11affiliationtext: Department of Physics and Astronomy, University of
Sheffield, Sheffield S3 7RH, UK22affiliationtext: Joint Quantum Institute,
University of Maryland, College Park, MD 20742, USA33affiliationtext: Centre
for Nanophotonics, Department of Physics, Engineering Physics and Astronomy,
Queen’s University, Kingston, Ontario, Canada K7L 3N644affiliationtext: EPSRC
National Epitaxy Facility, University of Sheffield, Sheffield S1 4DE,
UK**affiliationtext<EMAIL_ADDRESS>these authors
contributed equally to this work

# Topological and conventional nano-photonic waveguides for chiral integrated
quantum optics

N.J Martin  M. Jalali Mehrabad  X. Chen  R. Dost  E. Nussbaum  D. Hallett  L.
Hallacy  A. Foster  E. Clarke  P.K. Patil  S. Hughes  M. Hafezi  A.M Fox  M.S.
Skolnick  L.R. Wilson

(August 27, 2024)

###### Abstract

Chirality in integrated quantum photonics has emerged as a promising route
towards achieving scalable quantum technologies with non-linearities at the
single photon level. Topological photonic waveguides have been proposed as a
novel approach to harnessing such chiral light-matter interactions on-chip.
However, uncertainties remain regarding the strength of the chiral coupling of
embedded quantum emitters to topological waveguides in comparison to
conventional line defect waveguides. In this work, we present an investigation
of chiral coupling in a range of waveguides using a combination of
experimental, theoretical, and numerical analyses. We quantitatively
characterize the position-dependence of the light-matter coupling on several
topological photonic waveguides, and benchmark their chiral coupling
performance against conventional line defect waveguides. We conclude that
topological waveguides under perform, in comparison to conventional line
defect waveguides, casting their directional optics credentials into doubt. To
demonstrate this is not a question of the maturity of the field, we show that
state of the art inverse design methods, while capable of improving the
directional emission of these topological waveguides, still places them
significantly behind the operation of a conventional (glide-plane) photonic
crystal waveguide. Our results and conclusions pave the way towards improving
the implementation of quantitatively-predicted quantum nonlinear effects on-
chip.

##  1 Introduction

Integrated nano-photonic platforms, in which embedded quantum emitters are
interfaced with optical waveguides and cavities on-chip, are a promising route
towards scalable quantum technologies. An attractive property of nanophotonic
waveguides is their support for chiral light-matter interactions, whereby an
emitter with a circularly polarised transition dipole moment couples
unidirectionally at the single photon level to a single photonic waveguide
mode [1, 2, 3, 4]. Such interactions have previously been demonstrated on-chip
using semiconductor quantum dots (QDs) coupled to photonic crystal (PhC) line
defects such as W1 [3] and glide plane [2, 5, 6] waveguides.

Recently, PhC topological waveguides have received significant interest for
integrated nanophotonics due to their attractive properties, which include
robust transmission around tight bends [7, 8, 9, 10, 11, 12, 13, 14, 15],
opening up their applications to more complex guided structures, such as ring
resonators, beam splitters and filters that require sharp bends [16, 17, 18,
19].

More pertinently for chiral quantum optics, topological waveguides were
envisioned to have two key properties, 1. robustness to in-plane back-
scattering and 2. intrinsic unidirectional emission for embedded emitters,
with the unidirectional emission expected to arise from the intrinsically
helical edge modes which arise at the interface between two topologically-
distinct PhCs [7, 20, 15].

Despite intense research on the application of these systems, experimental
measurement of the degree of robustness to back-scattering of these waveguides
was only very recently examined using external light sources [21], the results
casting considerable doubt to the resistance to back scattering. However, an
experimental investigation into the envisioned property of intrinsic
unidirectional emission of topological waveguides, key to their chiral quantum
optics applications, remains missing.

Figure 1: (a-e) (i), schematics of the (a) W1 waveguide, (b) Glide plane
waveguide, (c) Zig-Zag interface VH topological waveguide, (d) Un-optimised
Bearded interface VH topological waveguide and (e) a Bearded interface VH
topological waveguide optimised using an inverse design technique for a more
favourable band structure and improved electric field and $S_{3}$ overlap.
(a-e) (ii) simulated band structures of the waveguides, with the single mode
region of interest highlighted in blue, and the specific frequency of single
mode operation chosen for the electric field plots in the rest of the figure
highlighted on the band structures in red. (a-e) (iii) Simulated electric
field profiles and (a-e) (iv) $S_{3}$ maps for the waveguides. Within (a-e)
(iv), encircled white regions indicate regions where chiral contrast is
expected to be 95% and above. Band structures and electric field plots were
simulated by guided mode expansion. (a-e) (v) Shows the spatial dependence
within the waveguides of the $\beta$-factor at the same frequency as the
electric field plots, simulated using FDTD. Black regions indicate the
position of the waveguide apertures. (a-e) (vi) Probability density plots
showing the likelihood of a set of randomly positioned dots possessing a given
$S_{3}$ value for different electric field strengths within the waveguide
interface. Areas of high probability density indicate combinations of chiral
contrasts and electric field strengths that are more likely. The apertures of
the waveguides are excluded from the calculations.

As in conventional, non-topological waveguides, chiral coupling of an embedded
emitter to a topological photonic waveguide is position dependent. In the most
extreme case, the direction of emission from a circularly polarised transition
can be completely reversed by moving it within a unit cell of the PhC lattice.
While the properties of conventional waveguides and topological waveguides
have been compared in simulations [22], here we use both simulations and
experiment to compare these approaches to realising chiral light-matter
interactions on-chip.

Within the main text we present data on conventional W1, glide plane, zig-zag
interface valley-Hall (VH) and bearded interface VH waveguides. In addition we
include a novel version of the bearded interface VH waveguide, optimised using
inverse design techniques. (See supplementary information for a discussion of
slab waveguides and an optimised glide plane waveguide[6] waveguides) For each
waveguide we present the dispersion relation of the guided mode, the spatial
electric field intensity and degree of circular polarisation as well as the
spatial and wavelength dependence of the $\beta$-factor (see section 3.1), for
emitters within the waveguides. We use this simulated data to make predictions
for the expected distribution of the chiral contrast of embedded QDs and then
determine experimentally the chiral contrast for a large number of QDs in each
type of structure, showing good agreement with the simulations. While the
topological waveguides, optimised through inverse design, demonstrate
promising improvements in chiral properties in simulation, our experimental
results also serve to highlight the limitations of current approaches. In
particular we stress the need to develop waveguides which support high $\beta$
factors whilst simultaneously showing near-unity chiral coupling.

##  2 Photonic crystal waveguide designs for chiral quantum optics

The PhC waveguides considered in this work are described schematically in Fig
1 (a-e). We consider two conventional (i.e. non-topological) and three non-
trivial topological designs. For each type of waveguide we show:

  1. i

The waveguide design.

  2. ii

The band structure.

  3. iii

The electric field profile.

  4. iv

The chirality ($S_{3}$).

  5. v

The $\beta$-factor.

  6. vi

$S_{3}$ and E-Field probability density plots.

For each type of waveguide we focus on the parts of the band structure shaded
in blue, and the field profiles in panel (iii-v) are calculated for the points
shown by the red circle in panel (ii). The first conventional design (Shown in
Fig 1(a)) is that of a W1 waveguide, comprising a triangular lattice of
circular holes etched into a thin dielectric membrane, with one row of holes
omitted in the $\Gamma$-K direction to form a line defect. Shown in Fig 1 (a)
(ii), several guided Bloch modes can be observed within the PhC bandgap of the
W1 waveguide; we focus on the lowest frequency mode, with the field profile
shown in Fig 1 (a) (iii), which has an electric field antinode at the centre
of the waveguide.

The second conventional design studied here is the glide plane waveguide,
which is formed by displacing the holes on one side of a W1 waveguide by half
the lattice period along the waveguide. The TE like mode dispersion diagram
for the glide plane structure is shown in Fig 1 (b) (i). We focus on the
single mode region of the higher frequency mode, as the multimode region where
modes the two modes overlap spectrally prevents chiral coupling being realised
[23]. The field profile for this mode is shown in Fig 1 (b) (iii).

The conventional waveguides introduced above are compared in this work with
three topologically non-trivial structures. We have chosen valley-Hall
waveguides as their interfaces support guided optical modes lying below the
light line, unlike the alternative spin-Hall approach [7, 17, 24]. VH
waveguides can be formed by interfacing two VH photonic crystals where the
unit cell is a rhombus containing a pair of apertures of differing sizes, in
two distinct ways. The first, the zig-zag interface, is an interface of these
two crystals with mirror symmetry. For this work, a design for the zig-zag
interface comprised of triangular apertures, with the larger triangles at the
interface. This design is chosen for its favourable band structure, with the
dispersion diagram in Fig 1 (c) (ii) showing that the waveguide supports a
single guided TE mode, with the mode profile given in Fig 1 (c) (iii). The
bearded interface in contrast, is formed by interfacing two valley hall
photonic crystals such that it forms a glide-plane symmetric interface. For
this work, the original bearded interface waveguide is formed of circular
apertures, with the smaller of the apertures at the interface. We have also
included an optimised version of the bearded interface within our
investigation. Here, an inverse design algorithm was used to optimize the
geometrical parameters of a bearded interface VH nano-photonic waveguide for
high $\beta$-factors and strong chiral light-matter interactions (see
supplementary section S1 for more details on the inverse design process)

##  3 Simulation of the waveguide $\beta$-factor, E-field properties and
chirality

Figure 2:  (a) Schematic diagram showing the positional dependence of the
direction of emission of a circularly polarised dipole source within a VH
waveguide; labels show the polarisation of the local field at the different
positions, with the waveguide interface highlighted in magenta. (b) Level
diagrams and optical transitions for an neutral exciton QD state showing left
and right circularly polarised transitions. (c) $|E|^{2}$ field profile within
the VH waveguide interface. (d) $S_{3}$, degree of circular polarisation at
different points within the VH waveguide interface. Encircled white regions
indicate regions where $|S_{3}|\geq 0.95$ (e) FDTD simulations showing the
positional dependence of emission from a circularly polarised dipole at points
1-5 within the ziz-zag VH waveguide interface. The top of the figure shows the
position of the waveguide interface between two topologically distinct
photonic crystals (shown in orange and blue) as well as the central position
of the right circularly polarised dipole source used in the simulations.

This section presents simulation results, that quantify the spatial dependence
of the $\beta$-factor, E-field and chiral contrast. This includes a
description of the model used to predict the chiral statistics i.e. the
expected statistical distribution of chiral contrast for a large number of
randomly positioned QDs.

###  3.1 $\beta$-Factor

We define our $\beta$-factor as the ratio of the spontaneous emission rate of
an emitter into an intended guided mode compared to the total decay rate. To
calculate the $\beta$-factor we used finite-difference-time-domain (FDTD)
simulations to calculate the fraction of radiative power coupled into
propagating modes over the total power injected from the emitter. To determine
the spatial distribution of the $\beta$-factor, we vary the dipole position
within a region from approximately -2a to +2a from the waveguide centre,
excluding regions within the apertures of the waveguides (see supplementary
section S3-5 for more information on the calculation of the $\beta$ factor as
well as its spectral dependence).

The relationship between the intensity of the electric field at a given point
to the $\beta$-factor at that point can be seen by comparing the electric
field profiles of Fig 1 (a-e) (iii), and the $\beta$-factors of Fig 1 (a-e)
(v), with areas of high electric field intensity resulting in higher
$\beta$-factors. However, the enhancement of the $\beta$-factor arising from
the slow light regions of the waveguides can result in regions with relatively
low electric field intensities having high $\beta$-factors, with this effect
most notable within the W1. This wavelength dependence of the $\beta$-factor
is shown within the SI and offers waveguides with poor E-field and $S_{3}$
overlap a route to high $\beta$-factor at chiral points.

###  3.2 Stokes $S_{3}$ parameter and chiral contrast

The chiral properties of each waveguide were evaluated using guided mode
expansion simulations [25] by calculating the Stokes $S_{3}$ parameter defined
as:

$S_{3}=\frac{-2\textrm{Im}(E_{x}E_{y}^{*})}{\lvert E_{x}\rvert^{2}+\lvert
E_{y}\rvert^{2}}.$ (1)

The simulated Stokes $S_{3}$ parameter is shown in Fig 1 (a-e) (iv) for the
five waveguides. The Stokes parameter at the position of a QD determines the
chiral contrast of the emission. Fig 2 shows the relationship between the
$S_{3}$ parameter and the direction of emission of a circularly polarised
emitter, such as the Zeeman split lines of a neutral excition within a QD (Fig
2 b). The chiral contrast is defined as:

$C=\frac{T^{R}-T^{L}}{T^{R}+T^{L}},$ (2)

where $T^{R}$ and $T^{L}$ are the transmission right and left through a
waveguide respectively for an emitter embedded within a linear waveguide.
Within the FDTD simualtions of Fig 2, the chiral coupling of a circularly
polarised emitter (within the example of the VH zig-zag interface) is shown to
be position dependent, due to the different $S_{3}$ values at different points
within the waveguide. While all five waveguides show regions of high chiral
contrast, with contrasts above 95% indicated by the in-circled white regions
of Fig (a-e) (iv), S3 does not give the complete picture of a waveguide’s
suitability to chiral quantum optics, as discussed in the next section.

###  3.3 Coincidence of high chirality and E-field concentration

The likelihood of a set of randomly positioned QDs possessing a given $S_{3}$
value for different electric field strengths within the waveguide interface is
shown by the probability density plots of Fig (a-e) (iv). Areas of high
probability density indicate combinations of chiral contrasts and electric
field strengths that are more likely; the air regions of the waveguides where
there are no QDs are excluded from the calculations. Within these plots we see
that the glide plane waveguide has the best overlap of $S_{3}$ and $|E|^{2}$.
The W1 waveguide shows a high concentration of QDs at high $S_{3}$ for
$|E|^{2}=0.7$, which outperforms the distributions seen for the topological
waveguides, characterised by low density at the intersection of high $S_{3}$
and $|E|^{2}$ found in the dark regions at the top right of the plots. The
optimised bearded interface does, however, perform better than the un-
optimised version in this regard, with the optimised waveguide showing a
greater concentration of high chirality QDs at $|E|^{2}=0.3$.

Figure 3: SEM images of the fabricated waveguide for (a) W1, (b) Glide plane,
(c) zig-zag VH, (d) original bearded VH and (e) optimised bearded VH. Scale
bars are 0.4$\mu m$. Examples of dot lines split by B$\neq$0 magnetic fields,
with (f) high chiral contrast, (g) medium contrast, (h) low contrast and (i)
asymmetric contrast. (j) Schematic of the on chip device layout, indicating
the location of the left out-coupler collection (red) and the right out-
coupler collection (blue)

###  3.4 Simulated Chiral Statistics

The chiral statistics of the waveguides can be modelled by considering the
$S_{3}$ field maps for the waveguides over the selected waveguide regions
highlighted in blue in Fig 1 (a-e) (ii). By calculating the electric field
data using guided mode expansion for the middle of the slab (where the QDs are
located) for multiple points within a 10a super cell of the waveguide, the
distribution of the chirality can be calculated. With FDTD simulations
providing data for the $\beta$-factor (see Fig 1 (a-e) (v)) and Purcell
factor, a threshold was applied to the points that were included within the
chiral statistics. This threshold was set so that only points within the
waveguide that had a combined value for Purcell factor multiplied by
$\beta$-factor above 0.5 were included ( $F_{P}\cdot\beta\geq 0.5$).
Additionally, to account for the ‘dead-zones’ for QD emission that exist
around the apertures of the waveguides. The extend of this dead-zone can
depend on the wafer properties and the presence of fabrication processes such
as surface passivisation [26], for this reason we have calculated the expected
statistics for the waveguides excluding a region of both 15 and 30 nm [27]
around the apertures. The absence of QD emission in proximity to the etched
surfaces is likely to arise from detrimental effects of surface defects and
interface roughness. The simulations of the chiral statistics for the five
types of waveguides are shown in Fig 4(a)-(e). These results will be discussed
in the next section when comparing to the experimental data.

Figure 4: (a-e) Modeled predictions for the chiral statistics of the
waveguides for a 0nm and 15nm and 30nm dead-zone with the plots normalised to
the 0nm case to show the reduction in the expected number of dots that arise
from these dead-zones. (f-j) Chiral contrast measured experimentally for
randomly positioned QDs within the different waveguides. Left out-coupler data
is presented in red and Right out-coupler data in blue. All experimental data
were recorded when an external magnetic field of $B_{Z}$=3T was applied. The
grey background plot shows a smooth distribution of the chirality data where
no distinction between the results of the left and right out-couplers are
made.

##  4 Method

###  4.1 Experimental comparison of the photonic crystal waveguides

The experimental results were obtained for devices fabricated in a 175nm-thin
GaAs p-i-n membrane containing a single layer of InAs QDs. Representative
scanning electron microscope (SEM) images of the waveguide interfaces are
shown in Fig 3 (a-e). Each waveguide was terminated on both ends with a
grating coupler for light extraction into external optics, as shown in 3 (j).
The sample was cooled to 4.2K in a superconducting magnet cryostat. To
determine the operation bandwidth of each waveguide, broadband
photoluminescence (PL) was generated by exciting the ensemble of QDs located
within one grating coupler using non-resonant excitation
($\lambda_{\textrm{laser}}=810$nm). PL emission was then detected from the
other outcoupler, with representative transmission spectra shown in SI Fig S2.
A combination of the identification of a sharp decrease in transmission
resulting from the band edge of a guided mode and an analysis of the Fabry-
Pérot fringes of the waveguide were used to identify the single mode regions
of interest. (More information on the identification of the single mode
regions of the waveguides can be found in the supplementary section S2).

Next, we measured the chiral contrast independently for a large number of
individual QDs in each waveguide. To do so, we used low power micro-
photoluminescence ($\mu\mathrm{PL}$) measurements, exciting non-resonantly
from above the waveguide and collecting emission independently from the two
out-couplers. We focused on QDs spectrally located in the single mode regions
of the waveguides, highlighted in blue in Fig 1 (a-e). In the presence of a
Faraday-geometry magnetic field, the circularly polarised dipole transitions
of the QD split energetically, allowing for their spectral resolution. To
allow for the possibility of dissimilar out-coupler collection efficiency, the
chiral contrast was evaluated independently for each out-coupler using the
relationship:

$C_{i}=\frac{I_{i}^{\sigma^{+}}-I_{i}^{\sigma^{-}}}{I_{i}^{\sigma^{+}}+I_{i}^{\sigma^{-}}},$
(3)

where $I_{i}^{\sigma^{j}}$ ($j=+,-$ ) represents the intensity of $\sigma^{j}$
polarised light emitted by the QD and collected from OC $i$ (= left, right).

###  4.2 Experimental chiral statistics

We quantify the degree of chiral coupling in each nano-photonic waveguide
experimentally, using randomly distributed self-assembled InAs QDs,
distributed by growth in the $xy-$plane at $z=0$ of the GaAs membrane (see Fig
3 (j) for axes). The guided PL of single QDs is detected from right and left
out-couplers (denoted as Left OC and Right OC in Fig 4 (f-j)) at opposite ends
of each waveguide. Self-assembled growth leads to random positions of QDs
within the xy-plane, which leads to coupling of QDs placed at areas of the
waveguide with a range of chiral contrast. The results are shown in histogram
graphs for the experimentally measured chiral contrasts for each case. The
binned histograms from the $S_{3}$ parameters calculated via FDTD simulations,
as discussed in section 3.4, are also shown for comparison.

For the W1 waveguide we see that, in general, there is predicted to be a low
proportion of QDS with a chirality > 80%, but a high proportion of dots with
low chiral contrast. This can be explained by both the concentration of the
electric field being at points of low circular polarisation, as seen in Fig 1
(f,k), and the annihilation of chiral points at the band-edge in a W1
waveguide [28]. The experimental results broadly agree with this prediction.

The glide-plane PhC waveguide exhibits the best chiral coupling for QDs in
both simulation and experiment. Nevertheless there is a lower proportion of
high contrast QDs in experiment than expected. This may be explained by the
glide plane’s high chiral contrast regions being located near the waveguides’
etched areas.

The topological VH Zig-Zag interface achieves a high proportion of QDs with
high contrast, but not as high as the glide plane, and as can be seen in Fig 1
(e) (vi), these high contrast dots are unlikely to be at points of high
electric field concentration. The original bearded interface, however, while
predicted to have a better distribution than the W1 waveguide, performs the
worst out of all of the waveguides in experiment. This is likely related to
the interface holes in the design being a source of fabrication error due to
their very small size, and also to the surface proximity issues that they
introduce beyond those included in our model. The optimised bearded VH
waveguide shows a significant improvement in the experimental results in
comparison to its un-optimised counterpart, but not the improvement implied by
the simulation data. This is likely due a combination of two factors. The
first is the high group indexes of the ideal design, not being replicated in
experiment, resulting in a lower probability of high contrast QDs being
measured. The second is the interface holes again leading to a reduction in
the contrast for reasons described above.

A feature that is present within the statistics for all of the waveguides, and
has been seen in other experiments and QD wafers [3], is the asymmetry in the
chirality of the QDs measured from the left and right out-couplers of the
waveguide. Fig 3 (f-i) shows examples of QD spectra that fit closely with the
predictions of the FDTD simulations shown in Fig 1 (a). These spectra show the
expected behaviour of the spectra of the right out-coupler (red) and the left
out-coupler (blue) are symmetric, mirrored versions of each other, displaying
the same chiral contrast with opposite intensities. However, Fig 3 (f) shows
an example of asymmetric behaviour where the contrast as viewed from the left
out-coupler is greater than is seen in the right out-coupler. (A discussion on
the extent and nature of this asymmetry can be found in the SI section S6).

##  5 Conclusion

We conducted a comparative analysis of conventional and topological waveguides
for chiral coupling of embedded quantum emitters. Through a combination of
experimental characterization of chiral coupling and a simulation-based
analysis of electric field properties and $\beta$-factors, we have gained a
comprehensive view of the waveguides’ individual suitability for chiral
quantum optics applications. Among the waveguides examined, the glide plane
demonstrated the highest proportion of high chiral contrast QDs, making it
well-suited for achieving such QDs within a linear waveguide system with
randomly positioned QDs. In contrast, while the zig-zag interface VH waveguide
exhibited good experimental performance by producing a large portion of high
chiral contrast QDs, our modelling showed that regions with high chiral
contrast within the zig-zag waveguide are unlikely to have high
$\beta$-factors.

When considering more complex structures like ring resonators that require
robust transmission around tight bends, topological waveguides emerge as a
promising platform for chiral quantum optics. Despite the discussed
$\beta$-factor limitations of the zig-zag interface, it is still possible to
achieve high $\beta$-factor within topological waveguides by using slow light.
By combining the high $\beta$-factors resulting from the slow light region of
the inverse designed bearded VH waveguide with appropriate QD registration [3,
29, 30, 27] or site-controlled growth [31, 32] techniques, it may be possible
to achieve deterministic positioning of QDs at points within the waveguide
that exhibit both high chirality and high $\beta$-factors. This approach
presents a pathway towards achieving high beta-factor, high chiral contrast QD
emission within more complex photonic crystal device geometries.

Exciting future prospects of this research include the realization of
separation-independent QD-QD interactions [1, 33], super and subradiant many-
body states [34, 35, 36], and the formation of large-scale chiral spin
networks [37] using a conventional or topologically-protected photonic
platform. Spin [38] and vorticity [39] selective light-matter coupling in the
quantum Hall regime is also an intriguing new platform, in which the
topological interplay between light and matter can be explored.

## Acknowledgements

This work was supported by EPSRC Grant No. EP/N031776/1, EP/V026496/1, the
Quantum Communications Hub EP/T001011/1 and the natural sciences and
engineering council Canada (NSERC). The authors would like to acknowledge
helpful discussions with Edo Waks, Kartik Srinivasan, Nir Rotenberg, Peter
Millington-Hotze and Hamedreza Siampour.

## Author contributions statement

N.J.M., M.J.M., and E.N. designed the photonic structures, which R.D.
fabricated. E.C. and P.K.P. grew the sample. N.J.M., M.J.M., X.C., L.H., A.F.
and D.H. carried out the measurements and simulations. L.R.W., A.M.F, S.H.,
M.H., and M.S.S. provided supervision and expertise. N.J.M., M.J.M., X.C. L.H.
and E.N. wrote the manuscript, with input from all authors.

## Data Availability

Data supporting this study are openly available from the authors upon
reasonable request.

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Supplementary Materials

##  S1 Waveguide Schematics and optimisation

Figure S1: Schematics and parameters for the waveguides used in the main text.
(a) Schematic for the W1 waveguide where $t=0.763a_{\mathrm{W1}}$ and
$r=0.3a_{\mathrm{W1}}$. (b) Schematic for the glide-plane waveguide, where
$t=0.69a_{\mathrm{GP}}$ and $r=0.35a_{\mathrm{GP}}$. (c) Schematic for the
zig-zag VH waveguide, where $t=0.64a_{\mathrm{VH}}$,
$r_{1}=0.4a_{\mathrm{VH}}$ and $r_{2}=0.6a_{\mathrm{VH}}$. (d) Location and
radius of the 12 holes, closest to the interface in the initial bearded
interface VH waveguide design. (e) Location and radius of the 12 holes,
closest to the interface in the optimised bearded interface VH waveguide
design in terms of changes relative to the original waveguide. A dielectric
constant of $\epsilon=11.6$ was used throughout.

To optimise the bearded interface waveguide, an inverse design technique is
used, comprised of Figures of merit (FOM) used in combination to yield more
favourable properties for the topologically non-trivial mode including higher
group indexes within a single mode region, and better overlap of the electric
field and chiral points. A two step optimization is run to obtain the improved
design. In the first step $\mathcal{F}_{1}$ is maximized and in the second
step $\mathcal{F}_{2}$ is minimized. $\mathcal{F}_{1}$, is:

$\mathcal{F}_{1}={\rm max}(G_{S}M\cdot C\cdot F_{p}^{R})F_{\rm SM},$ (S1)

where $G_{S}M$ is a smooth step function (equal to 0 inside a hole edge and
equal to 1 more than 40 nm from the closest hole edge), $C$ is the chiral
contrast, $F_{p}^{R}=\Gamma^{f}/\Gamma^{\hom}$ is the Purcell factor, and
$F_{\rm SM}$ is a function that encourages a certain magnitude and position of
single mode bandwidth. The second phase of the optimization is focused on
improving the dispersion, with a figure of merit, $\mathcal{F}_{2}$, of:

$\mathcal{F}_{2}=\omega_{bb}-\omega_{{\rm SM},\max}+\frac{A}{\max_{\rm
SM}(n_{g})},$ (S2)

where $\omega_{bb}$ is the frequency where the topologically non-trivial band
first bends back, else, in the absence of a bend, it is the frequency of the
topologically non-trivial band at $k_{x}a=\pi$. The parameter $\omega_{{\rm
SM},\max}$ is the maximum frequency of the topologically non-trivial mode’s
lowest frequency single mode bandwidth, $A$ is a weight to balance the third
term, and $\max_{\rm SM}(n_{g})$ is the maximum $n_{g}$ in the topologically
non-trivial mode’s lowest frequency single mode bandwidth. This FOM works to
eliminate any bending back of the topologically non-trivial mode and to
increase the maximum single mode group index. The improved guided mode group
index can be seen in the flatter band of Fig 1 (e) (ii) in comparison to (d)
ii. More information about this optimisation and design of the optimised
bearded interface waveguide can be found in Ref [PhysRevA.106.033514].

##  S2 Identification of Waveguide Regions

Figure S2:  (a-c) Transmission of QD photoluminescence (PL) through an (a) W1,
(b) glide-plane, and (c) VH waveguide, respectively. In each case, PL was
generated using high-power non-resonant excitation of QDs located in a grating
coupler at one end of the waveguide, and collected from the coupler at the
opposite end of the device after transmission through the waveguide. The inset
in (a) provides a reference PL spectrum for QDs in the bulk. The inset in (b)
shows the PL signal measured from one outcoupler when exciting the QDs located
in the glide-plane waveguide, revealing high finesse peaks within the slow
light (and multimode) spectral window.

In this work, we identified the regions of the waveguides by considering both
the transmission properties and the Fabry-Pérot (FP) fringes of the waveguide.
By exciting the waveguide with either an above band light source to create
broad QD emission, or the use of a broadband light source, we observed the FP
fringes in the transmission spectrum. Fig S2 (a-c) shows transmission spectra
for the W1, glide-plane and zig-zag VH waveguides. As expected, the spectra
contain sharp peaks originated from the FP resonances sustained by the
reflection at the waveguide ends. From the spacing between the fringes, we
were able to determine the group velocity of the waveguide at different
spectral points.

Considering the case of the W1 waveguide, we can see a prominent drop in
transmission at $\sim$ 920nm, an indication of the band edge. For the glide-
plane waveguide, we observed high finesse modes in the wavelength range of
$955-975$nm. This occurrence can be attributed to the overlapping slow light
(flat band) spectral window for the two modes of the glide-plane waveguide. By
identifying the regions of low and high group velocity, and locating the band
edge from regions of low transmission we were able to work out the operational
single mode bandwidths of the various waveguides from which we exclusively
collected chiral data.

##  S3 Calculation of $\beta$-factor

Figure S3:  (a)-(e) Detailed view of dipole positions (yellow crosses) used in
calculating the positional and spectral dependence of the $\beta$-factor with
regard to a) W1, b) glide-plane, c) zig-zag VH, d) original bearded VH and e)
optimised bearded VH waveguide respectively. The irreducible repeating unit of
each waveguide is shaded in blue. (f) Schematic of the FDTD $\beta$-factor
simulations carried out. A dipole source with in-plane circular polarisation
($\sigma^{+}$) is surrounded by a series of transmission boxes that record the
radiative power into the waveguide mode, in-plane loss, and out-of-plane loss.

The $\beta$-factor of a waveguide in the contect of chiral quantum optics,
quantifies the coupling efficiency of a quantum emitter into cavity/waveguide
mode. It is defined as the spontaneous emission rate into guided modes over
total decay rate. That is,

$\beta=\frac{\Gamma_{bound}}{\Gamma_{bound}+\gamma_{ng}}\>,$ (S3)

where $\Gamma_{bound},\gamma_{ng}$ are decay rates into a guided and non-
guided mode respectively. The calculation of the $\beta$-factor for this work
is done through finite-difference-time-domain (FDTD) simulations, using the
the commercial FDTD software package, Lumerical [Lumerical]. The simulations
consist of measuring the power throughput of a circularly polarised dipole
emitter, collected from a series of transmission/power monitors surrounding
the source. The $\beta$-factor is then calculated as the fraction of power
coupled into the propagating modes over the total power injected from the
emitter. Transmission monitors that account for out-of-plane and in-plane
losses are placed two lattice constants above the waveguide and six lattice
constants away from the waveguide centre respectively, while those for the
waveguide modes are placed at 1µm, 2µm, 3µm and 4µm away from the dipole
source for transmission comparison as well as uncertainty calculations, as is
shown in Fig. S3. The simulation region is $20a$ in $x$, $16a$ in $y$ and $5a$
in $z$, with $42,8,8$ layers of perfect matched layers (PML) at the x, y and z
boundaries. The simulation is run with a mesh of 20 cells per lattice
constant. A refractive index of $3.4$ and $1$ is used for the GaAs dielectric
slab and air holes respectively. We vary the circularly polarised dipole’s
position within the least irreducible unit of the waveguide, up to $1.5a\sim
2a$ away from the waveguide centre (see Fig.S3 for more details with regard to
each waveguide).

During the $\beta$-factor calculations, we observed minor, high frequency
fringes in the transmission data caused by weak reflection at the PML region
and the finite sampling rate inherent to the nature of the Fourier transforms
of FDTD. In order to remove the fringes, we apply a Savitzky–Golay Filter. The
filter is applied to all three elements (i.e. forward, backward and total
electric field power) used for the $\beta$ factor calculation.

##  S4 Loss into free space modes at chiral points

Since the $\beta$-factor characterises the fraction of spontaneous emission
into waveguide propagating mode, it is therefore closely associated with the
Purcell-factor, which describes the spontaneous emission rate enhancement of a
dipole transition by its surrounding environment in the weak coupling regime.
For waveguides the Purcell factor is proportional to the group index and the
electric field intensity, and can be written as: [beta]

$F_{P}(r)=\frac{3\pi
c^{2}a}{\omega^{2}\sqrt{\epsilon_{r}(r)}}\cdot\frac{1}{v_{g}}\cdot\frac{|\hat{e}_{k}(r)\cdot\hat{n}|^{2}}{\int\epsilon_{r}(r)|\hat{e}_{k}(r)|^{2}\rm{d^{3}}r}\>,$
(S4)

where $\hat{e}_{k}(r)$ is Bloch mode profile at position $r$, $\hat{n}$ is a
unit vector that denotes dipole orientation, $v_{g}$ is group velocity,
$\epsilon_{r}$ is dielectric permittivity, $a$ is the period of the waveguide.
The relation between Purcell Factor and $\beta$-factor is given by,

$\beta=\frac{F_{P}}{F_{P}+F_{r}}\>,$ (S5)

where $F_{P}$ is Purcell factor, $F_{r}$ is a factor representing the loss
into free space modes, which takes a value between 0 and 1. By calculating
both the $\beta$-factor as described in the previous section and the Purcell
enhancement of the dipole sources at chiral points, we are able to extract the
loss into free space modes at chrial points for each waveguide at the
frequency point indicated by the red circles in Fig.1 in the main text. If the
waveguide contained multiple locations where the chiral contrast was 1, we
choose the point with the highest Purcell enhancement. For each waveguide we
extract values for $F_{r}$ of:

Waveguide Type | $F_{r}$
---|---
W1 | $0.05\pm 0.01$
Standard GP | $0.05\pm 0.01$
Optimised GP | $0.06\pm 0.01$
Zig-zag VH | $0.10\pm 0.02$
Original bearded VH | $0.06\pm 0.02$
Optimised bearded VH | $0.05\pm 0.01$

##  S5 Spectral and spatial dependence of $\beta$-factor

Figure S4: Frequency (horizontal-axis) and position (vertical-axis) dependence
of $\beta$-factor for a) W1, b) glide-plane, c) zig-zag VH, d) original
bearded VH and e) optimised bearded VH waveguide. See Fig.S3 (a-e) for
detailed dipole positions. Range of the colour-map is set to $0.5\sim 1$ so as
to highlight the contrast.

The spectral and position dependence of $\beta$-factor for the five waveguides
explored in the main text is shown in Fig S4. Dipole positions are labelled
first along the vertical axis (upward, away from the waveguide centre), then
the horizontal axis (rightward, along the propagation direction). The
frequency range for each waveguide is deduced from the single-mode regime of
its band structure (see Fig. 1 in the main text).

For the W1 waveguide, the single-mode region goes across the whole band, with
a gradual decrease in group velocity, which reaches zero at the Brillouin Zone
(BZ) edge. It has a near-invariant mode profile, contributing to a monotonic
increase in the Purcell Factor against mode frequency. It is interesting to
see a considerable $\beta$-factor in the presence of slow-light even at
positions in the waveguide that have an absence of high electric field
intensity.

As for the standard glide-plane waveguide, its single-mode region only covers
a small fraction of the upper band. Since there is no slow-light effect in
this region and the electric field is well concentrated in the waveguide
centre, the $\beta$-factor is lower than the W1 waveguide. Dispersion is
approximately linear in the frequency range of interest, resulting in a near-
constant $\beta$-factor across the single-mode region.

The Zig-Zag Topological VH waveguide is a single-mode waveguide. The modes
possess a broad mode profile, with a mode waist radius of up to $3a\sim 6a$
depending on the relative contrast of two triangular holes [taper]. Although
this waveguide has a single mode slow light region, this region overlaps with
the bulk crystal band and so can not confine the modes within the waveguide
centre (see the dark region of short wavelength modes in Fig.S4). Furthermore,
the broad mode profile makes the mode less concentrated, giving rise to low
$\beta$-factors generally throught the waveguide’s operation.

Similar to the standard glide-plane waveguide, the original bearded VH
waveguide has a slow-light region but unfortunately, once again it is multi-
mode. Given that these modes tend to possess opposite polarisation, chirality
is compromised and the slow-light region is no longer appropriate for chiral
applications. With the additional consideration of its broad mode profile, the
original bearded VH waveguide performs the worst in the $\beta$-factor
comparison.

A significant improvement of the optimised bearded VH waveguide is the
breaking of glide-plane symmetry which resolves the multi-mode issue by
shifting the high-order mode upward. The introduction of a slow-light region
away from the BZ edge brings about high Purcell enhancement for topologically-
protected, gap guided modes with high electric field concentration. As a
result, the optimised version of the topological Bearded VH outperforms all
other waveguides explored in the main text in terms of $\beta$-factor .

##  S6 QD chirality asymmetry

Figure S5: (a) Frequency of the asymmetry in the chiral contrast measured from
the left and right out-coupler for the W1 waveguide. (b) spectra of an
asymmetric QD, showing the difference in contrast viewed from the right out-
coupler in red and the left out-coupler in blue.

A notable characteristic observed across all waveguides, consistent with
findings from other experiments and QD wafers[Coles_thesis, Barik2020, 18], is
the presence of asymmetry in the chirality of the QDs when comparing the
measured chirality from the left and right out-couplers of the waveguide. In
Fig 3 (f-i), we can observe examples of QD spectra that closely align with the
predictions generated by the FDTD simulations illustrated in Fig 2 (j). These
spectra exhibit the anticipated behavior, where the outputs from the right
out-coupler (represented in red) and the left out-coupler (represented in
blue) are symmetric, acting as mirrored versions of each other. They display
the same chiral contrast but with opposite intensities. However, the spectra
of Fig S5 (b) presents an instance of asymmetric behavior, where the chiral
contrast observed from the left out-coupler surpasses that observed from the
right out-coupler. Fig S5 (a) shows the distribution of the asymmetry, showing
the number of QDs that were found within the W1 waveguide with a given
difference between the contrast as calculated from the left out-coupler, and
the contrast as calculated from the right out-coupler. This discrepancy in
behavior warrants a thorough investigation into its possible origin and the
implications it holds for the practical applications of these devices.

##  S7 Slab Waveguide

Figure S6: (a) Top down view of 3D-FDTD simulation with 2 transmission
monitors $T^{R}$ and $T^{L}$ and a 3D optical power monitor $P_{n}$. (a)
average chiral contrast and (b) average $\beta$-factor with respect to the dot
position (starting in the middle of the waveguide at 0nm and moving towards
the edge at 140nm).

While the bulk of our work has focused on photonic crystal waveguides, for a
complete comparison, we consider a simple slab waveguide. Here we analyse how
the properties of chiral QD emission in a slab waveguide changes according to
its position along the width of the structure. Using FDTD simulations and
employing a circularly polarised dipole source, we can compare changes in
average chiral contrast ($C_{av}$) and average $\beta$-factor ($\beta_{av}$)
from normalised transmission monitors placed either side of the dipole
($T^{L}$ and $T^{R}$) with a 3D power monitor placed around the dipole source
(layout shown in Fig S6 (a)).

With simulated results we can calculate $C_{av}$ and $\beta_{av}$ using:

$C_{av}=\frac{T^{R}_{av}-T^{L}_{av}}{T^{R}_{av}+T^{L}_{av}}\;\;\;\text{and}\;\;\;\beta_{av}=\frac{T^{R}_{av}+T^{L}_{av}}{P_{norm}},$
(S6)

where $T^{R}_{av}$ and $T^{L}_{av}$ is the mean transmission over the
wavelength range 900-960 nm and $P_{norm}$ is the ratio of the injected power
into the simulation region with the power that couples into the waveguide. To
ensure the best possible coupling of a single fundamental mode into the
waveguide, we restrict out width $w=280$ nm (with a depth of 170 nm) in
addition to mode expansion monitors which discard the minor coupling of
higher-order modes.

From Fig S6 (c), we can see that circular dipole sources within the slab
waveguide can show high contrast (max($C_{av}$) = 92%), but the maximum
$\beta$-factors possible are significantly lower than that of photonic
crystals.

##  S8 Optimised Glide-plane

Figure S7: (a) Band structure for the optimised glide-plane wavetguides
showing superated upper and lower guided modes, without a multi mode region.
(b) The electric field intensity at the interface of the waveguide (c) The
degree of circular polarisation of the electric field at the centre of the
slab ($S_{3}$) (d) $\beta$-factor of the waveguide at different physics and
spectral points in the waveguide.

For the main text, the simplest versions of the conventional waveguides were
used, for a fair comparison to the topological waveguide designs that haven’t
been optimised. However in this supplementary section we highlight a key
candidate waveguide, the optimised glide-plane waveguide. Waveguides of this
nature have featured predominately in work in quantum optics [6], [5], with
the design that is explored here coming from reference [5].

The optimisation of this waveguide, is mostly focused on creating a favourable
band structure. The $S_{3}$ and electric field map of the waveguide interface
is shown in Fig S7 (b-c), which is a very similar field profile as seen in Fig
2 (b) for the un-optimsied waveguide.

In comparison to the un-optimised version, this waveguide has no multi mode
region, and instead has single-mode operation from both guided modes of the
waveguide. In the un-optimised waveguide, the slow-light region of operation,
is also in a multi mode region. Because the two guided modes often have
opposite circular polarisations of their electric fields for a single point at
the interface of the waveguide, the multi mode region renders highly
directional chiral emission impossible.

The slow light region of the waveguide provides a significant improvement in
the $\beta$-factor of the waveguide at all points within the interface. The
drawbacks of the standard glide-plane waveguide design are addressed by the
optimised version. The $\sim 5nm$ wide slow-light region at BZ edge leads to
high $\beta$ even if the emitter is placed $2a$ away from the centre. In
figure S7 (d), we see a line of high $\beta$-factor at 0.282 a/$\lambda$
corresponding to the slow light region.


Rare kaon decay $K^{+}\rightarrow\pi^{-}\mu^{+}\mu^{+}$ as the key event

for the right-handed weak interaction effects

Yoshio Koide

a Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan

E-mail address<EMAIL_ADDRESS>
> We discuss on the search for the right-handed weak interaction effects in
> the SU(2)${}_{L}\times$SU(2)R model with lepton doublets
> $(\nu_{\ell},\ell^{-})_{L}$ and $(N_{\ell},\ell^{-})_{R}$
> ($\ell=e,\mu,\tau)$. We will point out that only the chance of the
> observation of the right-handed weak interaction effect will be in the rare
> decay $K^{+}\rightarrow\pi^{-}\mu^{+}\mu^{+}$.

1\. Introduction

We are interested in the right-handed weak interaction effect in the
SU(2)${}_{L}\times$SU(2)${}_{R}\times U(1)_{B-L}$ model [1]. Let us denote the
lepton doublets as

$\left(\begin{array}[]{c}\nu_{\ell}\\\ \ell^{-}\\\ \end{array}\right)_{L},\ \
\ \left(\begin{array}[]{c}N_{\ell}\\\ \ell^{-}\\\ \end{array}\right)_{R},\ \ \
\ \ (\ell=e,\mu.\tau).$ $None$

Here, the neutral leptons $\nu_{\ell}$ and $N_{\ell}$ have Majorana masses
separately.

Of course, the right-handed lepton doublets $(N_{\ell},\ell^{-})_{L}$ interact
with the right-handed weak boson $W_{R}$. At present, it is known [2] that the
mass of $W_{R}$ is experimentally

$M(W_{R})>3\ {\rm TeV}.$ $None$

Therefore, events induced by the right-handed weak interaction are highly
suppressed by the factor

$\left(\frac{M(W_{L})}{M(W_{R})}\right)^{4}<\left(\frac{0.1\ {\rm TeV}}{4\
{\rm TeV}}\right)^{4}\sim 10^{-6}.$ $None$

Here and hereafter, for simplicity, we take $g_{L}=g_{R}$ for the gauge
coupling constants with the weak bosons $W_{L}$ and $W_{R}$.

For example, we know that the $Ds$ meson (mass is 1.9684 GeV) decays into muon
and neutrino with the branching ratio $Br(Ds\rightarrow\mu+\nu_{\mu})=5,50\pm
0.23$ % [3] . If there is the right-handed weak boson $W_{R}$, the leptonic
decay is also possible by mediating $W_{R}$.

In general, the momenta $p_{1}$ and $p_{2}$ in the final state at the rest
frame of parent particle are given by

$p_{1}=p_{2}=\frac{1}{M}\left[M^{4}-2M^{2}(m_{1}^{2}+m_{2}^{2})+m_{1}^{4}+m_{2}^{4}-2m_{1}^{2}m_{2}^{2}\right]^{1/2}.$
$None$

Here, $M$ is a mass of the ps-meson in the initial state, and $m_{1}$ and
$m_{2}$ are lepton masses of the final state. Therefore, the momenta of the
final state muon are quite different from each other:

Event via $W_{L}$

$(p_{\mu})_{L}\simeq\frac{1}{2M}\left(M^{4}-2M^{2}m_{\mu}^{2}+m_{\mu}^{4}\right)^{1/2}=\frac{M}{2}\left(1-\frac{m_{\mu}^{2}}{M^{2}}\right),$
$None$

Event via $W_{R}$

$(p_{\mu})_{R}\simeq\frac{1}{2M}\left(M^{4}-2M^{2}m_{N}^{2}+m_{N}^{2}\right)^{1/2}=\frac{M}{2}\left(1-\frac{m_{N}^{2}}{M^{2}}\right),$
$None$

where $M$ is the mass of $Ds$ meson. Thus, by measuring the momentum of the
muon, i.e. (1.5) or (1.6), we can know that the case is due to $W_{L}$ or
$W_{R}$.

However, it is noted that the events (1.6) are highly suppressed by the factor
(1.3), so that we cannot observe the events substantially.

In this paper, we would like point out that only the visible effect induced by
$W_{R}$ is a rare decay $K^{+}\rightarrow\pi^{-}\mu^{+}\mu^{+}$.

2\. Rare decay $K^{+}\rightarrow\pi^{-}\mu^{+}\mu^{+}$

The rare decay $K^{+}\rightarrow\pi^{-}\mu^{+}\mu^{+}$ is only possible event
which we can observe under the right-handed weak interaction. In this section,
we will discus the detail.

For convenience, let us give only the leptonic part of the matrix element. As
we show in Appendix A, the matrix element of the lepton part due to the
exchange of $W_{L}$ for $K^{+}\rightarrow\pi^{-}\mu^{+}\mu^{+}$ is given by

${\cal
M}_{L}=m_{\nu}\overline{v^{c}}(k_{1})q\\!\\!/\frac{iC^{-1}}{k^{2}-m_{\nu}^{2}}\frac{1-\gamma_{5}}{2}p\\!\\!/v(k_{2})$
$None$

and that due to the exchange of $W_{R}$ is given by

${\cal
M}_{R}=M_{N}\overline{v^{c}}(k_{1})q\\!\\!/\frac{iC^{-1}}{k^{2}-M_{N}^{2}}\frac{1+\gamma_{5}}{2}p\\!\\!/v(k_{2}).$
$None$

Then, for the case $M_{N}^{2}\ll k^{2}$, we get

[Case A]

$R_{R/L}^{A}\equiv\frac{{\cal M}_{R}}{{\cal
M}_{L}}\sim\frac{M_{N}/k^{2}}{m_{\nu}/k^{2}}=\frac{M_{N}}{m_{\nu}}.$ $None$

On the other case, for the case $M_{N}^{2}\gg k^{2}$, we get

[Case B]

$R_{R/L}^{B}\equiv\frac{{\cal M}_{R}}{{\cal
M}_{L}}\sim\frac{1/M_{N}}{m_{\nu}/k^{2}}=\frac{k^{2}}{m_{\nu}M_{N}}.$ $None$

Note that the case $M_{N}^{2}\gg k^{2}$ does not always give a large
enhancement of $R_{R/L}$ as we see in Eq.(2.4).

Let us see the numerical value of the ratio $R_{R/L}$. For example, we take
$m_{\nu_{\mu}}\sim 10\ {\rm meV}\sim 10^{-11}$ GeV, and $k^{2}\sim 0.2$ GeV2.

Since we want as possible as large value of $R_{R/L}$, for the case A, we take
$M_{N}\sim 10^{-1}$ GeV. Then, we obtain

[Case A]

$R_{R/L}^{A}\sim\frac{M_{N}}{m_{\nu}}\sim\frac{10^{-1}}{10^{-11}}\sim
10^{10}.$ $None$

On the other hand, for the case B, for the $M_{N}$ mass $M_{N}\sim 10$ GeV, we
obtain

[Case B]

$R_{R/L}^{B}\sim\frac{k^{2}}{m_{\nu}M_{N}}\sim\frac{10\ {\rm
GeV}^{2}}{10^{-11}\ {\rm GeV}\ 10\ {\rm GeV}}\sim 10^{11}.$ $None$

Of course, since the ratio $R_{R/L}$ is the ratio for the lepton part, so that
we must multiply the weak boson mass ratio $(M_{WL}/M_{WR})^{4}\sim 10^{-6}$.

$\left(\frac{M(W_{L})}{M(W_{R})}\right)^{4}R_{R/L}^{A}\sim 10^{4},\ \ \ \ \ \
\ \left(\frac{M(W_{L})}{M(W_{R})}\right)^{4}R_{R/L}^{B}\sim 10^{5}.$ $None$

(Note that this diagram has two weak boson exchanges. But the ratios (2.3) and
(2.4) are ones for the amplitude.) Therefore, our result (2.7) shows that the
right-handed weak interaction effect is enhanced in the rare decay
$K^{-}\rightarrow\pi^{-}\mu^{+}\mu^{+}$. We can confirm by seeing whether the
muons in the final state are $\mu_{R}$ not $\mu_{L}$.

4\. Conclusion

In conclusion, only the chance of the search for the right-handed weak
interaction will be in the rare decay $K^{+}\rightarrow\pi^{-}\mu^{+}\mu^{+}$.
If we observe the decay $K^{+}\rightarrow\pi^{-}\mu^{+}\mu^{+}$ in a near
future, it will be caused by the exchange of the right-handed weak boson. This
will be confirmed by the helicity of muons in the final state, i.e.
$(\mu^{+})_{R}$.

Acknowledgements

The author is grateful to Prof. M. Tanaka for his enjoyable discussion and his
helpful comments. And, also, the author is grateful to Prof. T. Yamashita for
his helpful assistance in preparing this manuscript. And he is also grateful
to Prof. A. Sato for his valuable advice on the experimental behavior of muon.
This work is supported by JPS KAKENHI Grant number JP1903826.

Appendix A – Decay $K^{+}\rightarrow\pi^{-}\mu^{+}\mu^{+}$

There are two diagrams for the $K^{+}\rightarrow\pi^{-}\mu^{+}\mu^{+}$ decay
as thown in Fig.1 snd Fig.2.

$\overline{s}$ $\overline{u}$

$u$ $d$

$\nu$ $\bar{\nu}$

$\mu^{+}(k_{1})$ $\mu^{+}(k_{2})$

Fig.1 Box-type diagram for
$K^{+}(q)\rightarrow\pi^{-}(p)+\mu^{+}(k_{1})+\mu^{+}(k_{2})$ decay

$\bar{s}$ $\bar{u}$

$u$ $d$

$\nu$ $\bar{\nu}$

$\mu^{+}(k_{1})$ $\mu^{+}(k_{2})$

Fig.2 Tree-type diagram for
$K^{+}(q)\rightarrow\pi^{-}(p)+\mu^{+}(k_{1})+\mu^{+}(k_{2})$ decay

However, our interest is only in the lepton part whose matrix element is given
by the following:

${\cal M}_{L}\equiv
q_{\nu}p_{\mu}\,\overline{v^{c}}(k_{1})\gamma^{\nu}\frac{1-\gamma_{5}}{2}\left(\frac{iC^{-1}(k\\!\\!/+m_{\nu})}{k^{2}-m_{\nu}^{2}+i\varepsilon}\right)\gamma^{\mu}\frac{1+\gamma_{5}}{2}v(k_{2})$
$=\overline{v^{c}}(k_{1})q\\!\\!/\frac{1-\gamma_{5}}{2}\frac{iC^{-1}}{k^{2}-m_{\nu}^{2}}(k\\!\\!/+m_{\nu})p\\!\\!/\frac{1+\gamma_{5}}{2}v(k_{2})$
$=\overline{v^{c}}(k_{1})q\\!\\!/\frac{iC^{-1}}{k^{2}-m_{\nu}^{2}}\frac{1-\gamma_{5}}{2}(k\\!\\!/+m_{\nu})\frac{1-\gamma_{5}}{2}p\\!\\!/v(k_{2})$
$=m_{\nu}\overline{v^{c}}(k_{1})q\\!\\!/\frac{iC^{-1}}{k^{2}-m_{\nu}^{2}}\frac{1-\gamma_{5}}{2}p\\!\\!/v(k_{2})$
$None$

Here, we have used the following formulae

$\Psi^{c}\equiv C\bar{\Psi}^{T},$ $None$
$\bar{\Psi}^{c}=(\Psi^{c})^{\dagger}\gamma^{0}=(C\bar{\Psi}^{T})^{\dagger}\gamma^{0}=\Psi^{T}C(\gamma^{0})^{2}=-\Psi^{T}C,$
$None$ $C^{-1}\gamma^{\mu}C=-(\gamma^{\mu})^{T},$ $None$
$C^{-1}\gamma^{5}C=\gamma^{5},$ $None$ $C=C^{-1}\ \ \ \ \ (C^{2}={\bf 1}).$
$None$

The propagator of Majorana neutrino was quoted from the Haber and Kane’s paper
[5].

Similarly to the case of the $W_{L}$ exchange, (A.1), the result in the
$W_{R}$ exchange is given as follows:

${\cal
M}_{R}=M_{N}\overline{v^{c}}(k_{1})q\\!\\!/\frac{iC^{-1}}{k^{2}-M_{N}^{2}}\frac{1+\gamma_{5}}{2}p\\!\\!/v(k_{2}).$
$None$

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