, J. Qiu, X. Hong, Y. Zuo, H. Guo, W. Tong, and J. Wu, Mmonitoring and automatic tuning and stabilization of, Appl. Sci. 9, 2233 (2019).
  * Noordegraaf  _et al._ [2009] D. Noordegraaf, P. M. W. Skovgaard, M. D. Nielsen, and J. Bland-Hawthorn, Efficient multi-mode to single-mode coupling in a photonic lantern, Opt. Express 17, 1988 (2009).
  * AlTaha  _et al._ [2019] M. W. AlTaha, H. Jayatilleka, Z. Lu, J. F. Chung, D. Celo, D. Goodwill, E. Bernier, S. Mirabbasi, L. Chrostowski, and S. Shekhar, Monitoring and automatic tuning and stabilization of a 2x2 mzi optical switch for large-scale wdm switch networks, Opt. Express 27, 24747 (2019).
  * Guo  _et al._ [2023] H. Guo, L. Chen, Z. Shi, W. Chang, L. Gu, Z. Wang, and Y. ge Liu, Ultra-low-loss 5-lp mode selective coupler based on fused biconical taper technique, Opt. Express 31, 18050 (2023).
  * Clements  _et al._ [2016] W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, Optimal design for universal multiport interferometers, Optica 3, 1460 (2016).
  * Fontaine  _et al._ [2015] N. K. Fontaine, R. Ryf, H. Chen, A. V. Benitez, J. E. A. Lopez, R. A. Correa, B. Guan, B. Ercan, R. P. Scott, S. J. B. Yoo, L. Grüner-Nielsen, Y. Sun, and R. J. Lingle, 30×30 mimo transmission over 15 spatial modes, in  _Optical Fiber Communication Conference Post Deadline Papers_ (Optica Publishing Group, 2015) p. Th5C.1.
  * Noordegraaf  _et al._ [2012] D. Noordegraaf, P. M. W. Skovgaard, R. H. Sandberg, M. D. Maack, J. Bland-Hawthorn, J. S. Lawrence, and J. Lægsgaard, Nineteen-port photonic lantern with multimode delivery fiber, Opt. Lett. 37, 452 (2012).
  * Noordegraaf  _et al._ [2010] D. Noordegraaf, P. M. W. Skovgaard, M. D. Maack, J. Bland-Hawthorn, R. Haynes, and J. Lægsgaard, Multi-mode to single-mode conversion in a 61 port photonic lantern, Opt. Express 18, 4673 (2010).


# Achiral words

Shrinit Singh and A. Satyanarayana Reddy
Department of Mathematics,
Shiv Nadar Institution of Eminence, India-201314
(e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A word $w$ in a free group is achiral if for every group $G,$
$G_{w}=G_{w^{-1}},$ where $G_{w}$ is the image of the word map $w$ on $G.$ We
will give few classes of examples of achiral words. Cocke and Ho asked whether
Engel words are achiral or not. We will prove that it is enough to apply
Whitehead’s algorithm to check the same.

Key Words: Free group, Word map, Chiral word
AMS(2020):20F10

##  1 Introduction

Let $F_{n}$ be a free group on $n$ generators
$\\{x_{1},x_{2},\ldots,x_{n}\\}.$ A non-empty reduced word $w\in F_{n}$ is an
expression $\prod_{j=1}^{s}x_{i_{j}}^{a_{j}}$, where
$i_{j}\in\\{1,2,\ldots,n\\},a_{j}\in{\mathbb{Z}}\setminus\\{0\\}$ and for
every $j\in\\{1,2,\ldots,s\\},$ $x_{i_{j}}\neq(x_{i_{j+1}})^{-1}.$ In this
paper, word, we mean a reduced word. The length of the reduced word $w$ is
defined by $\Sigma_{j=1}^{s}|a_{j}|.$ For any group $G$, let $G^{(n)}$ denote
the direct product of $n$ copies of $G$. Corresponding to the word $w$, the
word map $w$ on $G$ is an evaluation map from $G^{(n)}$ to $G$, defined as

$(g_{1},g_{2},\ldots,g_{n})\mapsto
w(g_{1},g_{2},\ldots,g_{n})=\prod_{j=1}^{s}g_{i_{j}}^{a_{j}}.$

The image of the word map $w$ on $G$ is denoted by $G_{w}.$ It is clear that
$G_{w}$ is nonempty as the identity element $e\in G_{w}.$ For example, if
$w=x_{1}x_{2}x_{1}^{-1}x_{2}^{-1}\in F_{2}$ is a word of length $4$ and $G$ is
an abelian group, then $G_{w}=\\{e\\}=G_{w^{-1}}.$

Word maps have become an important tool to explore more about groups. One can
refer [2], [6], [13], [14] to see the applications of word maps on groups. In
particular, the image of a word map on a finite simple group is studied by
[8], [9], [12], [15].

It is easy to see that the image of a word map on a group $G$ corresponding to
any given word is invariant under endomorphism of $G$. A group $G$ is said to
satisfy the property $\mathcal{P}$ if for every subset $S$ of $G$ with $e\in
S$ and $S$ is invariant under every automorphism of $G$ can be seen as an
image of a word map. In 2012, Alexander Lubotzky [10] proved that all finite
simple groups satisfy the property $\mathcal{P}.$ Carolyn Ashurst [3], in her
thesis, asked that for any finite group $G$, whether under every word map, the
cardinality of preimage of an element $g$ is same as the cardinality of
preimage of $g^{-1}.$ Cocke and Ho [5] by using Lubotzky’s result showed that
this is not true and defined chiral word. A word $w\in F_{n}$ is said to be
chiral if there exists a group $G$ such that $G_{w}\neq G_{w^{-1}}$. We define
a word to be achiral if it is not chiral. A group $G$ is said to be chiral if
there exists a word $w\in F_{n}$ for some $n$, such that $G_{w}\neq
G_{w^{-1}}$. If $G$ is not chiral then we call $G$ as achiral. Let $X$ be a
subset of a group $G$. Then $X$ is inverse closed if $X=X^{-1},$ where
$X^{-1}=\\{g^{-1}|g\in X\\}.$ It is clear that ${G_{w}}^{-1}=G_{w^{-1}}$.
Hence a word $w$ is achiral if for every group $G,$ $G_{w}=(G_{w})^{-1}.$ That
is a word $w$ is achiral if and only if $G_{w}$ is inverse closed for every
group $G.$ For example, the sets $G^{k}=\\{g^{k}|g\in G\\},k\in{\mathbb{Z}}$
are inversely closed for any group $G,$ hence the words
$w=x_{1}^{k},k\in{\mathbb{Z}}$ are achiral. Consequently, every word from
$F_{1}$ is achiral. Let $G$ be a group then the subset
$S=\\{ghg^{-1}h^{-1}|g,h\in G\\}$ of $G$ is inverse closed. Hence the word
$w=x_{1}x_{2}x_{1}^{-1}x_{2}^{-1}$ is achiral. If the word map $w$ is onto for
every group $G$ i.e., $G_{w}=G,$ then $w$ is achiral. For example,
$w=x_{3}x_{1}x_{2}x_{1}^{-1}x_{2}^{-1}\in F_{3}$ and
$w=x_{1}x_{2}x_{1}x_{2}^{2}\in F_{2}$ are achiral. Let

$A_{n}=\\{w\in F_{n}|\;\mbox{$w$ is achiral}\\}$ $B_{n}=\\{w\in
F_{n}|\;\mbox{for every group}\;G,\;G_{w}=G\\}.$

Then it is clear that $B_{n}\subseteq A_{n}.$ It is known that the Nielsen
transformations do not change the image of a word map in a group. Let
$Aut(F_{n})$ denote the set of all automorphisms of $F_{n}.$ Thus if $w\in
F_{n}$ and $\sigma\in Aut(F_{n}),$ then $G_{w}=G_{\sigma(w)}.$ Hence the
automorphic image of an achiral word is achiral. We will prove this result as
a consequence of Proposition 2 in section 2. This observation will help us to
provide a few elements in $B_{n}.$ A word $w\in F_{n}$ is said to be primitive
if $w$ can be a part of a basis of $F_{n}$. A primitive word will always be
achiral as it is an automorphic image of achiral word $x_{1}$. Since $x_{1}\in
B_{n},$ the set of all primitive words is a subset of $B_{n}.$ But there are
words in $B_{n}$ which are not primitive , for example $x_{1}^{2}x_{2}^{3}$.
Akbar Rhemtulla [16] gave a complete description of $B_{n}$ which is as
follows.

######  Lemma 1 (Akbar Rhemtulla [16]).

Let $w\in F_{n}.$ Then $w\in B_{n}$ if and only if there exists integers
$e_{1},\ldots,e_{n}$ with $gcd(e_{1},e_{2},\ldots,e_{n})=1$ such that
$w\in{x_{1}}^{e_{1}}\ldots{x_{n}}^{e_{n}}{F_{n}}^{\prime},$ where
${F_{n}}^{\prime}$ is the commutator subgroup of $F_{n}.$

It is known [1] that the probability of an element
$(a_{1},a_{2},\ldots,a_{k})\in{\mathbb{Z}}^{n}$ such that
$gcd(a_{1},a_{2},\ldots,a_{k})=1$ is $\frac{1}{\zeta(k)}.$ Hence the
probability for a word $w\in F_{n}$ belongs to $A_{n}$ is positive. In section
2, we provide a necessary and sufficient condition for a word $w\in F_{n}$ is
achiral. By using this characterization, one can get more elements in $A_{n}.$
In the last subsection we will prove that it is enough to apply Whitehead’s
algorithm for Engel word’s achirality.

##  2 Main Results

The following result was stated in the thesis of William Cocke [4]. We are
giving the proof for the sake of completeness.

######  Proposition 2.

Following two statements are equivalent for a word $w\in F_{n}$:

  1. 1.

$w\in A_{n}.$

  2. 2.

There exists an endomorphism $\psi$ of $F_{n}$ such that $\psi(w)=w^{-1}.$

###### Proof.

Let $w\in F_{n}.$ We can view $w$ as a word map on $F_{n}$ as
$w:{F_{n}}^{n}\longrightarrow F_{n}$ and $w(x_{1},\ldots,x_{n})=w$ i.e.,
$w\in(F_{n})_{w}.$ Suppose $w$ is achiral. If we take $G=F_{n}$, then
$w^{-1}\in{(F_{n})}_{w^{-1}}={(F_{n})}_{w}.$ It implies that there exists
$w_{1},\ldots,w_{n}\in F_{n}$ such that $w(w_{1},\ldots,w_{n})=w^{-1}$. This
suggests that the endomorphism $\psi$ of $F_{n}$ defined by the map
$x_{i}\mapsto w_{i}$ $\forall i\in\\{1,2,\ldots,n\\}$ will give
$\psi(w)=w^{-1}.$

Let $\psi$ be an endomorphism of $F_{n}$ such that $\psi(w)=w^{-1}.$ Let $G$
be a group and $g\in G_{w}.$ Then there exists $g_{1},\ldots,g_{n}\in G$ such
that $w(g_{1},\ldots,g_{n})=g.$ We have

$\psi(w(x_{1},x_{2},\ldots,x_{n}))=\psi(w)=w^{-1}.$

Thus
$g^{-1}=w^{-1}(g_{1},\dots,g_{n})=w((\psi(x_{1}),\ldots,\psi(x_{n}))(g_{1},\ldots,g_{n}))\in
G_{w}.$ ∎

Let $End(F_{n})$ denote the set of all endomorphisms of $F_{n}.$ Then $w\in
F_{n}$ is achiral if and only if there exist $\sigma\in End(F_{n})$ such that
$\sigma(w)=w^{-1}.$ The following observations are immediate from the
Proposition 2.

######  Corollary 3.

Let $w\in F_{n}.$

  1. 1.

If $w$ is achiral, then $\sigma(w)$ will also be achiral for all $\sigma\in
Aut(F_{n}).$

  2. 2.

$w^{n}$ is achiral for all $n\in{\mathbb{Z}}.$

  3. 3.

If $w^{n}$ for $n\in{\mathbb{Z}}\setminus\\{0\\}$ is achiral then so is $w$.

Proof of Part 1.

Since $w$ is achiral, there exists an endomorphism $\phi$ taking $w$ to
$w^{-1}$. Then the endomorphism defined by $\sigma\circ\phi\circ\sigma^{-1}$
takes $\sigma({w})$ to $\sigma(w)^{-1}.$

Proof of Part 2.

Let $f:F_{n}\to F_{n}$ be an endomorphism such that $f(w)=w^{-1}$. Then
$f(w^{k})={f(w)}^{k}=w^{-k}\;\;\forall k\in{\mathbb{Z}}.$

Proof of Part 3.

Let $\phi$ be endomorphism inverting the word $w^{n}$. Suppose $\phi(w)=u$,
then we have $\phi(w^{n})=u^{n}=w^{-n}$. Hence $u=w^{-1}$.

We have seen that the automorphic image of an achiral word will always be
achiral. Cocke and Ho [5] proved that a homomorphic image of achiral group
will always be achiral group. But a homomorphic image of achiral word need not
be achiral. In particular, let $w$ be a chiral word, then we can construct a
required homomorphism $\phi$ such that $\phi(x_{1})=w.$ The next result states
that a word being achiral is independent of free groups.

######  Proposition 4.

Let $w\in F_{m}\subseteq F_{n}$ for $m\leq n.$ If $w$ is achiral (chiral) in
$F_{m}$ if and only if $w$ is achiral (chiral) in $F_{n}$.

###### Proof.

We will prove for achiral, chiral part will automatically follow. Let
$x_{1},\ldots,x_{m}$ be a basis of $F_{m}$ and $x_{1},\ldots,x_{n}$, extending
the basis of $F_{m}$, be the chosen basis of $F_{n}$. Let $i$ be the inclusion
map $F_{m}\overset{i}{\hookrightarrow}F_{n}$ and $\alpha$ be the surjection
map $F_{n}\overset{\alpha}{\twoheadrightarrow}F_{m}$ defined by the map
$\alpha(x_{i})$ goes to $x_{i}$ if $1\leq i\leq m$, $1$ otherwise.

Suppose $w$ is achiral in $F_{m}.$ Then there exists an endomorphism $\phi$ of
$F_{m}$ such that $\phi(w)=w^{-1}.$ We can extend this endomorphism to an
endomorphism $\bar{\phi}$ of $F_{n}$ sending each basis element which is not
in $F_{m}$ to identity. Thus we have $\bar{\phi}(w)=w^{-1}.$

Let $w$ be achiral in $F_{n}.$ Then there exists an endomorphism $\psi$ of
$F_{n}$ such that $\psi(w)=w^{-1}.$

$F_{m}{\overset{i}{\hookrightarrow}}F_{n}{\overset{\psi}{\rightarrow}F_{n}\overset{\alpha}{\twoheadrightarrow}}F_{m}$

The composition of these maps $\alpha\circ\psi\circ i$ is an endomorphism of
$F_{m}$ such that $\alpha\circ\psi\circ i(w)=w^{-1}.$ ∎

For a shorter proof one can use chiral words.

######  Remark 5.

In fact the proof of above proposition also suggests that if there exists an
automorphism of $F_{m}$ inverting the word $w$ if and only if there exists an
automorphism of $F_{n}$ inverting the word $w.$

######  Corollary 6.

There exists an automorphism inverting the following words.

  1. 1.

Every palindromic word is achiral.

  2. 2.

Every word of type $x_{1}^{m_{1}}x_{2}^{m_{2}}$, where
$m_{1},m_{2}\in{\mathbb{Z}}$, is achiral.

  3. 3.

Every word of type
$x_{1}^{m}x_{2}^{\epsilon_{1}}x_{1}^{n}x_{2}^{\epsilon_{2}}$ where
$m,n,\epsilon_{i}\in{\mathbb{Z}}$ and $\epsilon_{1}=\pm\epsilon_{2}$ is
achiral.

Proof of Part 1.

Let $w\in F_{n}$ be a palindromic word. Take the automorphism generated by
$f(x_{i})={x_{i}}^{-1}$ for $i\in\\{1,2,\ldots,n\\}.$ Then $f(w)=w^{-1}.$

Proof of Part 2.

Here we prove the existence of automorphism inverting the word for $F_{2}$.
Hence by remark of previous proposition the result holds. It is enough to
prove for $x_{1}^{m_{1}}x_{2}^{m_{2}}$. Take automorphism $f$ generated by
sending $x_{1}$ to $x_{2}^{-m_{2}}x_{1}^{-1}x_{2}^{m_{2}}$ and $x_{2}$ to
$x_{2}^{-1}$. Then
$f(x_{1}^{m_{1}}x_{2}^{m_{2}})=x_{2}^{-m_{2}}x_{1}^{-m_{1}}.$

Proof of Part 3.

Here we prove the existence of automorphism inverting the word for $F_{2}$.
Hence by remark of previous proposition the result holds. When
$\epsilon_{1}=\epsilon_{2}$, take automorphism generated by sending $x_{1}$ to
$x_{1}^{-1}$ and $x_{2}$ to $x_{1}^{m}x_{2}^{-1}x_{1}^{-m}$. If
$\epsilon_{1}=-\epsilon_{2}$, take automorphism generated by sending $x_{1}$
to $x_{1}^{-1}$ and $x_{2}$ to $x_{1}^{m}x_{2}x_{1}^{-m}.$

Makanin [11] has proved that whether a system of equations over a free group
has a solution or not is algorithmically decidable. The following result shows
that the achirality of a word is a decidable problem.

######  Lemma 7.

Showing achirality of a word $w$ in a free group $F_{n}$ is equivalent to
solving an equation over $F_{n}.$

###### Proof.

Let $w=\prod_{j=1}^{l}x_{i_{j}}^{t_{j}}\in F_{n}$ be achiral. From the
Proposition 2 there exist an endomorphism $\phi$ of $F_{n}$ such that
$\phi(w)=w^{-1}.$ Since endomorphism of a free group is determined by the
images of its generators, let $\phi(x_{i})=w_{i}$ So
$\phi(w)=\prod_{j=1}^{l}w_{i_{j}}^{t_{j}}$. Thus $w$ is achiral if and only if

$\prod_{j=1}^{l}x_{i_{j}}^{t_{j}}\prod_{j=1}^{l}w_{i_{j}}^{t_{j}}=1.$

Hence checking a given word $w\in F_{n}$ is achiral is equivalent to solving
an equation in $F_{n}.$ ∎

###  2.1 Achiral words in $F_{2}$

Let $F_{2}$ be a free group of rank $2$ with alphabet $X=\\{x_{1},x_{2}\\}.$
If $w\in F_{2},$ then $w$ is reduced word with letters
$x_{1},x_{2},x_{1}^{-1},x_{2}^{-1}.$ We denote the length of word $w$ by
$\ell(w).$ Let ${\mathcal{W}}_{n}$ denote the set of all words of length $n$
i.e.,

${\mathcal{W}}_{n}=\\{w\in F_{2}|\ell(w)=n\\}.$

It is easy to see that $|{\mathcal{W}}_{n}|=4\cdot 3^{n-1}.$ For example,

${\mathcal{W}}_{1}=\\{x_{1},x_{2},x_{1}^{-1},x_{2}^{-1}\\}.$
${\mathcal{W}}_{2}=\\{x_{1}^{2},x_{2}^{2},x_{1}^{-2},x_{2}^{-2},x_{1}x_{2},x_{1}^{-1}x_{2}^{-1},x_{2}x_{1},x_{2}^{-1}x_{1}^{-1},x_{1}^{-1}x_{2},x_{1}x_{2}^{-1},x_{2}^{-1}x_{1},x_{2}x_{1}^{-1}\\}.$

Let

${\mathcal{W}}^{n}=\\{w\in{\mathcal{W}}_{n}|\exists\sigma\in\;Aut(F_{2}),\sigma(w)=w^{-1}\\}.$

Hence ${\mathcal{W}}^{n}\subseteq A_{2}.$ Before we proceed, given an
automorphism $f\in\;Aut(F_{2})$, we define

${\mathcal{W}}_{f}^{n}=\\{w\in{\mathcal{W}}^{n}|f(w)=w^{-1}\\}.$

If
$f(x_{1})=x_{1}^{-1},f(x_{2})=x_{2}^{-1};g(x_{1})=x_{2}^{-1},g(x_{2})=x_{1}^{-1}$
and $h(x_{1})=x_{2},h(x_{2})=x_{1},$ then it is easy that

${\mathcal{W}}_{f}^{2}=\\{x_{1}^{2},x_{2}^{2},x_{1}^{-2},x_{2}^{-2}\\},{\mathcal{W}}_{g}^{2}=\\{x_{1}x_{2},x_{1}^{-1}x_{2}^{-1},x_{2}x_{1},x_{2}^{-1}x_{1}^{-1}\\},{\mathcal{W}}_{h}^{2}=\\{x_{1}^{-1}x_{2},x_{1}x_{2}^{-1},x_{2}^{-1}x_{1},x_{2}x_{1}^{-1}\\}.$

Thus ${\mathcal{W}}_{2}={\mathcal{W}}^{2}.$ That is every word of length $2$
over $F_{2}$ is achiral. From the observations listed in Corollary 3 and 6, it
is easy to see that ${\mathcal{W}}_{i}={\mathcal{W}}^{i}$ for
$i\in\\{1,2,3,4,5\\}$ and $W_{6}=A_{6}$ but the word
$x_{1}^{2}x_{2}^{2}x_{1}x_{2}^{-1}\in{\mathcal{W}}_{6}\setminus{\mathcal{W}}^{6}$
and it is an achiral word of shortest length such that it is not inverted by
any automorphism (see [7], Remark 3.7). In short, Our result above with
Rhemtulla’s result are enough to prove that all words of length $6$ are
achiral.

###  2.2 Test word and achiral word

Let $w\in F_{n}.$ We define

$End_{w}=\\{f\in End(F_{n})|f(w)=w^{-1}\\}\;\;\;Aut_{w}=\\{f\in
Aut(F_{n})|f(w)=w^{-1}\\}.$

It is clear that $Aut_{w}\subseteq End_{w}.$ From Proposition 2 we have $w\in
A_{n}$ if and only if $End_{w}\neq\emptyset.$ Let $w\in F_{n}$ be a primitive
word. Then we can construct an endomorphism $\sigma$ satisfying $\sigma\in
End_{w}\setminus Aut_{w}.$ As discussed earlier if $w=x^{2}y^{2}xy^{-1},$ then
$w\in A_{2}$ that is $End_{w}\neq\emptyset$ but $Aut_{w}=\emptyset.$ Let
$D_{n}=\\{w\in A_{n}|End_{w}=Aut_{w}\\}.$

######  Definition 8.

A word $w\in F_{n}$ is a test word if every endomorphism which fixes $w$ is an
automorphism.

We denote set of all test words of $F_{n}$ as $T_{n}$ that is

$T_{n}=\\{w\in F_{n}|\;\mbox{if $\sigma\in End(F_{n})$ with $\sigma(w)=w,$
then $\sigma\in Aut(F_{n})$}\\}.$

Turner [17] gave a criterion to check a given word is a test word or not.

######  Definition 9.

A subgroup $H$ of $G$ is said to be retract if there exists an endomorphism
$\phi$ such that $\phi:G\longrightarrow H$ with the property $\phi(h)=h$ for
all $h\in H.$

######  Lemma 10 (Turner [17]).

A word $w\in F_{n}$ is a test word if and only if it is not in any proper
retract of $F_{n}$.

Turner also showed that $T_{2}={F_{2}}^{\prime}\setminus\\{e\\}.$

######  Theorem 11.

If $n\in{\mathbb{N}},$ then $T_{n}\cap A_{n}=D_{n}.$

###### Proof.

Let $w\in(T_{n}\cap A_{n})\setminus D_{n}.$ Then there exists $\phi\in
End_{w}\setminus Aut_{w}.$ Which leads to a contradiction that $\phi^{2}$
fixes $w$ hence $\phi\in Aut_{w}.$ Hence $(T_{n}\cap A_{n})\setminus
D_{n}=\emptyset$ or equivalently $D_{n}\subseteq T_{n}\cap A_{n}.$ Let $w\in
T_{n}\cap A_{n}$ and $\phi\in End_{w}.$ Then $\phi^{2}$ fixes $w.$ Hence
$\phi^{2}\in Aut(F_{n})$ so is $\phi.$ Thus $\phi\in Aut_{w}.$ ∎

The above theorem suggests that all words in $T_{n}\setminus D_{n}$ are
chiral. Cocke and Ho has given an explicit example of chiral word
$[x^{440}(x^{440})^{(y^{440})}x^{440},(y^{440})^{(x^{440}y^{440})}y^{440}]$
which lies inside $T_{2}\setminus D_{2}$ in $F_{2}$. They asked whether engel
words $e_{n}=[x,_{n}y]$ are achiral or not. From Theorem 11 it is enough to
check for an automorphism inverting $e_{n}$. For this case, we have
Whitehead’s algorithm [18].

It is easy to see that $B_{1}=\\{x,x^{-1}\\}$ and
$D_{1}=T_{1}=F_{1}\setminus\\{e\\}.$ But if $n\geq 2,$ then we have $B_{n}\cap
T_{n}=\emptyset.$ As let $w\in B_{n}.$ Take $H=\langle w\rangle.$ We will show
that $H$ is a proper retract of $F_{n}$. Hence $w$ can’t be a test word. Write
$w=x_{1}^{r_{1}}\ldots x_{n}^{r_{n}}c$ where $c\in{F_{n}}^{\prime}$. Since
$w\in B_{n},$ we have $gcd(r_{1},\ldots,r_{n})=1.$ Hence there exists
$m_{1},\ldots,m_{n}$ such that $r_{1}m_{1}+\ldots+r_{n}m_{n}=1.$ Define a
homomorphism $\phi:F_{n}\longrightarrow H$ generated by
$\phi(x_{i})=w^{m_{i}}$ for all $i\in\\{1,\ldots,n\\}.$ So we get $\phi(w)=w.$
Hence $H$ is a proper retract of $F_{n}.$ So by Turner’s result, $w$ can not
be test word.

## References

  * [1] S. J. Benkoski, The probability that k positive integers are relatively r-prime, Journal of Number Theory, 8 (1976), pp. 218–223.
  * [2] J. N. Bray, J. S. Wilson, and R. A. Wilson, A Characterization of Finite Soluble Groups by Laws in Two Variables, Bulletin of the London Mathematical Society, 37 (2005), pp. 179–186.
  * [3] A. Carolyn, Fibres of words in finite groups, a probabilistic approach, Ph.D thesis, 2012.
  * [4] W. Cocke, Idle Words:Word Maps on Finite Groups, Ph.D thesis, 2019.
  * [5] W. Cocke and M.-C. Ho, On the symmetry of images of word maps in groups, Communications in Algebra, 46 (2018), pp. 756–763.
  * [6] , The probability distribution of word maps on finite groups, Journal of Algebra, 518 (2019), pp. 440–452.
  * [7] G. Giles, Encoding and Detecting Properties in Finitely Presented Groups, Ph.D thesis, 2017.
  * [8] S. Jambor, M. W. Liebeck, and E. A. O’Brien, Some word maps that are non-surjective on infinitely many finite simple groups, Bulletin of the London Mathematical Society, 45 (2013), pp. 907–910.
  * [9] M. Kassabov and N. Nikolov, Words with few values in finite simple groups, The Quarterly Journal of Mathematics, 64 (2012), pp. 1161–1166.
  * [10] A. Lubotzky, Images of word maps in finite simple groups, Glasgow Mathematical Journal, 56 (2014), p. 465–469.
  * [11] G. S. Makanin, Equations in a free group, Mathematics of the USSR-Izvestiya, 21 (1983), p. 483.
  * [12] M.Levy, Word maps with small image in simple groups, arXiv preprint: arXiv:1206.1206, (2012).
  * [13] N. Nikolov and D. Segal, On finitely generated profinite groups I: strong completeness and uniform bounds, Annals of Mathematics, 165 (2006), pp. 171–238.
  * [14] N. Nikolov and D. Segal, A characterization of finite soluble groups, Bulletin of the London Mathematical Society, 39 (2007), pp. 209–213.
  * [15] O. Ore, Some remarks on commutators, Proceedings of the American Mathematical Society, 2 (1951), pp. 307–314.
  * [16] A. H. Rhemtulla, A problem of bounded expressibility in free products, Mathematical Proceedings of the Cambridge Philosophical Society, 64 (1968), p. 573–584.
  * [17] E. C. Turner, Test Words for Automorphisms of Free Groups, Bulletin of the London Mathematical Society, 28 (1996), pp. 255–263.
  * [18] J. H. C. Whitehead, On equivalent sets of elements in a free group, Annals of Mathematics, 37 (1936), p. 782.