#  Nonlinear effects in Anderson localization of light by two-level atoms

Noel Araujo Moreira  Instituto de Física de São Carlos, Universidade de São
Paulo - 13566-590 São Carlos, SP, Brazil  Robin Kaiser  Université Côte
d’Azur, CNRS, INPHYNI, France  Romain Bachelard<EMAIL_ADDRESS>Université
Côte d’Azur, CNRS, INPHYNI, France  Departamento de Física, Universidade
Federal de São Carlos, Rodovia Washington Luís, km 235 - SP-310, 13565-905 São
Carlos, SP, Brazil

###### Abstract

While Anderson is a single-particle wave effect, guaranteeing a single
excitation in the system can be challenging. We here tackle this limitation in
the context of light localization in three dimensions in disordered cold atom
clouds, in presence of several photons. We show that the presence of these
multiple excitations does not affect substantially the abnormal intensity
fluctuations which characterize the Anderson localization transition, provided
that the radiated light is frequency filtered. Due to their narrow linewidth,
long-lived modes, and particularly the localized ones, are strongly saturated
even for a weak resonant pump, leading to a large increase of the inelastic
scattering and to reduced fluctuations in the total radiation. Yet the atomic
coherences and the resulting elastic scattering remain a proper witness of the
Anderson localization transition. Hence, frequency filtering allows one to
investigate the single-excitation sector, dismissing the many-body effects
showing up in the fluorescence spectrum.

Introduction.— Since its introduction in the context of the metal-to-insulator
transition for electronic transport [Anderson1958], Anderson localization by
disorder has been shown to be a general wave phenomenon. In three dimensions,
it has since been reported experimentally for elastic waves [Hu2008], atomic
matter waves [Chabe2008], and electrons [Ying2016]. Anderson localization
formally corresponds to the localization of a single excitation, that is, if
several waves/excitations are present, they do not interact with each other.

In this context, the localization of light seems particularly promising, since
photons are notoriously inefficient at interacting with each other. However,
the initial experimental reports of light localization in 3D [Wiersma1997,
Storzer2006] have been later reinterpreted [Scheffold1999, Scheffold2013,
Sperling2016], and an unambiguous observation is still missing
[Skipetrov2016]. From a theoretical point of view, the near-field terms
coupling the different polarization channels have been pointed at as an
obstacle to localization [Skipetrov2014, Naraghi2016, Cobus2022], challenging
the mere existence of Anderson localization of light in three dimensions.
These advances stimulated new proposals to restore light localization
[Skipetrov2015, Celardo2017], taking advantage of the tunability of the light-
emitter interaction in cold atom platforms, along with their relative absence
of decoherence mechanisms: Decoupling the polarization channels with a strong
external field [Skipetrov2015], or randomly shifting the atomic resonance of
each atom with a disordered field [Celardo2017] – one step closer from the
original Anderson model [Anderson1958].

Nonetheless, while the use of a weak drive may seem sufficient to guarantee
that the atoms will react linearly to the pump field, thus guaranteeing the
pristine condition of non-interacting waves, long-lived modes have actually
been reported to be particularly sensitive to nonlinear effects
[Williamson2020, Cipris2021]. Indeed, localized modes present lifetimes which
are order of magnitude larger than that of single atoms, so their effective
saturation may be equally larger. This questions the possibility to observe
light localization in cold atomic clouds, as nonlinear effects may arise even
for the weakest pumps.

In this work, we investigate the localization of light in disordered clouds of
two-level atoms submitted to a classical weak pump. A mean-field (MF)
approximation allows us to simulate large disordered systems, neglecting the
quantum correlations between the atomic dipoles, yet capturing collective
linewidths and frequency shifts. Close to the atomic resonance, where
localized and subradiant modes are most efficiently addressed, a strong
increase of the inelastic scattering is observed, which stems from the narrow
linewidth of these modes. Nevertheless, we show that the coherence stored in
the atomic dipoles preserves the signature of the localization transition,
even when the localized modes are strongly saturated: The resulting enhanced
intensity fluctuations can be monitored by frequency filtering the radiated
light, as the elastically-scattered signal exhibits these abnormal statistics
[Chabanov2000, Cottier2019]. Our work is thus a first step toward the
transition from single- to multi-excitation localization of light in three-
dimensional cold atom systems.

Single-Excitation vs. Mean-Field.— Let us here consider a cloud of $N$ two-
level atoms (ground and excited states $\ket{g_{j}}$ and $\ket{e_{j}}$,
respectively) with positions $\mathbf{r}_{j}$, with a transition characterized
by its frequency $\omega_{a}$, linewidth $\Gamma$, and raising/lowering
operators $\hat{\sigma}_{j}^{\pm}$ for atom $j$. The system is pumped by a
monochromatic classical field with a Gaussian profile of waist $w_{0}$, Rabi
frequency $\Omega_{0}$ at the waist, and detuned by
$\Delta=\omega_{\textrm{laser}}-\omega_{a}$ from the atomic transition. We
introduce the resonant saturation parameter at the beam waist:
$s_{0}=2\Omega_{0}^{2}/\Gamma^{2}$, and the non-resonant one
$s(\Delta)=2\Omega_{0}^{2}/(\Gamma^{2}+4\Delta^{2})$. Within the Born-Markov
approximation, the light-mediated interactions between the atomic dipoles
gives rise to a master equation of the form
$d\rho/dt=-\frac{i}{\hbar}\mathcal{H}[\rho]+\mathcal{L}[\rho]$ associated to
the following Hamiltonian and Lindbladian:

$\displaystyle\mathcal{H}[\rho]=-\frac{i}{\hbar}\sum_{j=1}^{N}[H_{j},\rho]-i\sum_{j,m\neq
j}^{N}\Delta_{jm}[\hat{\sigma}^{+}_{j}\hat{\sigma}^{-}_{m},\rho],$ (1)
$\displaystyle\mathcal{L}[\rho]=\frac{1}{2}\sum_{j,m}\Gamma_{jm}[2\hat{\sigma}^{-}_{j}\rho\hat{\sigma}^{+}_{m}-\hat{\rho}\sigma^{+}_{m}\hat{\sigma}^{-}_{j}-\hat{\sigma}^{+}_{m}\hat{\sigma}^{-}_{j}\rho],$
(2)

with
$H_{j}=-\frac{\hbar}{2}\Delta_{j}\hat{\sigma}^{z}_{j}+\frac{\hbar}{2}\Omega_{j}(\hat{\sigma}_{j}^{+}+\hat{\sigma}_{j}^{-})$
the single-atom Hamiltonian term ($\Omega_{j}$ the local Rabi frequency),
$\Delta_{jm}=-(\Gamma/2)\cos(k_{0}|\mathbf{r}_{j}-\mathbf{r}_{m}|)/k_{0}|\mathbf{r}_{j}-\mathbf{r}_{m}|$
and
$\Gamma_{jm}=\delta_{jm}\Gamma+(1-\delta_{jm})\Gamma\sin(k_{0}|\mathbf{r}_{j}-\mathbf{r}_{m}|)/k_{0}|\mathbf{r}_{j}-\mathbf{r}_{m}|$
the dipole-dipole interaction terms [Lehmberg1970]. Here we work in the scalar
wave approximation, in which localization occurs without resorting to external
fields [Skipetrov2015, Celardo2017].

Anderson localization refers formally to single-excitation dynamics, when the
waves do not interact with each other. This corresponds to the case of states
with at most one photon, of the form
$\ket{\psi}=\alpha\ket{g_{1}g_{2}..g_{N}}+\sum_{j=1}^{N}\beta_{j}\ket{g_{1}..e_{j}..g_{N}}$,
which leads to the following equation describing the evolution of the atomic
coherences:

$\frac{d\beta_{j}}{dt}=\left(i\Delta-\frac{\Gamma}{2}\right)\beta_{j}-i\frac{\Omega_{j}}{2}-\frac{\Gamma}{2}\sum_{m\neq
j}^{N}\frac{e^{ik_{0}|\mathbf{r}_{j}-\mathbf{r}_{m}|}}{ik_{0}|\mathbf{r}_{j}-\mathbf{r}_{m}|}\beta_{m},$
(3)

hereafter referred to as Coupled Dipole Equations (CDE). The set of equations
(3) describes the optical coherences,
$\beta_{j}=\langle\hat{\sigma}_{j}^{-}\rangle$, and contains no information on
the excited population. In order to investigate the role of a finite pump
strength, and thus the presence of multiple photons in the system, we resort
to the MF approach. It accounts for the finite atomic population,
$z_{j}=\langle\hat{\sigma}_{j}^{z}\rangle$, and neglects connected
correlations between the atoms:
$\langle\sigma_{j}^{\alpha}\sigma_{m}^{\beta}\rangle\equiv\langle\sigma_{j}^{\alpha}\rangle\langle\sigma_{m}^{\beta}\rangle$.
We then obtain the following $2N$ equations for the coherences $\beta_{j}$ and
the populations $z_{j}$ [Santo2020]:

$\displaystyle\frac{d\beta_{j}}{dt}$
$\displaystyle=\left(i\Delta-\frac{\Gamma}{2}\right)\beta_{j}+iW_{j}z_{j},$
(4a) $\displaystyle\frac{dz_{j}}{dt}$
$\displaystyle=-\Gamma(1+z_{j})-4\imaginary(\beta_{j}W_{j}^{*}),$ (4b)
$\displaystyle W_{j}$
$\displaystyle=\frac{\Omega_{j}}{2}-\frac{\Gamma}{2}\sum_{m\neq
j}^{N}\frac{e^{ik_{0}|\mathbf{r}_{j}-\mathbf{r}_{m}|}}{ik_{0}|\mathbf{r}_{j}-\mathbf{r}_{m}|}\beta_{m},$
(4c)

where $W_{j}$ is the effective Rabi frequency for atom $j$, composed of the
pump and of the radiation from other atoms. The MF approximation is necessary
to reduce drastically the complexity of the Hilbert space of a 3D system, yet
accounting for the saturation of the atoms. In particular, the set of
equations (4) is nonlinear, as one enters the realm of nonlinear optics where
waves can interact with each other through the atomic medium.

The far-field intensity of the light scattered by the atoms in a direction
$\hat{n}$,
$I_{\mathrm{tot}}=\sum_{j,m}e^{-ik\hat{n}.(\mathbf{r}_{j}-\mathbf{r}_{m})}\langle\hat{\sigma}_{m}^{+}\hat{\sigma}_{j}^{-}\rangle$,
can be decomposed into an elastically- and inelastically-scattered components,
$I_{\mathrm{tot}}=I_{\mathrm{el}}+I_{\mathrm{in}}$, given by:

$\displaystyle I_{\mathrm{el}}$
$\displaystyle=\bigg{|}\sum_{j}e^{-ik\hat{n}.\mathbf{r}_{j}}\beta_{j}\bigg{|}^{2},$
(5a) $\displaystyle I_{\mathrm{in}}$
$\displaystyle=\sum_{j}\frac{1+z_{j}}{2}-\left|\beta_{j}\right|^{2}.$ (5b)

Note that the inelastic component (5b) contains only single-atom contributions
due to the MF approximation, which neglects two-atom connected correlations.

Intensity fluctuations from saturated atoms.— Intensity fluctuations have been
reported to witness the Anderson localization transition in the single
excitation regime [Chabanov2000, Cottier2019]: Let us now probe these in the
multiple-excitation regime, using the MF approach (4). In Fig. 1, the
evolution of the intensity variance of the scattered light,
$\sigma_{I}=\langle I^{2}\rangle/\langle I\rangle^{2}$, computed over
different azimuthal angles and realizations, is presented for a saturation
parameter $s_{0}=0.1$: Panel (a) depicts the fluctuations of the elastically-
scattered intensity $I_{\textrm{el}}$ in a range of detuning and density for
which localization manifests [see panel (c) and Ref. [Cottier2019] for the
single-excitation case]. The radiation from the coherences (5a) thus presents
large intensity fluctuations in the localization region, despite the presence
of multiple excitations in the system. Indeed, at first order the number of
excitations $N_{e}$ in the cloud can be evaluated making an independent
scatterer hypothesis where the cloud holds $N_{e}=Ns/2(1+s)$ excitations. In
the case of Fig. 1, this corresponds to more than a hundred excitations.

Figure 1: Variance of the scattered intensity, computed from its fluctuations
over the azimuthal angle and over 50 realizations (a) for the elastically-
scattered light $I_{\textrm{el}}$, (b) for the total intensity
$I_{\textrm{el}}+I_{\textrm{in}}$, and (c) in the single-excitation regime
(3). Simulations realized for a cylindrical cloud with radius $R=3\lambda$ and
length $h=6\lambda$, pumped by a Gaussian beam with waist $w_{0}=1.5\lambda$
and saturation parameter $s_{0}=0.1$.

The total intensity scattered $I_{\textrm{el}}+I_{\textrm{in}}$ presents a
very different behaviour, with reduced fluctuations close to resonance, see
panel (b). This feature stems from the nature of the fluctuations investigated
here: As mentioned before, these fluctuations refer to variations over space
(azimuthal angle) and atomic realizations of the intensity computed as an
expected value $I\sim\langle\hat{E}^{\dagger}\hat{E}\rangle$. For a given
direction of observation and a given realization, this expected value formally
corresponds to an infinitely-long measurement for static atoms. Practically,
this measurement needs to be long enough to capture a large number of photons
and get a statistically representative intensity, yet short enough to prevent
the loss of coherence from mechanisms such as the atomic motion
[Lassegues2023]. However, spontaneous emission from the excited state brings
in a new time scale, that is, the excited state lifetime $1/\Gamma$. A proper
detection of the fluctuations in the spontaneously emitted field require a
treatment which addresses quantum fluctuations, so phenomena such as photon
bunching and antibunching on a time scale $1/\Gamma$ can be addressed. In the
context of the master equation used here, this would mean dealing with higher-
order atom-atom correlations [Pucci2017, Plankensteiner2022], which is beyond
the scope of our work.

Thus, the measurement we consider does not capture fluctuations stemming from
spontaneous emission, and the inelastic scattering contributes an homogeneous
background for the radiated light. This is illustrated in Fig. 2(a) where the
polar profile of the intensity is plotted. The total intensity (red dash-
dotted curve) presents the same fluctuations as the elastic component, yet
shifted by the inelastic homogeneous background. Note that the linear CDE
presents slightly different fluctuations from the MF approach: This is a
signature of the saturation of the atoms, that is, of the excited population,
which is accounted for in that model. The associated intensity probability
density functions are represented in panel (b), where both the elastic
component of the MF and the single-excitation signal exhibit increased
fluctuations, with tails larger than for Rayleigh law,
$P(I)\sim\exp(-I/\langle I\rangle)$, valid for uncorrelated scatterers. We
note that these tails are responsible for the enhanced fluctuations in
presence of Anderson localization [Cottier2019]. Due to the inelastic
background, the total intensity explores, relatively, a smaller range of
values, which results in reduced fluctuations.

Figure 2: (a) Intensity as a function of the azimuthal angle $\phi$,
considering the elastic component, the total field, and the single-excitation
sector. (b) Intensity probability distribution function $P(I)$ for the same
parameters, averaged over 100 realizations. Simulations realized for a
spherical cloud of $N=1500$ particles, density $\rho=25/\lambda^{3}$, detuning
$\Delta=0.8\Gamma$ and saturation parameter $s_{0}=0.1$.

The fact that the localized modes are able to contribute substantially with
elastically scattered light, and that strong intensity fluctuations can be
observed, is not trivial: Indeed, long-lived modes are saturated even for low
saturation parameters due to their narrow linewidth [Williamson2020,
Cipris2021]. This is confirmed by the spectral analysis of the scattered
power, monitoring the elastically and inelastically scattered powers:

$\displaystyle
P_{\mathrm{el}}=4\pi\sum_{j,m}\frac{\sin(k_{0}|\mathbf{r}_{j}-\mathbf{r}_{m}|)}{k_{0}|\mathbf{r}_{j}-\mathbf{r}_{m}|}\beta_{j}\beta_{m}^{*},$
(6) $\displaystyle
P_{\mathrm{in}}=4\pi\left[\sum_{j}\frac{1+z_{j}}{2}-|\beta_{j}|^{2}\right].$
(7)

These are obtained by integrating the intensity over all angles, and for
independent scatterers the ratio between them is simply given by the
saturation parameter: $P_{\textrm{el}}/P_{\textrm{in}}=1/s(\Delta)$. We thus
define the ratio
$R_{\textrm{el/in}}=s(\Delta)P_{\textrm{el}}/P_{\textrm{in}}$, which
quantifies the inelastic contribution beyond the single atom effect. Its
behavior as a function of the detuning and saturation parameter is presented
in Fig. 3(a): Close to resonance, where most long-lived modes are encountered
and populated [Guerin2017], spontaneous emission is stronger than for
independent scatterers, which can be interpreted as the fact that even
relatively low saturation parameters ($s\sim 10^{-4}$) are able to saturate
the localized modes and make them radiate inelastically. Oppositely, far from
resonance, the broad-linewidth superradiant modes are less saturated than
independent scatterers would be, which in turn results in a stronger elastic
scattering, yielding a ratio $R_{\textrm{el/in}}>1$.

Delving farther into collective scattering modes, we investigate the
contribution of the localized modes to the optical coherences $\beta_{j}$. The
modes are considered to be localized when their spatial shape presents an
exponential decaying profile (more precisely, when the logarithm of their
profile presents a linear decay with a $R^{2}$ Pearson parameter larger than
$0.5$ [Moreira2019]). We then decompose the vector of the steady-state
coherences $\beta_{j}$ onto the basis of eigenvectors from the single-
excitation sector, (that is, the eigenvectors $\hat{\Psi}_{n}$ of the
scattering matrix of (3)) as $\sum_{n}\alpha_{n}\hat{\Psi}_{n}$, and define
the weight of the each mode in the coherence vector as $|\alpha_{n}|^{2}$. The
map of this population is presented in Fig. 3(b), in the complex plane of
eigenvalues $\lambda_{n}=i\omega_{n}-\Gamma_{n}$, with $\gamma_{n}$ the mode
inverse lifetime and $\omega_{n}$ its shift from the atomic resonance –
superradiant modes thus correspond to $\gamma_{n}>\Gamma$. Localized modes are
weakly populated compare to superradiant ones, yet their large number makes up
for their weak coupling to the external world.

Figure 3: (a) Normalized ratio of elastically to inelastically scattered power
$R_{\textrm{El/In}}=s(\Delta)P_{\textrm{el}}/P_{\textrm{in}}$, as a function
of $\Delta$ and $s_{0}$, for a density $\rho=25/\lambda^{3}$. (b) Population
of each mode, $|\alpha_{n}|^{2}$, in the complex plane of eigenvalues
$\lambda_{n}=i\omega_{n}-\Gamma_{n}$, for a drive with $s_{0}=10^{-1}$ and
$\Delta=0.5\Gamma$. The upper and rectangle encompass the localized (Loc) and
superradiant (SR) states, respectively. (c) Evolution of the population of
localized and superradiant states in the atomic coherences, as a function of
the saturation parameter $s$. Simulation realized for a spherical cloud with
$N=1000$ particles, density $\rho=146/\lambda^{3}$, and a drive detuned by
$\Delta=1.27\Gamma$, which corresponds to the typical energy of localized
modes for these parameters), averaged of 20 realizations. (d) Relative weight
of the localized modes, defined as
$(\sum_{n\in\textrm{Loc/SR}}|\alpha_{n}|^{2})/\sum_{n\in\textrm{All}}|\alpha_{n}|^{2}$,
as a function of the saturation parameter $s(\Delta)$. Same parameters as for
(c).

.

Let us now define the weight of the localized (superradiant) modes in the
coherences as $W_{\textrm{Loc}}=\sum_{n\in\textrm{Loc}}|\alpha_{n}|^{2}$
($W_{\textrm{SR}}=\sum_{n\in\textrm{SR}}|\alpha_{n}|^{2}$). As shown in Fig.
3(c), a below-linear growth of the weight with the saturation parameter
$s_{0}$ is observed for localized modes.This is yet another evidence that the
localized modes are more easily saturated than superradiant ones, so their
population grow slower with the saturation parameter. Note that this situation
is different from the decay dynamics probed in Ref. [Cipris2021], where the
decay by collective spontaneous emission from multiple excitation states
toward few excitation ones actually increases the contribution of long-lived
states to the radiative dynamics: While the difference between spontaneously-
emitted light and the coherently-scattered one was not done in that work, we
here focus on the coherences, in the steady-state regime. In particular,
monitoring the relative weight of localized/superradiant modes in the
coherence vector (by normalizing the vector of $\\{\alpha_{n}\\}$), the
relative contribution of the localized ones is reduced by a factor three as
the saturation parameter increases by four orders of magnitude, for the
benefit of superradiant ones, see Fig. 3(d) – the remaining population lies in
subradiant extended modes [Moreira2019]. The present observation of largely
saturated localized modes even at low saturation parameter makes it all the
more remarkable that the elastic-scattering intensity fluctuations
characteristic of the localization transition be preserved for a finite-
strength drive.

Perspectives.— Although localized modes are effectively strongly saturated by
relatively weak probes, for which single atoms would remain very close to the
ground state, the signature of intensity fluctuations at the localization
transition is preserved, provided that the light is filtered to select the
elastic scattering component. This result is particularly important for setups
where the single-photon condition – the pristine condition for Anderson
localization of light – is challenging to achieve.

Our work paves the way to future studies on light scattering in presence of
multiple excitations [Rubies2023]. In particular, the fluorescence spectrum of
collective modes may also reveal precious information regarding the
correlations between the dipoles [Pucci2017]. Hence, delving deeper in the
hierarchy of quantum correlations is a next natural step to understand the
many-body regime of these disordered systems [Fayard2021].

###### Acknowledgements.

R.B. acknowledges the financial support of the São Paulo Research Foundation
(FAPESP) (Grants No. 2018/15554-5, 2019/13143-0 and 2022/00209-6), from the
Brazilian CNPq (Conselho Nacional de Desenvolvimento Científico e
Tecnológico), Grant No. 313886/2020-2, and from the French government, through
the UCA J.E.D.I. Investments in the Future project managed by the National
Research Agency (ANR) with the reference number ANR-15-IDEX-01. RB and RK
received support from the project STIC-AmSud (Ph879-17/CAPES
88887.521971/2020-00), CAPES-COFECUB (CAPES 88887.711967/2022-00) and from the
European project ANDLICA, ERC Advanced grant No. 832219.

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subsystems of $\mathbb{V}_{\mathbb{Z}}=R^{1}f_{*}\mathbb{Z}_{\mathcal{X}}$,
which we want to describe a little bit in the following:

The action of $\mbox{Aff}^{+}(X,\omega)$ on
$H_{\text{st}}^{1}(\mathcal{O},\mathbb{Z})$ under the identificaiton of the
span $\langle\mbox{Re}(\omega),\mbox{Im}(\omega)\rangle_{\mathbb{Z}}$ with
$\mathbb{Z}^{2}$ is just the standard action of the derivative
$D(\mbox{Aff}^{+}(X,\omega))\subset\mbox{SL}_{2}(\mathbb{Z})$ on
$\mathbb{Z}^{2}$. Hence the action $\rho_{\text{triv}}$ of $\pi_{1}(C,c)$ on
$H_{\text{st}}^{1}(\mathcal{O},\mathbb{Z})$ is also given by the standard
action. This shows that the local subsystem $\mathbb{L}^{\text{id}}$ of
$R^{1}f_{*}\mathbb{R}_{\mathcal{X}}$ from remark 18 is defined over
$\mathbb{Z}$ in the origami case $\mathcal{O}=(X,\omega)$. Thus the action
$\rho_{\text{triv}}$ corresponds to a $\mathbb{Z}$-local subsystem
$\mathbb{L}_{\mathbb{Z}}$ of $\mathbb{V}_{\mathbb{Z}}$ such that
$\mathbb{L_{\mathbb{Z}}}\otimes_{\mathbb{Z}}\mathbb{R}=\mathbb{L}^{\text{id}}$.

To describe the action
$\rho_{\text{sh}}\colon\pi_{1}(C,c)\to\mbox{Sp}_{\Omega^{*}}(H_{(0)}^{1}(\mathcal{O},\mathbb{Z}))$
and the corresponding local subsystem of
$\mathbb{V}_{\mathbb{Z}}=R^{1}f_{*}\mathbb{Z}_{\mathcal{X}}$, we will use
Theorem 17 in combination with the following Theorem of Gutkin and Judge:

######  Theorem 22 ([20],Thm. 5.5 and 7.1).

For the trace field $K(X,\omega)$ of an Origami $\mathcal{O}=(X,\omega)$, we
have $K(X,\omega)=\mathbb{Q}$.

From Theorem 17 we know that the local system
$\mathbb{V}_{\mathbb{Q}}=R^{1}f_{*}\mathbb{Z}_{\mathcal{X}}\otimes_{\mathbb{Z}}\mathbb{Q}$
splits over $\mathbb{Q}$ as

$\mathbb{V}_{\mathbb{Q}}=\mathbb{W}_{\mathbb{Q}}\oplus\mathbb{M}_{\mathbb{Q}}.$

We have a further splitting of $\mathbb{W}_{\mathbb{Q}}$ over the Galois
closure $F$ of the trace field $K(X,\omega)$ in
$\mathbb{W}_{F}=\oplus_{\sigma}\mathbb{L}^{\sigma}$ with
$\sigma\in\mbox{Gal}(F|\mathbb{Q}){/}\mbox{Gal}(F|K(X,\omega))$. But since in
the origami case $K(X,\omega)=\mathbb{Q}$ by Theorem 22, we get
$\mathbb{W}_{F}=\mathbb{L}_{F}^{\text{id}}$. We conclude that for an Origami
$\mathcal{O}$ of genus $g$ the local system
$\mathbb{V}_{\mathbb{Z}}=R^{1}f_{*}\mathbb{Z}_{\mathcal{X}}$ splits as

$\mathbb{V}_{\mathbb{Z}}=\mathbb{L}_{\mathbb{Z}}\oplus\mathbb{U}_{\mathbb{Z}},$

where $\mathbb{U}_{\mathbb{Z}}$ is the $\mathbb{Z}$-local system of rank
$2(g-1)$ corresponding to the action of $\pi_{1}(C,c)$ on
$H_{(0)}^{1}(\mathcal{O},\mathbb{Z})$ and $\mathbb{L}_{\mathbb{Z}}$ is the
local system correspnding to the trivial action of $\pi_{1}(C,c)$ on
$H_{\text{st}}^{1}(\mathcal{O},\mathbb{Z})$. Furthermore
$\mathbb{U}_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathbb{Q}\cong\mathbb{M}_{\mathbb{Q}}$
and therfore we know from Theorem 17 that $\mathbb{U}_{\mathbb{Z}}$ carries a
polarized VHS of weight one, which we denote by $(\mathbb{U}_{\mathbb{Z}},\
\mathcal{U}^{1,0},Q_{U})$.

######  Theorem 23.

Let $\mathcal{O}=(X,\omega)$ be an origami of genus $g=2$ with associated
family of curves $f\colon\mathcal{X}\to C$ as in Section B.1.4. Then
$\mbox{Sp}_{\Omega^{*}}(H_{(0)}^{1}(\mathcal{O},\mathbb{Z}))\cong\mbox{SL}_{2}(\mathbb{Z})$
and the image of the map

$\rho_{\text{sh}}\colon\text{SL}(\mathcal{O})\longrightarrow\mbox{SL}_{2}(\mathbb{Z})$

is a finite index subgroup of $\mbox{SL}_{2}(\mathbb{Z})$.

###### Proof.

In genus $g=2$ the module $H_{(0)}^{1}(\mathcal{O},\mathbb{Z})$ has rank two
and hence $\mbox{Sp}_{\Omega^{*}}(H_{(0)}^{1}(\mathcal{O},\mathbb{Z}))$ is
isomorphic to $\mbox{SL}_{2}(\mathbb{Z})$.

The period mapping of the polarized vHS $(\mathbb{U}_{\mathbb{Z}},\
\mathcal{U}^{1,0},Q_{U})$ is in our situation given by the following map
(compare [33] Section 4): There exists a global section $\omega$ of the
$(1,0)$-part of the pullback of
$\mathbb{U}_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathcal{O}_{C}$ to the universal
cover $\mathbb{H}$ of $C$ ([14], Theorem 30.3). Choose locally a symplectic
basis $\\{a,b\\}$ of the pull back of $\mathbb{U}_{\mathbb{Z}}^{\vee}\subset
R_{1}\mathbb{Z}_{\mathcal{X}}$ to $\mathbb{H}$ such that

$\int_{a(\tau)}\omega(\tau)\in\mathbb{H}\quad\text{and}\quad\int_{b(\tau)}\omega(\tau)=1$

for every $\tau\in\mathbb{H}$.

The period domain is in our situation analytic isomorphic to the Siegel upper
half space $\mathbb{H}$ (see Proposition 1.24 in [17]) and the period mapping
is given by

$P\colon\mathbb{H}\longrightarrow\mathbb{H},\quad\tau\longmapsto\int_{a(\tau)}\omega(\tau).$

By part b) of Proposition 20 the period mapping
$P\colon\mathbb{H}\to\mathbb{H}$ descends to a holomorphic map $p\colon
C\to\rho_{\text{sh}}(\pi_{1}(C,c)){\setminus}\mathbb{H}$. Recall that
$\rho_{\text{sh}}(\pi_{1}(C,c))$ acts discontinously on $\mathbb{H}$.

We show in the following that the map $p$ is not constant. Write
$\text{deg}(\mathcal{U}^{1,0})$ and $\text{deg}(\mathcal{L}^{1,0})$ for the
degrees of the line bundles $\mathcal{U}^{1,0}$ and $\mathcal{L}^{1,0}$, where
$\mathcal{U}^{1,0}$ and $\mathcal{L}^{1,0}$ are the $(1,0)$-parts of the Hodge
filtration of the Deligne extension of
$\mathbb{U}_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathcal{O}_{C}$ and
$\mathbb{L}_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathcal{O}_{C}$ to
$\overline{C}$. With Proposition 19 from above and Proposition 8.5 in [3] we
conclude for the positive Lyapunov exponent $\lambda_{U}$ corresponding to
$\mathbb{U}_{\mathbb{Z}}$:

$\lambda_{U}=\frac{\text{deg}(\mathcal{U}^{1,0})}{\text{deg}(\mathcal{L}^{1,0})}=\frac{1}{3}\quad\text{or}\quad\lambda_{U}=\frac{\text{deg}(\mathcal{U}^{1,0})}{\text{deg}(\mathcal{L}^{1,0})}=\frac{1}{2}$

Thus $\text{deg}(\mathcal{U}^{1,0})\neq 0$, the variation of Hodge structure
$(\mathbb{U}_{\mathbb{Z}},\ \mathcal{U}^{1,0},Q_{U})$ is non trivial and the
period mapping $p\colon
C\to\rho_{\text{sh}}(\pi_{1}(C,c)){\setminus}\mathbb{H}$ is non-constant and
thus open.

Next we want to show that
$\rho_{\text{sh}}(\pi_{1}(C,c)){\setminus}\mathbb{H}$ has finite hyperbolic
volume. By Theorem 9.5 in [19] we can extend the period mapping $p$
holomorphically to a map

$p\colon C\cup
S\longrightarrow\rho_{\text{sh}}(\pi_{1}(C,c)){\setminus}\mathbb{H},$

where $S\subset\partial C$ denotes those cusps, for which $\rho_{\text{sh}}$
maps the corresponding parabolic elements in the monodromy representation to
elements of finite order in $\mbox{SL}_{2}(\mathbb{Z})$. From Proposition 9.11
and the proof of Theorem 9.6 in [19] we conclude that $p\colon C\cup
S\to\rho_{\text{sh}}(\pi_{1}(C,c)){\setminus}\mathbb{H}$ is proper and since
$p\colon C\cup S\to\rho_{\text{sh}}(\pi_{1}(C,c)){\setminus}\mathbb{H}$ is
also holomorphic and non-constant, it is surjective. Furthermore Theorem 9.6
in [19] implies that
$\rho_{\text{sh}}(\pi_{1}(C,c)){\setminus}\mathbb{H}=p(C\cup S)$ has finite
hyperbolic volume or equivalently the group $\rho_{\text{sh}}(\pi_{1}(C,c))$
has finite index in $\mbox{SL}_{2}(\mathbb{Z})$ (see [37], Prop. 1.31). ∎

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