# Enhanced Early Galaxy Formation in JWST from Axion Dark Matter?

Simeon Bird<EMAIL_ADDRESS>Chia-Feng Chang<EMAIL_ADDRESS>Yanou
Cui<EMAIL_ADDRESS>Daneng Yang<EMAIL_ADDRESS>Department of Physics and
Astronomy, University of California, Riverside, CA 92521, USA

###### Abstract

We demonstrate that enhanced early galaxy formation can generically arise in
axion-like particle (ALP) dark matter (DM) theories with the kinetic
misalignment mechanism, potentially addressing the excess recently observed by
the James Webb Space Telescope (JWST), while being consistent with existing
constraints. We identify viable parameter regions with the ALP mass in the
range of $10^{-22}~{}{\rm eV}<m_{a}<10^{-19}~{}\rm eV$. In addition, we show
that the ALP parameter regions of interest can lead to intriguing
complementary signatures in the small scale structure of DM halos and existing
experimental searches for ALPs.

Introduction. The standard $\Lambda$CDM cosmology makes firm predictions for
the abundance of dark matter halos as a function of time. However, recent
James Webb Space Telescope (JWST) observations have revealed what may be an
unexpectedly large population of luminous galaxies at redshifts $10$ and above
Labbe et al. (2022); Atek et al. (2023); Finkelstein et al. (2022); Harikane
et al. (2023a); Bradley et al. (2022); Harikane et al. (2023b); Gardner et al.
(2006); Labbé et al. (2023). In particular, Ref. Harikane et al. (2023b)
reported $25$ spectroscopically confirmed galaxies at $z_{\rm
spec}=8.61-13.20$, two of which have $M_{\rm UV}<-19.8~{}\rm mag$ at $z>11$.
The reported number exceeds most predictions based on the $\Lambda$CDM
cosmology, and may be the harbinger of new fundamental physics and/or lead to
a new understanding of structure formation Dayal et al. (2014, 2019); Yung et
al. (2020); Behroozi et al. (2020); Wilkins et al. (2023); Mason et al.
(2023); Lovell et al. (2023); Boylan-Kolchin (2022).

These surprising JWST results have inspired many dedicated studies Sabti et
al. (2023); Lovell et al. (2023); Boylan-Kolchin (2022); Chen et al. (2023);
McCaffrey et al. (2023); Kannan et al. (2022). Proposed explanations for the
JWST excess include enhanced star formation, accelerated mass assembly, early
or clustering dark energies, large scale-dependent non-Gaussianities, etc.
Dekel et al. (2023); Nadler et al. (2023); Passaglia and Sasaki (2022);
Biagetti et al. (2023); Liu et al. (2022); Liu and Bromm (2022); Hütsi et al.
(2023); Batista and Marra (2017); Klypin et al. (2021). Among them, a
population of high $z$ heavy compact objects has emerged as a compelling
solution Zurek et al. (2007); Hütsi et al. (2023); Liu et al. (2022); Liu and
Bromm (2022). The presence of these objects in the radiation dominated era
introduces isocurvature perturbations which enhance the matter power spectrum
(MPS) with a shot-noise-like contribution up to a certain truncation scale
Carr and Silk (2018). This shot noise enhancement could then trigger higher
star formation efficiency, particularly in the most massive halos and at
earlier epochs Zurek et al. (2007); Dekel et al. (2023); Iršič et al. (2020).
To fully address the excess, however, these objects must be extremely massive
and contribute a substantial fraction to the total matter content in the
universe ($\Omega_{m}$) Hütsi et al. (2023); Liu et al. (2022); Liu and Bromm
(2022).

A well-motivated source of such heavy compact objects is axion DM, which is
the focus of this work. Large amplitude axion density fluctuations can
collapse into massive clusters by $z\sim 10$ Hogan and Rees (1988); Kolb and
Tkachev (1993, 1994a, 1994b); Niemeyer (2019); Barman et al. (2022); Enander
et al. (2017), with masses ranging from $10^{-12}~{}\rm M_{\odot}$ to
$10^{4}~{}\rm M_{\odot}$ Zurek et al. (2007); Fairbairn et al. (2018). The
mass of these clusters anti-correlates with the axion mass $m_{a}$, and the
$m_{a}$ range favored by the JWST excess is $10^{-19}~{}{\rm
eV}<m_{a}<10^{-16}~{}\rm eV$ in the Standard Misalignment Mechanism (SMM)
Hütsi et al. (2023). This mass range is excluded by black hole super-radiance
(BHSR) Arvanitaki et al. (2015); Hütsi et al. (2023); Stott and Marsh (2018);
Ünal et al. (2021); Kimball and van Bibber (2022), and in addition requires a
larger star formation efficiency $f_{*}$ than expected from low redshift
astrophysics.

In this work, we show that models with delayed axion oscillation, in
particular the recently proposed kinetic misalignment mechanism (KMM) Co et
al. (2020); Chang and Cui (2020a), enable efficient formation of field
fragments at sub-horizon scales Eröncel et al. (2022); Eröncel and Servant
(2023), opening up a parameter space safe from the BHSR constraint with
astrophysically plausible $f_{*}$. In particular, we reveal a large viable
parameter region with $10^{-22}~{}{\rm eV}<m_{a}<10^{-19}\rm eV$ that can
explain the JWST excess, while being consistent with relevant constraints such
as from BHSR, Lyman-$\alpha$ forest, and stellar dynamics Chabanier et al.
(2019); Brandt (2016); Koushiappas and Loeb (2017); Stott and Marsh (2018);
Ünal et al. (2021); Kimball and van Bibber (2022). Furthermore, we demonstrate
that the same ALP models that address the JWST excess may yield complementary
signals in a variety of current/future axion search experiments Berlin et al.
(2021); Graham et al. (2021); Bloch et al. (2020), and intriguingly have the
potential to alleviate puzzling features found in the small-scale structure of
DM halos Tulin and Yu (2018); Kaplinghat et al. (2016); Bullock and Boylan-
Kolchin (2017). In addition, we identify a parameter range that is consistent
with existing $\Lambda$CDM predictions, which is worth exploration even if the
current JWST excess resolves upon further investigations.

Axion clusters from kinetic misalignment. We first review some essential
aspects of axion physics that are relevant to this study. Axions are pseudo-
Nambu-Goldstone bosons arising from spontaneous breaking of global Peccei-
Quinn (PQ) symmetries, originally proposed to solve the strong CP problem in
QCD Peccei and Quinn (1977a, b); Wilczek (1978). Later on, they were found to
be attractive dark matter candidates, and more general non-QCD ALPs are also
well motivated from theoretical frameworks such as string theory and
supersymmetry Weinberg (1978); Abbott and Sikivie (1983); Dine and Fischler
(1983); Preskill et al. (1983). The very light $m_{a}$ needed to affect high
redshift structure growth as we found is incompatible with the QCD axion, but
is generic for ALPs.

Axions can be produced from both misalignment mechanisms (MM) and the decay of
axion topological defects. The axion energy density resulting from the latter
mechanism is contingent upon model specifics and still under development, with
uncertainties in part due to technical challenges in simulations (see e.g.
Auclair et al. (2022); Chang and Cui (2022, 2020b); Ellis et al. (2022);
Eggemeier et al. (2020); Buschmann et al. (2020); Chang and Cui (In
preparation), and the refs therein). In particular, the effect of topological
defects can be absent in certain cosmological scenarios. For example, if PQ
symmetry breaking occurs before inflation, axion cosmic strings would be
sufficiently diluted; on the other hand, the contribution from axion domain
walls can be inconsequential when compared to that from the MM mechanism, if
the lifetime of domain walls is sufficiently short. For these reasons, here we
choose to focus on the formation of axion clusters (ACs) via the misalignment
mechanism.

In the SMM, the axion field value is initially displaced from its true
potential minimum, with a zero initial velocity, then starts to oscillate when
it obtains a mass $m_{a}$ similar to the Hubble rate, $H_{*}\sim m_{a}$. The
oscillations of the axion field then result in its relic abundance today
($\Omega_{a}$) as a condensate of CDM. In contrast, in the recently proposed
KMM Co et al. (2020); Chang and Cui (2020a), a non-zero initial axion velocity
is considered, which can arise from models with asymmetries in the PQ charges
or some dimensional symmetry-breaking operators. This results in a larger
kinetic energy for the axion, with the possibility of $H_{*}\ll m_{a}$, which
delays the onset of axion field oscillation and thus alters the prediction for
$\Omega_{a}$. It was later shown in Eröncel and Servant (2023); Eröncel et al.
(2022) that such a delayed oscillation in the KMM also generically leads to a
phenomenon called axion field fragmentation, which is the exponential growth
of axion quanta via parametric resonances. The enhanced fluctuation due to the
fragmentation triggers the formation of axion clusters, in a way distinct from
that resulting from SMM.

Here we focus on the KMM scenario described in Eröncel and Servant (2023);
Eröncel et al. (2022), in the context of pre-inflationary PQ breaking, and
follow the approach outlined there for analyzing axion cluster formation. We
introduce a parameter $\eta\equiv m_{a}/H_{*}$ to characterize the degree of
the oscillation delay due to KMM, which plays a key role in determining the
scale of fragmentation, and consequently the mass function of the ACs. A
delayed oscillation corresponds to a larger value of $\eta$, compared to
approximately $3$ in SMM. The parameter space can be divided into regions of
incomplete fragmentation ($\eta\lesssim 40$), complete fragmentation
($40\lesssim\eta\lesssim\mathcal{O}(10^{3})$), and non-perturbative
($\eta\gtrsim\mathcal{O}(10^{3})$) Eröncel and Servant (2023); Eröncel et al.
(2022).

The fraction of the field which is fragmented saturates quickly with
increasing $\eta$, and we focus on the case of complete fragmentation which
spans the bulk of the parameter space of interest. While $\eta$ and $f_{a}$
are in principle independent model parameters, for a given $m_{a}$ and a
specified $\Omega_{a}$ output, they have a 1-1 relation. As an example, for a
simple, temperature-independent axion potential as used in most ALP models,
$f_{a}$ relates to $\eta$ in the following way Eröncel and Servant (2023):

$\displaystyle f_{a}\approx 10^{15}~{}{\rm GeV}\left(\frac{10^{-20}{\rm
eV}}{m_{a}}\right)^{\frac{1}{4}}\left(\frac{90}{\eta}\right)^{\frac{3}{4}}\left(\frac{g_{*}}{4}\frac{\Omega_{a}h^{2}}{0.12}\right)^{\frac{1}{2}},$
(1)

where $g_{*}$ is the effective degrees of freedom in the entropy evaluated at
the onset of fragmentation. Throughout our paper, we will assume that all the
DM is comprised of the ACs, i.e. $\Omega_{a}=\Omega_{\rm DM}$, to fix $\eta$,
which then fixes the 1-1 correspondence between $f_{a}$ and $\eta$ for a given
$m_{a}$.

We assume complete fragmentation, in which case the dark matter is comprised
of ACs with typical masses determined by the comoving scale around $k_{\rm
osc}\approx m_{a}a_{*}\kappa_{p}$, where $a_{*}$ is the expansion parameter at
the oscillation onset Eröncel et al. (2022). $\kappa_{p}\sim O(1)$ is a
correction factor that contains a mild $\eta$ dependence, as we will discuss
later in Eq. (8). The typical mass of the AC can be estimated as

$\displaystyle M_{0}$ $\displaystyle\approx$
$\displaystyle\bar{\rho}_{a}\frac{4\pi}{3}\left(\frac{\pi}{m_{a}a_{*}\kappa_{p}}\right)^{3}$
$\displaystyle\approx$ $\displaystyle 10^{4}~{}\rm
M_{\odot}\left(\frac{70}{\eta}\right)^{\frac{3}{2}+0.66~{}\theta(\eta-80)}\left(\frac{4\times
10^{-20}~{}{\rm eV}}{\it{m_{a}}}\right)^{\frac{3}{2}},$

where $\theta$ is the Heaviside Function, $\bar{\rho}_{a}$ is the average
axion energy density and is taken to be the average DM energy density today.
Both $\eta$ and $m_{a}$ anti-correlate with $M_{0}$, i.e. $M_{0}$ decreases as
$\eta$, $m_{a}$ increases. For a fixed $M_{0}$, increasing $\eta$ can
accommodate a smaller $m_{a}$, which can help alleviate the BHSR constraints.
These axion clusters are produced from the collapse of the axion fields that
fragmented at subhorizon scales.

The substantial mass of ACs naturally induces significant velocity dependent
gravitational scatterings Loeb (2022). In terms of the cluster mass, we have

$\displaystyle\frac{\sigma}{M_{0}}\approx 10{\ \rm
cm^{2}/g}\left(\frac{M_{0}}{10^{4}{\rm M_{\odot}}}\right)\left(\frac{10\ {\rm
km/s}}{v}\right)^{4},$ (3)

where $v$ is the relative velocity between the two ACs and anomalies in DM
small scale structure observations may be addressed by values of $O(1)~{}\rm
cm^{2}/g$ Tulin and Yu (2018); Kaplinghat et al. (2016); Bullock and Boylan-
Kolchin (2017); Boylan-Kolchin et al. (2011, 2012); Vogelsberger et al.
(2012); Rocha et al. (2013); Zavala et al. (2013); Peter et al. (2013); Wolf
et al. (2010); Kaplinghat et al. (2019); Yang et al. (2020); Yang and Yu
(2021); Turner et al. (2021); Correa et al. (2022); Yang et al. (2022). The
wave nature of ALPs in our considered mass range gives rise to pc to kpc scale
solitonic cores in ACs and ALP halos, which may also help alleviate small
scale challenges in $\Lambda$CDM Hu et al. (2000); Marsh (2016); Niemeyer
(2019). Given a large cross section, e.g., $10~{}\rm cm^{2}/g$, close
encounters between ACs can merge them into larger, more diffuse structures
within the ALP halos, enriching the small scale features of our model. The
diffuse nature of the ACs also alleviates the existing stringent constraints
from stellar dynamical, microlensing, and small-scale structures that apply to
massive compact halo objects such as primordial black holes Brandt (2016);
Koushiappas and Loeb (2017); Allsman et al. (2001); Tisserand et al. (2007);
Wyrzykowski et al. (2011); Blaineau et al. (2022); Esteban et al. (2023).
Depending on the shapes of the ALP potential, ACs could further evolve into
clusters of smaller objects such as oscillons, solitons, and axion stars Ollé
et al. (2020); Marsh (2016); Niemeyer (2019); Bogolyubsky and Makhankov
(1976); Copeland et al. (1995); Fodor et al. (2006); Chavanis (2011).

Figure 1:  Example MPSs in KMM axion DM models (colored curves) compared to
the MPS from a standard $\Lambda$CDM model. Data points with error bars
represent Lyman-$\alpha$ measurements (black) from Ref.Chabanier et al. (2019)
and HST UV luminosity function measurements (green) from Ref.Sabti et al.
(2022). The red dashed curve denotes the maximal cut-off scale for $k^{4}$
growth in the adiabatic curvature power spectrum based on the COBE/FIRAS bound
from Ref.Chluba and Grin (2013). Higher resolution Lyman-$\alpha$ surveys can
extend the scales on which the power spectrum is measured to $k\sim 10$ h/Mpc,
similar to the UV luminosity data Iršič et al. (2020); Rogers and Peiris
(2021).

Matter power spectrum. We decomposed the density perturbation into linear and
quadratic components following the methodology outlined in Ref. Eröncel and
Servant (2023). The linear component approximately corresponds to the
adiabatic perturbation in $\Lambda$CDM cosmology, denoted as $\mathcal{P}_{\rm
CDM}(k)$. On the other hand, the quadratic component introduces a shot noise-
like contribution referred to as $\mathcal{P}_{\rm AC}(k)$ which can be
estimated as Liu and Bromm (2022); Inman and Ali-Haïmoud (2019); Meszaros
(1974); Dike et al. (2023)

$\displaystyle{\cal P}_{\rm AC}(k)\approx(D_{\rm iso}(0)-1)^{2}\frac{f_{\rm
AC}M_{0}}{\rho_{c}\Omega_{a}}\propto M_{0}\text{,~{} for }k<k_{\rm osc}.$ (4)

Here $\rho_{c}$ is the critical energy density today, $f_{\rm AC}$ is the
fraction of dark matter comprised by ACs which we fix to be $1$, $f_{\rm
AC}\rho_{c}\Omega_{a}/M_{0}$ is the number density of the ACs, and $D_{\rm
iso}(z)$ is the growth factor of isocurvature perturbations which can be
parameterized as Meszaros (1974); Inman and Ali-Haïmoud (2019); Dike et al.
(2023)

$\displaystyle D_{\rm iso}(z)$ $\displaystyle=$
$\displaystyle\left(1+\frac{\Omega_{a}}{\Omega_{m}a_{-}}\frac{3a}{2a_{\rm
eq}}\right)^{a_{-}},$ (5)

where

$\displaystyle a_{-}$ $\displaystyle=$
$\displaystyle(\sqrt{1+24\Omega_{a}/\Omega_{m}}-1)/4.$ (6)

The $D_{\rm iso}$ does not evolve at $a\ll a_{\rm eq}$ and grow as
$3a/(2a_{\rm eq})$ (identical to the adiabatic one) for
$\Omega_{a}=\Omega_{m}$ at $a\gg a_{\rm eq}$. If ACs comprise only a fraction
of dark matter, then $f_{\rm AC}<1$, resulting in a smaller ${\cal P}_{\rm
AC}(k)$. Note that the AC number density is independent of $\Omega_{a}$
because $M_{0}\propto\Omega_{a}$. However, ${\cal P}_{\rm AC}$ decreases as
$\Omega_{a}$ decreases through the $D_{\rm iso}$ in Eq. 5 and Eq. 4, resulting
in fewer high-z massive galaxies.

We numerically calculate the leading order ${\cal P}_{\rm AC}(k)$, including
backreaction, following Eq. $4.11$ in Ref. Eröncel and Servant (2023), and
parameterize the obtained results as 111 We solved for the relic abundance
neglecting the effect of fragmentation which could induce a sub-O(1)
uncertainty Eröncel et al. (2022). A more accurate determination of the relic
abundance requires dedicated study which is beyond the scope of this work. :

$\displaystyle\mathcal{P}_{\rm
AC}(k)=\frac{2\pi^{2}}{k^{3}}\frac{f(\eta)(D_{\rm iso}(0)-1)^{2}}{(D_{\rm
iso}(z_{\rm
eq})-1)^{2}}\left(\frac{\kappa}{\kappa_{p}(\eta)}\right)^{3}\theta(\kappa_{p}-\kappa),$
(7)

where $\kappa\equiv\frac{k}{m_{a}a_{*}}$, $\kappa_{p}(\eta)$ and $f(\eta)$ are
numerical factors parameterized as

$\displaystyle\kappa_{p}(\eta)\simeq\left\\{\begin{aligned}
&0.9,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;40\leq\eta\leq
80,\\\
&0.9\left(\frac{\eta}{80}\right)^{0.22},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,80<\eta\leq
10^{3},\end{aligned}\right.$ (8)

and

$\displaystyle f(\eta)\simeq
1.19\left(\frac{\eta}{80}\right)^{-0.1},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,40<\eta\leq
10^{3}.$ (9)

The spectrum $\mathcal{P}_{\rm AC}(k)$ becomes dominant at scales larger than
$k_{*}\approx a_{*}H_{*}$ and is truncated at $k_{\rm osc}$. For $k>k_{\rm
osc}$, the spectrum becomes negligibly small and decreases as $1/k^{2}$
Eröncel and Servant (2023); Eröncel et al. (2022).

In Fig. 1, we show the MPS for three benchmark examples. The $M_{0}$ is
$1.6\times 10^{3}$, $1.5\times 10^{4}~{}\rm M_{\odot}$, and $4.9\times 10^{5}$
for the orange, golden, and magenta cases, respectively. We will show in Fig.
3 that the case in gold color is favored in order to explain the JWST excess,
the orange case is consistent with $\Lambda$CDM, while the magenta case
overproduces structures and is excluded by measurements of the UV luminosity
function. In all cases, ${\cal P}_{\rm AC}(k)\gg{\cal P}_{\rm CDM}(k)$ for
$k>k_{*}$, where the spectrum plateaus, before being truncated at $k\sim
k_{\rm osc}$ (beyond the range of Fig. 1). The case in gold color,
specifically, has a transition around $k_{*}\sim 10~{}{\rm h/Mpc}$, which is
close to the scale associated with the JWST excess.

Figure 2:  The halo mass function for the benchmark cases of Fig. 1 at
redshifts $z=0$ and $z=12$.

Halo Mass Function.  The clustering of ACs in the matter era leads to the
formation of halos. We use the Press-Schechter formalism to estimate the halo
mass function (HMF) from the MPS Press and Schechter (1974). Given an MPS
${\cal P}(k,z)$ at redshift $z$, we compute the mass variance using a top-hat
window function as

$\sigma^{2}_{M}(R)=\dfrac{1}{2\pi^{2}}\int dk{\cal
P}(k,z)\left(3\dfrac{\sin(kR)-kR\cos(kR)}{kR}\right)^{2}k^{2},$

where $M$ and $R$ are halo mass and radius and are related through
$M=4\pi\bar{\rho}_{m}R^{3}/3$, with $\rho_{m}$ being the mean matter density
today. The HMF is computed in terms of
$\nu\equiv\delta_{c}^{2}(z)/\sigma^{2}_{M}$ as

$\displaystyle\frac{dn}{dM}=\frac{\bar{\rho}_{m}}{M}\nu f_{\rm
EPS}(\nu)\frac{d\ln\nu}{d\ln M},$ (10)

where $\nu f_{\rm EPS}(\nu)$ is a shape function including the effect of
ellipsoidal collapse from Ref. Sheth and Tormen (1999). This approach has been
widely applied in the literature because of its simplicity. Ref. Liu et al.
(2022) demonstrated that this technique can produce reliable approximations of
the N-body reconstructed HMFs, even in the presence of massive primordial
black holes.

In Fig. 2, we show the obtained HMFs for the $\Lambda$CDM and the three
benchmarks in Fig. 1 at $z=0$ (left) and $z=12$ (right). We see that the HMFs
at $z=0$ are relatively similar across all cases; however, at $z=12$, the KMM
cases exhibit significant enhancements towards larger halo masses, increasing
the likelihood of finding halos hosting the very massive galaxies. Once small
halos virialize, they almost decouple from the background evolution. Hence the
abundance of massive halos is largely unaffected at low redshifts. As a
result, the effect of our model becomes less significant today, which
alleviates constraints from low redshift measurements Dike et al. (2023);
Esteban et al. (2023).

Figure 3:  The viable model parameter space given the observations in Ref.
Harikane et al. (2023a), considering galaxies with $M_{\rm UV}<-19.8~{}\rm
mag$ at $z\geq 11$. The 68% CL preferred region shaded in blue corresponds to
a prediction of $0.74<N_{\rm exp}<4.3$. It is obtained for a star formation
efficiency $f_{*}=0.14$ and would shift up (down) if $f_{*}$ increases
(decreases). Black lines represent contours of the gravitational scattering
cross section per mass for values of $1,~{}10,$ and $100\rm cm^{2}/g$,
evaluated at a velocity of $v=10~{}\rm km/s$. At larger velocities, the cross
section quickly reduces due to the $v^{-4}$ dependence. Constraints from
various sources exclude certain regions: the Lyman-$\alpha$ forest data
Chabanier et al. (2019) disfavors the orange area; the region to the left of
the red curve is inconsistent with measured UV luminosity functions Sabti et
al. (2022); and the grey area to the right is excluded by BHSR constraints
Arvanitaki et al. (2015); Stott and Marsh (2018); Ünal et al. (2021); Kimball
and van Bibber (2022).

Parameter space favored by JWST excess.  The enhanced massive halo population
as we see in Fig. 2 can trigger earlier galaxy formation and enhance the
formation efficiency Zurek et al. (2007); Dekel et al. (2023). To convert the
model prediction for HMF into the number of observed galaxies, we consider a
simplified approach assuming a constant star formation efficiency $f_{*}$. We
compute the expected number of observed galaxies with $M_{\rm UV}<-19.8~{}\rm
mag$ at $z>11$ as considered in Fig. 13 of Ref. Harikane et al. (2023b), where
the number of observed galaxies $N_{\rm obs}=2$ is systematically higher than
theoretical predictions from the literature Dayal et al. (2014, 2019); Yung et
al. (2020); Behroozi et al. (2020); Wilkins et al. (2023); Mason et al.
(2023), e.g., $0.3$ from Ref. Yung et al. (2023) as a recent estimate. We
convert the requirement of $M_{\rm UV}<-19.8~{}\rm mag$ into a minimum halo
mass by using the $M_{*}-M_{\rm UV}$ relation of Ref. Yung et al. (2023).
Assuming a constant star formation efficiency $f_{*}$, we set the minimum halo
mass $M_{\rm min}=2.2\times 10^{9}e^{-0.13z_{\rm min}}/(f_{*}f_{b})~{}\rm
M_{\odot}$, where $z_{\rm min}=11$, $f_{b}\approx 0.16$ is the baryon fraction
in the matter energy density, and we set $f_{*}=0.14$ so that the $\Lambda$CDM
prediction is for an expected $0.3$ galaxies at $z>11$ Yung et al. (2023). The
value of $f_{*}=0.14$ is relatively large compared to expectations at low
redshifts, but small compared to what has been proposed to explain the JWST
excess solely based on $f_{*}$ Dekel et al. (2023); Boylan-Kolchin (2022);
Lovell et al. (2023); Mason et al. (2023); Mirocha and Furlanetto (2023). It
has been noted that an excessively high $f_{*}$ could introduce new tensions
with the cosmic reionization history Gong et al. (2023), making our obtained
value a more desirable option. The number of expected galaxies count is
calculated as

$\displaystyle N_{\rm exp}=\frac{A_{\rm eff}}{4\pi}\int_{11}^{z_{\rm
max}}dz\frac{dV_{c}}{dz}\int_{M_{\rm min}}^{M_{\rm max}}dM\frac{dn}{dM},$ (11)

where $A_{\rm eff}=45~{}{\rm arcmin^{2}}$ is the effective area of the
observation following Ref. Harikane et al. (2023b) and $V_{c}$ is the comoving
Hubble volume at $z$. $dn/dM$ is the HMF in Eq. 10. Our result is not
sensitive to $z_{\rm max}$, $M_{\rm max}$, provided that they are sufficiently
large.

Fig. 3 presents the preferred $68\%$ CL parameter regions for a constant
$f_{*}=0.14$ as a blue band in the $m_{a}-\eta$ plane. We obtain the blue band
considering $0.74<N_{\rm exp}<4.3$ which arises from a Feldman-Cousins
analysis for two observations and zero background Feldman and Cousins (1998),
and the lower limit happens to coincide with that from Ref. Harikane et al.
(2023b). For a higher (lower) value of $f_{*}$, the band would shift upward
(downward). The upper-right corner corresponds to lower prediction for $N_{\rm
exp}$ values that are consistent with $\Lambda$CDM predictions within the
uncertainties Dayal et al. (2014, 2019); Yung et al. (2020); Behroozi et al.
(2020); Wilkins et al. (2023); Mason et al. (2023). The lower-left region has
higher $N_{\rm exp}$ and is constrained by Lyman-$\alpha$ and UV luminosity
measurements Chabanier et al. (2019); Sabti et al. (2022). We also plot the
contours of the gravitational scattering cross section of Eq. (3) evaluated at
$v=10~{}\rm km/s$ in Fig. 3. In the viable region, we see that the
gravitational cross section takes values in $(1-100)~{}\rm cm^{2}/g$, which
could lead to novel signal predictions that can address the small-scale
observations, e.g., the core-cusp problem and the too-big-to-fail problem
Tulin and Yu (2018); Kaplinghat et al. (2016); Bullock and Boylan-Kolchin
(2017); Boylan-Kolchin et al. (2011, 2012); Vogelsberger et al. (2012); Rocha
et al. (2013); Zavala et al. (2013); Peter et al. (2013); Wolf et al. (2010);
Kaplinghat et al. (2019); Yang et al. (2020); Yang and Yu (2021); Turner et
al. (2021); Correa et al. (2022); Yang et al. (2022).

Figure 4:  Axion-photon coupling versus axion DM mass $m_{a}$ (assuming
$\Omega_{a}=\Omega_{\rm DM}$ in Eq. 1). The blue and the lighter-blue regions
correspond to the regions favoured by matching the JWST excess, with
$C_{a\gamma}=1$ and $C_{a\gamma}=10^{3}$, respectively. The blue regions are
bounded by two dashed black lines, corresponding to the requirement of
perturbative $\eta$ with complete fragmentation. Constraints from various
existing searches (solid shaded regions) and forecasts (colored dashed lines)
are also illustrated. See text for further details. Figure 5:  Axion-nucleon
coupling versus axion DM mass $m_{a}$ (assuming $\Omega_{a}=\Omega_{\rm DM}$
in Eq. 1). The blue and the lighter-blue regions correspond to the JWST excess
favored regions with $C_{aN}=1$ and $C_{aN}=10^{3}$, respectively. The blue
regions are bounded by two dashed black lines, corresponding to the
requirement of perturbative $\eta$ with complete fragmentation. Constraints
from various existing searches (solid shaded regions) and forecasts (colored
dashed lines) are also illustrated. See text for further details.

Implications for Axion Models and Complementary Searches. As shown in the last
section, axion models with KMM can address the JWST excess, with a preferred
parameter space spanning $4\times 10^{-22}~{}{\rm eV}<m_{a}<10^{-19}~{}\rm eV$
and $40<\eta<1000$. Existing ALP searches may provide valuable complementary
probes for such ALPs, albeit in a model-dependent way. While the impact of ALP
on structure formation is largely independent of the non-gravitational axion
coupling to the Standard Model (SM) particles, most other ALP searches do
depend on the specific coupling patterns. Here we focus on axion interactions
with photons and nucleons ($N$), as described by

$\displaystyle\mathcal{L}\,\in\,\frac{g_{a\gamma}}{4}aF_{\mu\nu}\tilde{F}^{\mu\nu}+g_{aN}\partial_{\mu}a\bar{N}\gamma^{\mu}\gamma^{5}N,$
(12)

with effective couplings

$\displaystyle g_{a\gamma}=C_{a\gamma}\alpha/(2\pi
f_{a}),\;\;\;\;\;\;\;\;\;\;g_{aN}=C_{aN}/f_{a},$ (13)

where $F_{\mu\nu}$ represents the electromagnetic field strength, and
$C_{a\gamma}$, $C_{aN}$ are model-dependent parameters. In standard QCD axion
inspired models such as KSVZ or DFSZ Kim (1979); Shifman et al. (1980); Dine
et al. (1981); Zhitnitsky (1980), $C_{a\gamma}$, $C_{aN}\sim O(1)$. Meanwhile,
other recently proposed well-motivated axion theories such as those with
multiple PQ fermions, vector kinetic mixing, and axion clockworks, can
generically predict significantly larger coupling $C$’s, up to
$\mathcal{O}(10^{2}-10^{3})$ Dror and Leedom (2020); Agrawal et al. (2018a,
b). According to Eqs. 1 and 13, with our assumption of $\Omega_{a}=\Omega_{\rm
DM}$, $f_{a}$ and the corresponding $\eta$ are fixed for given
$g_{a\gamma}~{}(g_{aN})$ and $C_{a\gamma}~{}(C_{aN})$. Thus we may map the
parameter regions in Fig. 3 onto the $m_{a}-g_{a\gamma,N}$ space as in Fig. 4
and Fig. 5. In light of the aforementioned theoretically motivated range for
$C_{a\gamma},~{}C_{aN}$, in Fig. 4 and Fig. 5, the JWST excess favored regions
are illustrated for two benchmark values: $C_{a\gamma},~{}C_{aN}=1$ (blue
band) and $C_{a\gamma},~{}C_{aN}=1000$ (light blue band). The viable regions
for other values of $1\lesssim C_{a\gamma},~{}C_{aN}\lesssim 1000$ are
expected to lie between these two bands. These bands are truncated from above
and below (denoted by dashed black lines) by requiring that the corresponding
$\eta$ are within the perturbative regime with complete fragmentation,
$40\lesssim\eta\lesssim 1000$ (see discussion above Eq. 1), which we have
chosen to focus on in our analysis. As $C_{a\gamma}$ or $C_{aN}$ increases,
the blue and orange bands, along with the black dashed lines, would linearly
shift upwards. We also show the current constraints (solid shaded regions) and
future sensitivity forecast (bounded above by colored dashed lines) for a
variety of experiments/observations, including Lyman-$\alpha$ Chabanier et al.
(2019), BHSR Stott and Marsh (2018); Ünal et al. (2021); Kimball and van
Bibber (2022), Supernova-1987A Payez et al. (2015), neutron star cooling (NS)
Chang et al. (2018); Beznogov et al. (2018); Hamaguchi et al. (2018);
Sedrakian (2016), protoplanetary disk polarimetry (PPD) Fujita et al. (2019),
active galactic nuclei (AGN) Ivanov et al. (2019), and old comagnetometer
Bloch et al. (2020).

As can be seen from Fig. 4 and Fig. 5, upcoming experiments can offer
complementary probes for KMM axion parameter range that could address the JWST
excess. For example, for the benchmark of $C_{a\gamma}=1000$ the reach of
future heterodyne experiments Berlin et al. (2021) overlaps with the JWST
excess favored region of ($g_{a\gamma}$, $m_{a}$). Similarly, future storage
ring and comagnetometer experiments would be able to explore the ($g_{aN}$,
$m_{a}$) range motivated by the JWST excess Graham et al. (2021); Bloch et al.
(2020). It is also worth noting that the experiments covering the further
upper regions in Fig. 4 and Fig. 5, such as interferometry Graham et al.
(2018) and linearly polarized pulsar light (Pulsar I in Fig.4) Liu et al.
(2020), can potentially probe KMM axions with $\eta>10^{3}$ and $C_{a\gamma}$
(or $C_{aN}$) $>10^{3}$. However, these regions require a non-perturbative
analysis of fragmentation ($\eta\gtrsim 10^{3}$), which is beyond the scope of
this study.

Conclusion. We demonstrated that in the framework of KMM there is a large
parameter space of axion dark matter with $10^{-22}\rm
eV\lesssim{\it{m_{a}}}\lesssim 10^{-19}\rm eV$ that can address the JWST
excess while being compatible with all existing constraints. The delayed onset
of the axion field oscillation in KMM allows for efficient axion field
fragmentation at subhorizon scales. The fragmented axion field collapses into
a large population of massive ACs, which leads to more massive galaxies at
high redshift, and thus can potentially address the excess observed by JWST.
Upcoming experiments can provide complementary probes for the ALP parameter
range favored by the JWST excess. Near-future Lyman-$\alpha$ forest surveys
such as Weave-QSO Pieri et al. (2016) or DESI Karaçaylı et al. (2023); Ravoux
et al. (2023) will extend the scales on which the matter power spectrum can be
measured by a further factor of $2-3$. Future surveys of strong lensing
caustics could directly detect the predicted signature from small halos Daylan
and Birrer (2023). We also identified sizable gravitational scattering in our
model parameter space, which enriches small scale structure formation in our
model that is worth further investigation Kaplinghat et al. (2016); Bullock
and Boylan-Kolchin (2017). In addition to the complementary astrophysical
probes related to structure formation, we demonstrated that the JWST excess
favored parameter region can be probed by existing or planned axion search
avenues, e.g. heterodyne, comagnetometer and storage ring experiments,
assuming certain patterns of ALP-SM couplings (i.e. fixed
$C_{a\gamma},~{}C_{aN}$). This study identifies new avenues for probing axion
DM, and would stand as a worthwhile addition to the literature even if the
current JWST excess resolves after further investigation.

##  I Acknowledgment

We thank Yuichi Harikane, Huangyu Xiao, and Hai-Bo Yu for the helpful
discussion. SB is supported by NASA ATP-80NSSC21K1840. CC and YC are supported
by the US Department of Energy under award number DE-SC0008541. DY is
supported by the John Templeton Foundation under grant ID #61884 and the U.S.
Department of Energy under grant No. de-sc0008541. The opinions expressed in
this publication are those of the authors and do not necessarily reflect the
views of the John Templeton Foundation.

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# Reflection by two level system: phase singularities on the Poincaré
hypersphere

Ben Lang1, Edmund Harbord2 and Ruth Oulton2 1George Green Institute for
Electromagnetics Research, Faculty of Engineering, The University of
Nottingham, NG7 2RD, UK
2Quantum Engineering Technology Labs, H. H. Wills Physics Laboratory and
Department of Electrical & Electronic Engineering, University of Bristol, BS8
1FD, UK

###### Abstract

We consider the reflection of a photon by a two-level system in a quasi-one-
dimensional waveguide. This is important in part because it forms the backdrop
for more complicated proposals where many emitters are coupled to the
waveguide: leading to super and subradiant coupling even when the emitters are
distant. The incorporation of chiral effects, for example unidirectional
emission of dipole emitters, has already led to rich physics such as dimer
coupling. However, chirality is not the only effect of the dipole, as we
explore from a phase singularity perspective. We demonstrate that control of
the dipole allows a rich variety of control of the phase and amplitude of the
scattered light in both directions. This expands the scope for the physics of
1D chains of emitters.

The exploration of quantum emitter systems coupled to “one-dimensional”
waveguide-like photonic structures has developed into a wide field over the
past few years. The modified photonic density of states allows near-perfect
coupling between quantum emitters in the waveguide that does not decay with
distance. Studies of arrays of atoms or quantum emitters in such systems
predict a rich variety of physics. For instance, carefully positioned emitters
are predicted to show superradiance effects González-Tudela and Porras (2013).
Unidirectional emission and scattering leads to symmetry breaking in the
coupling of arrays of atoms, and the formation of “dimers” of many-body dark
states Ramos  _et al._ (2014). In those studies, one exploits the interference
between back-scattered and forward-scattered light. However, these studies
make a priori assumptions, for example that the phase change on reflection is
always $\pi$ Lodahl  _et al._ (2017), or only considering the specific case of
forward scattering for chiral emitters at specific points in the waveguide.

In this work we explore generalised complex dipole systems in a waveguide with
generalised polarization texture. The result is that one has signficant
control over the phase and amplitude of the scattered light, particularly in
reflection. Such generalized dipole properties will diversify the capabilites
of one-dimensional atom-chain systems, and relax the positioning requirements
for those emitters.

For simplicity we consider the reflection of a photon from just one two level
system (TLS), a classic problem in 1D quantum optics Shen and Fan (2005). A
photon incident on the TLS can be reflected, transmitted or scattered out of
the waveguide, ie. lost (although in a 1D system these losses are assumed to
be small). These possibilities are summarised using complex reflection and
transmission coefficients, $r$, $t$, with $|r|^{2}+|t|^{2}\leq 1$ with
equality occurring only at zero loss. The situation is depicted in fig.1,(a).

Figure 1: (a) Schematic cartoon of the scattering process. The input is split
between reflection, transmission and loss, with the coefficients determined by
the waveguide electric field at the location of the TLS (red), and the
transition dipole of the TLS (black). (b) Polarisation structure in a simple
waveguide, consisting of a high-index rectangular block (grey) in air (white).
Polarisation ellipses are shown through a cut, with light propagating up the
page (frame arrow).

The generalised nature of our study is that it includes both the waveguide
polarisation at the TLS location (which, even for simple waveguide is very
sensitive to location, fig.1(b)) and the electric dipole of the TLS. Recently
there has been interest in so called “chiral” coupling, which exploits the
longitudinal component of the waveguide electric field to make the TLS couple
asymmetrically to the waveguide modes in either direction Petersen  _et al._
(2014); Rodríguez-Fortuño  _et al._ (2013); Aiello  _et al._ (2015); Coles
_et al._ (2016); Picardi  _et al._ (2018). Chiral coupling enables systems
where the reflection/transmission coefficients are different for light
incident on the TLS from either side. For example a TLS decoupled from the
forwards mode will have $t=+1$ for a photon injected forwards, but a photon
incident in the backward direction may have $t$ anywhere between $+1$ and $-1$
depending on the coupling strengths to the backward and loss modes (the
applications of $t=0$ are discussed in Gonzalez-Ballestero  _et al._ (2016);
Sayrin  _et al._ (2015) and $t=-1$ in Young  _et al._ (2015); Söllner  _et
al._ (2015)).

The complex coefficients $r$ and $t$ depend on both the exact location in the
waveguide and the properties of the dipole. In nanophotonic systems the
electric E-field polarization at any specific point $r$ varies such that one
can describe a polarization ellipse at each point $r$ Young  _et al._ (2015).
The ellipse gives the relative magnitudes of the $E_{x}$ and $E_{y}$
components of the field and their relative phase. It is common to see purely
linear E-field points, purely circular (C) points, where the $E_{x}$ and
$E_{y}$ components are equal and out of phase by $\pi/2$, and arbitrary
elliptical points in between those two extremes. Unidirectional chiral
coupling occurs when a circular dipole emitter (ie a linear combination of
$d_{x}$ and $d_{y}$ dipole out of phase by $\pi/2$) is placed at a C-point
(the direction of emission is dictated by the sign of the phase between the
dipole components). However one can construct the arbitrary complex dipole
$\mathbf{d}=\alpha d_{x}+\beta d_{y}$. Our previous work Lang  _et al._
(2022), has highlighted that even when emitters are placed at an elliptical
point, emission may be made unidirectional by matching the helicity and
eccentricity of the ellipse and the dipole, but putting the long axes of the
two ellipses orthogonal. This negates the backscattered component so that only
forward scattering can occur.

We now explore the case where reflection is desired, by considering an
arbitrary dipole where the linear component is able to couple to the
backscattered direction. What is interesting here is the phase of the
reflection. While input-output models give only a $\pi$ phase shift to the
backscattered light, we show here that the orientation of the ellipse governs
the phase of the reflected light, giving rise to a rich structure. This is an
extremely useful property as it allows one to tune the phase delay when
considering a chain of atoms. This means that one is no longer constrained to
precise positions of the emitters in a 1D chain. Also, by dynamically
controlling the dipole ellipse orientation, one may control the reflectivity
in-situ.

We consider electric dipole interactions between the TLS and the waveguide,
and assume a narrow-band photon (narrow in frequency, long in time). The
crucial parameters determining the single photon $t$, and $r$ coefficients are
the local polarisation of the forwards (backwards) waveguide mode at the TLS
position $\mathbf{E}_{f(b)}$ and the electric dipole of the TLS transition
$\mathbf{d}$. Both $\mathbf{E}_{f}$ and $\mathbf{d}$ are complex vectors in
space. Without loss of generality we assume a basis in which both are 2D. The
time-reversal symmetry of Maxwell’s equations requires that
$\mathbf{E}_{f}=\mathbf{E}_{b}^{*}$, which amounts to reversing the ellipse
arrowheads.

The $r$, $t$ coefficients are given by Lang  _et al._ (2022):

$t=1-\frac{\mathbf{d}\cdot\mathbf{E}_{f}^{*}\,\,\mathbf{d}^{*}\cdot\mathbf{E}_{f}}{D}\,,$
(1)
$r=-\frac{\mathbf{d}\cdot\mathbf{E}_{b}^{*}\,\,\mathbf{d}^{*}\cdot\mathbf{E}_{f}}{D}\,,$
(2)

with

$\begin{split}D=&\frac{1}{2}\left(|\mathbf{d}\cdot\mathbf{E}_{f}^{*}|^{2}+|\mathbf{d}\cdot\mathbf{E}_{b}^{*}|^{2}\right)+\zeta\left(\mathbf{d}\cdot\mathbf{G}_{\text{l}}\cdot\mathbf{d}^{*}+\text{i}\hbar\epsilon_{0}\delta\right)\,.\end{split}$
(3)

We will assume resonance, and thus $\delta$, the detuning between the incident
photon and the TLS transition frequency, is set to zero.
$\mathbf{G}_{\text{l}}$ is the Green’s function controlling the interaction of
the TLS with non-guided modes Manga Rao and Hughes (2007), and the parameter
$\zeta$ characterises the inverse of the waveguide mode coupling strength to
the TLS. For simplicity we set
$\zeta\mathbf{d}\cdot\mathbf{G}_{\text{l}}\cdot\mathbf{d}^{*}=L$, with $L$ a
scalar controlling loss. If the TLS is prepared in its excited state the
fraction of the radiated intensity (photon probability) to enter the guided
modes is given by $\beta=W/(W+L)$ with
$W=\frac{1}{2}(|\mathbf{d}\cdot\mathbf{E}_{f}^{*}|^{2}+|\mathbf{d}\cdot\mathbf{E}_{b}^{*}|^{2})$.
By making loss independent of $\mathbf{d}$ we are implicitly assuming at least
two loss modes, with polarisations orthogonal at the TLS location.

First, we consider the case of a linear dipole. Here it is well-known that the
TLS acts as a mirror, with $|r|$ approaching unity for low loss Shen and Fan
(2005). However, the phase of the reflection has received less attention.

It should first be clarified what is meant by “the phase of the reflection”.
Naturally this phase must be determined relative to the phase of the input and
at some particular location in space. The location is important because the
input signal (travelling forwards) will have a phase that changes spatially
like $e^{ikx}$, while the backwards travelling reflection will instead evolve
as $e^{-ikx}$. As the distance between the phase reference point and the
location of the reflecting TLS is increased the phase of the reflection will
change according to the additional round trip distance - the mechanism of a
Michelson interferometer.

However, if some arbitrary (but fixed) location is picked at which to compare
the input and output phases it is then possible to calculate how changing
other parameters alters the phase of the reflected light.

Figure 2: (a) Michelson interferometer phase from mirror motion. (b)
Comparable effect from rotating a reflecting dipole to interact with a delayed
polarisation component.

Consider a TLS at a point of circular polarisation,
$\mathbf{E}_{f}=(1,\text{i})$. Here the $y$ component of the electric field is
a quarter-wave delayed relative to the $x$ component. Thus a linear dipole
aligned along the $y$ axis is effectively a quarter wavelength “further away”
than one along $x$. Here rotating a dipole some angle $\Delta\theta$,
transforms the reflectivity as $r=r_{0}\exp(2\text{i}\Delta\theta)$, with
$r_{0}$ the reflectivity for $\Delta\theta=0$. The mechanism is reminiscent of
a Michelson interferometer, where moving the mirror by distance $\Delta x$
delays the phase by the additional round-trip distance, $2\Delta x/\lambda$,
depicted in fig.2. Here we effectively move the mirror by rotating it.

Figure 3: $r$ as a function of $\mathbf{d}$. First column: hue (opacity)
indicates the phase (amplitude) of the reflection from the dipole on that
point of the Poincaré sphere, for the polarisation depicted on the left. These
polarisations are $\mathbf{E}_{f}=[\text{i}\cos(n\pi/12),\sin(n\pi/12)]$ with
(a,b,c,d) having $n=(3,4,5,6)$ respectively. Second column: streamlines
indicating the phase gradient. In (d) the phase is constant so streamlines
cannot be plotted. Instead a key indicates the locations of cardinal dipoles
on the sphere. $S_{1-3}$ mark the axes of the the Stokes Parameters Collett
(2005).

This phase is physically meaningful. One could build a interferometer with a
rotating dipole replacing the moving mirror and measure the effect. The phase
may also be important in other contexts. For example, if multiple TLSs are
coupled to a single waveguide their spacing is of critical importance, as it
determines whether emission and scattering adds constructively or
destructively, thereby controlling the superradiance Jones  _et al._ (2020),
dipole-dipole frequency shifts Dzsotjan  _et al._ (2011); Jones  _et al._
(2018), reflectivity Zhou  _et al._ (2020) and interatom entanglement Mirza
and Schotland (2016); Pichler  _et al._ (2015). But, as we have just
motivated, the phase delay is also dependent on the transition dipoles, so
that the effective emitter spacing depends on how the TLS dipoles are
oriented. Thus, the realisation of proposals that depend on the phase delay
between adjacent TLSs Holzinger  _et al._ (2022), is not determined entirely
by the TLS separation.

The ability of both distance and dipole to control phase motivates a mention
of superconducting giant atoms. While chirality normally exploits the relative
phase between two polarisation components at a single location, these giant
atoms achieve a similar effect by reaching spatially to exploit the relative
phase between two locations in the waveguide Wang  _et al._ (2021).

Requiring that $|\mathbf{E}_{f}|=|\mathbf{d}|=1$ for simplicity and neglecting
global phases both vectors can be written in the form:
$[\cos(\theta),\sin(\theta)\exp(i\phi)]$. The angles $\theta$, $\phi$ can be
used to represent the vectors as points on the surface of the Poincaré sphere.
In the general case a dipole can be picked from anywhere on the surface of
this sphere. The aforementioned linear dipoles comprise the equator, with
circular dipoles at the poles and ellipses elsewhere.

In fig.3,(a) we fix a circular polarisation, and calculate $r$ for all
possible dipoles. Each dipole, $\mathbf{d}$, corresponds to a point on the
sphere with its own value of $r$. In the first column of spheres the hue
indicates the phase $\angle r$, with the opacity indicating $|r|$ [with
$r=|r|\exp(\text{i}\angle r)$]. In the second column the phase gradient is
plotted. Following any line in the arrowhead direction the phase changes by
$-2\pi$ in a complete cycle. The Michelson-like phase motivated above appears
on the equator. We have set $|\mathbf{E}_{f}|=|\mathbf{d}|=1$ and $L=0.01$ to
consider the situation where coupling to loss modes is weak compared to
typical waveguide coupling. Such low loss is appropriate to, for example,
photonic crystal waveguide systems Scarpelli  _et al._ (2019). Due to this low
loss $|r|$ is close to unity almost everywhere on the sphere, becoming
significantly less only when the dot products
$|\mathbf{d}^{*}\cdot\mathbf{E}_{f}|^{2}$ or
$|\mathbf{d}^{*}\cdot\mathbf{E}_{b}|^{2}$ are comparable to $L$.

Plotting the data on a sphere highlights important topological restrictions
that are not obvious from inspection of equ.(1). The phase swirling about the
equator directly requires that in each hemisphere there is a dipole such that
$r=0$. This can be seen from the figure, after fixing the equator one cannot
find a smooth function without at least one zero in each hemisphere.

The zero points are _phase singularities_ , a ubiquitous wave phenomenon that
occur where a complex scalar field takes value $0$ at some location, with the
phase angle varying by $2\pi m$ in a circuit of the zero point Nye and Berry
(1974); Berry (2000). The total phase change along a closed curve is equal to
the $\sum_{n}2\pi m_{n}$ with $n$ counting over the singularities enclosed.
Thus, our $2\pi$ variation along the equator is enough to ensure that both
hemispheres contain at least one phase singularity with $r=0$.

With circular polarisation these points are the poles. At one
$\mathbf{d}^{*}\cdot\mathbf{E}_{f}=0$ and the dipole decouples from the
forwards mode, so the TLS cannot in any way interact with the input photon. At
the other $\mathbf{d}^{*}\cdot\mathbf{E}_{b}=0$ so that whatever the TLS does
to the photon it cannot involve any scatting to the backward mode (hence
$r=0$).

Moving to parts (b, c, d) of the figure we vary the polarisation of the
waveguide. This deforms the phase structures continuously, preserving the two
singularities until they mutually annihilate on the equator for a linear
polarisation. This highlights that for any polarisation there is a dipole that
decouples from the forward mode and another the backward one (the
singularities), with the two coinciding only for a linear polarisation Lang
_et al._ (2022).

The phase gradient can be considered a vector field. As this field lives on
the sphere it is subject to the “hairy ball theorem” which requires that it
cannot be smooth and nonzero everywhere. The theorem name refers to a
consequence of this, that a hairy ball cannot be combed to have all the hair
lie flat. More precisely the theorem requires the total Poincaré-Hopf indices
of the vector field’s zero points equal the sphere’s Euler characteristic of
+2 Berry  _et al._ (2004). The vector fields depicted by streamlines in fig.3
each have two singular points where the vector winds in a circle (the phase
singularities), such circles have an index of +1 irrespective of the arrowhead
directions, so that the pair has the required +2 total.

Figure 4: Each small sphere denotes $r$ as a function of $\mathbf{d}$ as in
fig.3. For each the polarisation, $\mathbf{E}_{f}$ used is indicated by its
location on the larger (wire frame) sphere.

The polarisation is able to explore its own Poincaré sphere of possible
values, such that the space of $\mathbf{E}_{f}\otimes\mathbf{d}$ has the form
$\mathbb{S}^{2}\otimes\mathbb{S}^{2}$, a “sphere of spheres” which we term a
Poincaré hypersphere. This full space is depicted in fig.4, with the larger
sphere indicating the polarisation and the smaller ones the dipole. One sees
that (starting from the north pole of the larger sphere) stretching our
initially circular polarisation brings the polar phase singularities closer to
one another, and that the line of latitude moved along is determined by the
latitude line on the big sphere. Opposite points on the Poincaré sphere
correspond to orthogonal polarisations, so that one phase singularity on each
sub-sphere points to the centre of the larger sphere. This corresponds to the
$\mathbf{d}^{*}\cdot\mathbf{E}_{f}=0$ case. The other phase singularity’s
location on each sub-sphere is given by reflecting the first through the
equator to set $\mathbf{d}^{*}\cdot\mathbf{E}_{b}=0$.

Taking a tangent, it is interesting to consider the geometry. $\mathbf{E}_{f}$
and $\mathbf{d}$ are each 2D, giving us a total of 4 dimensions. A phase
singularity has 2 dimensions fewer than the space it is embedded in, so that
the singularities on our 2-spheres were pointlike (0D), and in 3D space
singularities represent lines of darkness or silence in fields Padgett  _et
al._ (2011). Our singularities in the 4D space are 2D surfaces. There are two
such surfaces, each corresponding to $\mathbf{d}^{*}\cdot(\mathbf{E}_{f}\text{
or }\mathbf{E}_{b})=0$. These surfaces map to spherical shells, and the
intersection of the two maps to a circle. On this circle both dipole and
polarisation are linear and are orthongonal to one another, for example
resembling a “$\times$” in real space. The circular nature of the intersection
relates to the fact that the “$\times$” can be rotated freely, eg. “$+$”.

The transmission, $t$ can be equally assessed on the sphere (or hypersphere).
However $t$ lacks the complex structure seen in $r$. For the most part, a $t$
equivalent of fig.3 simply shows a phase of $\angle t=\pi$ in one hemisphere
and $\angle t=0$ in the other, the two separated by a $|t|=0$ equator. This
$|t|=0$ equator can be considered a phase singularity by re-introducing the
detuning, $\delta$. For a fixed $\mathbf{E}_{f}$, within the 3D space defined
by $\mathbf{d}\otimes\delta$ it takes the form of a 1D line phase singularity,
looped into a circle (near the equator) in the $\delta=0$ slice.

In conclusion, we demonstrate a rich behaviour of the reflection coefficient,
$r$, of a complex dipole TLS in a one-dimensional photonic waveguide. $r$ is a
complex field supporting phase singularities that live on the Poincaré
(hyper)sphere. The existence of two dipoles for each polarisation such that
$r=0$ and the existence of a Michelson-interferometer like phase gradient on
the equator can both be motivated by physical arguments. We showed that the
two effects are intimately linked, due to the topological constraints faced by
phase singularities living on the surface of a sphere. The rich dependence of
$r$ on dipole and waveguide polarisation for a single TLS will offer new
avenues for exploitation of chains of TLSs in waveguides.

Acknowledgement: We acknowledge support from EPSRC grants no. EP/N003381/1
“One-dimensional quantum emitters and photons for quantum technologies” and
EP/M024156/1 “Spin Space” and PHD funding DTA-1407622.

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