# Collective nuclear excitation dynamics in mono-modal x-ray waveguides

Leon M. Lohse<EMAIL_ADDRESS>Georg-August-Universität Göttingen,
37077 Göttingen, Germany  Deutsches Elektronen-Synchrotron DESY, 22607
Hamburg, Germany  Petar Andrejić  Friedrich-Alexander-Universität Erlangen-
Nürnberg, 91058 Erlangen, Germany  Sven Velten  Deutsches Elektronen-
Synchrotron DESY, 22607 Hamburg, Germany  Malte Vassholz  Georg-August-
Universität Göttingen, 37077 Göttingen, Germany  Charlotte Neuhaus  Georg-
August-Universität Göttingen, 37077 Göttingen, Germany  Ankita Negi  Deutsches
Elektronen-Synchrotron DESY, 22607 Hamburg, Germany  Anjali Panchwanee
Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany  Ilya Sergeev
Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany  Adriana Pálffy
Julius-Maximilians-Universität Würzburg, 97074 Würzburg, Germany  Tim Salditt
Georg-August-Universität Göttingen, 37077 Göttingen, Germany  Ralf Röhlsberger
Friedrich-Schiller-Universität Jena, 07743 Jena, Germany  Helmholtz-Institut
Jena, 07743 Jena, Germany  GSI Helmholtzzentrum für Schwerionenforschung GmbH,
64291 Darmstadt, Germany  Deutsches Elektronen-Synchrotron DESY, 22607
Hamburg, Germany

###### Abstract

Ensembles of identical atoms exhibit peculiar collective properties in their
interaction with radiation depending on geometry and environment where they
are embedded in. A remarkably clean and versatile platform to study collective
effects in resonant light scattering are Mössbauer nuclei placed in planar
x-ray waveguides. Here we conceive and demonstrate experimentally distinct
temporal emission characteristics in these systems, ranging from a tunable
accelerated exponential decay all the way to a pronounced oscillatory emission
pattern, depending on the waveguide geometry and mode of excitation. The
observed temporal and spatial emission characteristics of the collectively
excited nuclear state in the waveguide – the nuclear exciton – are well
reproduced by a unified theoretical model. Our findings pave the way for
applications ranging from fundamental studies of cooperative emission at hard
x-ray frequencies up to new methods of narrowband x-ray control via the
engineering of collective radiation patterns.

Ensembles of excited identical two-level atoms exhibit a collective dynamics
during their spontaneous decay that is fundamentally different from that of
the constituent individual atoms [1]. Collective effects such as directed
emission [2, 3], collective frequency shifts, superradiant decay [4], and
collective oscillations emerge already in the regime of weak excitation from
as little as a single photon [5]. They have been observed over a large range
of platforms, from one-dimensional chains of atoms coupled to optical
waveguides [6, 7, 8, 9, 10] to Mössbauer nuclei driven by hard x-rays [11, 12,
13, 14, 15, 16, 17]. In spatially extended ensembles, the geometry and spatial
distribution distinctively affects the dynamics [18, 19].

A well-suided platform to observe and harness collective effects are ensembles
of Mössbauer isotopes in solid state systems, where nuclear transitions
resonant to hard x-ray frequencies have extremely narrow linewidths [20],
making exceptionally clean quantum-optical systems [21, 22]. X-ray pulses
produced by synchrotron or x-ray free electron laser light sources are several
orders of magnitude shorter than the natural life time of the excited states.
Thus, excitation and decay can be considered as separate processes and the
collective excitation of the nuclear ensemble, the “nuclear exciton” [11, 12],
evolves freely after impulsive excitation.

The temporal evolution of the nuclear exciton distinctively depends on the
geometrical shape of the nuclear ensemble and its environment. In a thick slab
or foil illuminated in normal incidence, it radiates so-called dynamical beats
(DB) in forward direction [13, 23] – also known as nuclear forward scattering
(NFS). On the other hand, as a thin film embedded in an x-ray waveguide (WG)
and excited in grazing-incidence (GI) illumination it decays exponentially and
emits into the direction of specular reflection – which we call reflection
geometry – with superradiant speedup and a shifted resonance frequency
compared to the single atom [15]. Combining multiple such films in stacked
WGs, the exciton has even been observed to oscillate between them [16]. Both
scattering geometries have been studied extensively, yet their “overlap”,
bounded WGs, still remain elusive.

In this Letter, we investigate the dynamical evolution of the nuclear exciton
of an ultrathin layer of ^57Fe embedded in an x-ray WG. We present two
experiments, in which we excite the sample either in front-coupling (FC)
geometry [Fig. 1(a)] [24] or in GI geometry [Fig. 1(b)] [25] and observe the
photons that have been emitted into resonant modes of the WG leaving its back
end. Depending on the mode of excitation, the exciton either exhibits the same
dynamical beats as in NFS [Fig. 1(a) right] or decays exponentially with
superradiant speedup [Fig. 1(b) right]. The back (and front) boundaries of the
WG break spatial translational symmetry and demand an interpretation different
from models established so far [26, 27, 28] for the reflection geometry. Our
observations can be well modelled by applying a recently introduced approach
based on a real-space Green’s function formalism [29]. We obtain a one-
dimensional equation of motion that governs both geometries in terms of a few
palpable coefficients, which can be solved analytically. It provides a natural
and quantitative interpretation of the nuclear exciton dynamics in both
geometries in a unified _waveguide_ picture (including the spatial
propagation).

Figure 1: A short excitation pulse (blue) couples into a planar WG and creates
a nuclear exciton in the ^57Fe layer (red) that radiatively decays (green).
(a) FC excitation creates an exciton that emits pronounced temporal beatings.
(b) GI excitation of a _long_ WG: the exciton decays exponentially.
Superradiant speedup and shifted resonance, compared to the natural
exponential decay with rate $\gamma$ (dotted green line), depend on the
incidence angle $\theta_{\mathrm{in}}$.

Hard x-rays are an extreme regime for waveguiding. In the energy regime above
$10\text{\,}\mathrm{keV}$, refractive indices of matter are written as
$n=1-\delta+i\beta$, where $\delta,\beta<${10}^{-5}$$. X-ray waveguiding
manifests when lighter core materials are surrounded by denser cladding
materials. Because of the small contrasts in $n$, x-ray WGs are weakly guiding
(modes typically extend over a few 100 wavelengths in the transversal
direction) and cause significant photo-electric absorption in the dense
cladding. Consequently, the Purcell enhancement is negligible and coupling to
guided – or more generally resonant – modes is small [30]. However, dense
cladding strongly attenuates everything but the lowest few resonant modes
[31], so that fields generated by emitters in the WG can be asymptotically
approximated by only their resonant components [30].

We have prepared two planar x-ray WGs, both containing a few-atomic-layers-
thin iron film, to $95\text{\,}\mathrm{\char 37\relax}$ enriched in ^57Fe, in
the center of its guiding layer (see Supplemental Material [32]). The ^57Fe
does not develop long-range magnetic order in this thin film and hence
exhibits essentially a single resonance line. The experiments were performed
at the dynamics beamline P01 at PETRA III (DESY, Hamburg). The samples were
illuminated by $100\text{\,}\mathrm{ps}$-long synchrotron pulses at a
repetition period of $192\text{\,}\mathrm{ns}$, creating nuclear excitons. The
photons leaving the WG at its back end were detected with a stack of avalanche
photo diodes (APDs). By temporal gating, the (delayed) temporal emissions of
the nuclear exciton are detected independently from the (prompt) synchrotron
pulse.

Figure 2: Layer design and experimental data for FC excitation. (a)
Transversal mode profiles $u_{m}(z)$ of the 2 guided modes supported by the
WG. (b) Experimentally observed emissions as a function of delay $t$ for
different WG lengths $L$ (colored points). The intensities are normalized and
vertically shifted. The dashed lines show simulations of NFS through ^57Fe-
foils, using parameters (including foil thickness $L_{\mathrm{foil}}$) that
were estimated from the experimental data. (c) Extracted foil thickness and
resonant optical depth $L_{\mathrm{foil}}/\Lambda_{\mathrm{res}}$ for
different $L$ (colors indicate 2 independent measurements).

First, we have prepared a layer system optimized for FC excitation, consisting
of a $0.6\text{\,}\mathrm{nm}$-thin layer of ^57Fe, embedded in a planar x-ray
WG with $20\text{\,}\mathrm{nm}$ B_4C core, and symmetric
$30\text{\,}\mathrm{nm}$-thick Mo cladding as sketched in Fig. 2(a). The layer
structure was deposited onto a $1\text{\,}\mathrm{mm}$-thick Ge wafer and
capped with a second Ge wafer after deposition as detailed in Ref. [33]. The
synchrotron beam was focussed with two elliptical mirrors in Kirkpatrick-Baez
geometry to a spot of about $7\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diameter
into the WG entrance at the focal position. The Ge wafers were used to absorb
the tails of the focus to reduce background signal, but they do not affect the
resonant mode structure due to the thick cladding [34]. The wafer sandwich was
cut to triangular shape, allowing us to change the effective WG length $L$ by
translating it along $y$ transversal to the beam (see Supplemental Material
[32]). The WG length was calibrated by measuring the absorption in the wafer
as a function of $y$-translation. Figure 2(b) shows the temporal evolution of
the emitted x-rays for several WG lengths up to $2\text{\,}\mathrm{mm}$
(limited by off-resonant absorption). The emissions clearly exhibit dynamical
beats resembling those known from NFS through a single line resonant absorber
_foil_. The dashed lines show simulations of NFS through a thick foil of
enriched ^57Fe including slight inhomogeneous hyperfine-splitting in the order
of $6$ natural linewidths, in a 4-parameter model (see Supplemental Material
[32]), which accurately describes the data. The optimal parameters were found
using a maximum-likelihood estimation assuming Poisson statistics. The
magnitude of hyperfine splitting is expected from similar amorphous thin films
[15, 35]. The extracted optical depth and effective foil thickness [Fig. 2(c)]
is strictly proportional to the WG length $L$.

Figure 3: Off-resonant intensity of the GI sample, illuminated at an incidence
angle of $\theta_{\mathrm{in}}$ and recorded $3\text{\,}\mathrm{m}$ downstream
of the sample with a pixel detector. Vertical cuts correspond to observed far-
field patterns. The 4 resonant modes, which are supported by the layer
structure, are clearly visible in the region between the primary beam
($\theta_{\mathrm{in}}+\theta_{\mathrm{out}}=0$) and the specularly reflected
beam ($\theta_{\mathrm{out}}=\theta_{\mathrm{in}}$). The modes are well
separated in $\theta_{\mathrm{in}}$.

For the second experiment, we have prepared a layer system optimized for GI
excitation [Fig. 1(b)], having a $40\text{\,}\mathrm{nm}$-thick B_4C core and,
in contrast to the FC experiment, a top Mo cladding layer of only
$8\text{\,}\mathrm{nm}$ to allow incident light to evanescently couple into
the WG. We used a $10\text{\,}\mathrm{mm}$ long Si wafer as substrate, so that
the WG is significantly longer than the attenuation length of the resonant
modes and thus has effectively infinite extent. The back end of the sample was
broken off post deposition to provide a clean exit face. The sample was placed
onto an eulerian cradle allowing to tune the incidence angle
$\theta_{\mathrm{in}}$. Figure 3 shows the off-resonant far-field intensity
distribution, detected $3\text{\,}\mathrm{m}$ downstream of the sample by a
time-insensitive pixel detector as a function of $\theta_{\mathrm{in}}$. We
then aligned the APDs with the WG exit [$\theta_{\mathrm{out}}\approx$$$, see
Fig. 1(b)] and collected the emitted photons for several values of
$\theta_{\mathrm{in}}$ in the vicinity of the third resonance $\theta_{3}=$$$.
We selected the 3rd mode for practical reasons, because the divergence of the
incident beam was $None$, which is wider than the resonance of the 1st mode.
Figure 4(a) shows the emitted intensity as a function of time for three
different $\theta_{\mathrm{in}}$. The intensity exhibits an initial
superradiant decay and a slow-down for longer delays, which is caused by a
residual small hyperfine splitting of the nuclear levels. The initial decay is
plotted in panel (b), showing a decay rate of $30\gamma$ at $\theta_{3}$,
which gradually diminishes as $\theta_{\mathrm{in}}$ moves away from
resonance. The red curve shows a simulation in reflection geometry [32] that
considers divergence of the incoming beam and hyperfine splitting of the
transitions (red). The hyperfine parameters for the simulation were extracted
from measurements at $\theta_{\mathrm{in}}=$$$, far from the WG modes, and are
consistent with the FC experiment [32].

Figure 4: Experimentally measured emissions of the nuclear exciton in GI
excitation. (a) Decay pattern for 3 angles in the vicinity of the 3rd resonant
mode and initial exponential decay (black dashed line). (b) Initial decay rate
as a function of angle detuning: extracted from the experimental data (black
diamonds); calculated with the simple model (6) (purple dashed line);
simulation including small hyperfine splitting of the transition energies as
well as the angular divergence of the experiment (red line).

To qualitatively model the experimental observations, we employ the theory
developed in Ref. [29] (see Supplemental Material [32]) based on macroscopic
quantum electrodynamics, which describes the atom-light interaction using the
classical electromagnetic Green’s function. Here, we neglect the nearly
degenerate sublevels of the ground ($I_{g}=1/2$) and excited ($I_{e}=3/2$ )
state of ^57Fe ($\hbar\omega_{0}=$14.4\text{\,}\mathrm{keV}$$,
$\hbar\gamma=$4.7\text{\,}\mathrm{neV}$$, M1-transition) and describe the
nuclear state by a single nuclear transition operator for each atom
$\hat{\sigma}^{i}_{ge}=\outerproduct{g}{e}$. The excitation pulses have very
small pulse areas so that the nuclear populations are close to the ground
state ($\expectationvalue{\hat{\sigma}^{i}_{ee}}\approx 0$) and the response
is linear. The expectation values of the transition operators
$\sigma^{i}_{ge}:=\expectationvalue{\hat{\sigma}^{i}_{ge}}$ evolve, in the
rotating frame of the nuclear transition frequency $\omega_{0}$, according to
[6]

$\begin{split}-i\omega\sigma^{i}_{ge}(\omega)&=-\frac{\gamma}{2}\sigma^{i}_{ge}(\omega)+i\Omega(\mathbf{r}_{i},\omega)\\\
&+i\sum_{j\neq i}g_{ij}(\omega)\sigma^{j}_{ge}(\omega),\end{split}$ (1)

where
$\Omega(\mathbf{r},t)=\mathbf{m}^{*}\cdot\mathbf{B}_{\mathrm{in}}(\mathbf{r},t)/\hbar$
is the propagating excitation pulse expressed as a Rabi frequency and
$\mathbf{m}$ the magnetic dipole moment. The total single-atom spontaneous
decay rate $\gamma$ includes self-interaction and is approximately equal to
its value in a homogeneous environment (negligible Purcell factor). Note that
$\mathbf{B}_{\mathrm{in}}$ is the _free_ incident magnetic field and
“scattering” from the nuclei is fully accounted for by the coupling
coefficients $g_{ij}$. The coupling is mediated by the WG environment via

$g_{ij}(\omega)=\frac{\mu_{0}k_{0}^{2}}{\hbar}\,\mathbf{m}^{*}\cdot\overset{\leftrightarrow}{\mathrm{G}}_{\mathrm{m}}(\mathbf{r}_{i},\mathbf{r}_{j},\omega_{0}+\omega)\cdot\mathbf{m},$
(2)

where $\mu_{0}$ is the vacuum permeability and
$\overset{\leftrightarrow}{\mathrm{G}}_{\mathrm{m}}$ the Green’s function for
the magnetic field in the WG (see Supplemental Material [32]).

We convert (1) into a one-dimensional _macroscopic_ equation. To that end, we
assume homogeneously distributed nuclei in a single thin layer, insert the
analytic expression for the Green’s function, and assume interaction via a
_single_ resonant mode. We obtain

$\dot{\sigma}_{ge}(x,t)=-\frac{\gamma}{2}\sigma_{ge}(x,t)+i\Omega(x,t)\\\
-\frac{\zeta_{m}\gamma}{4\Lambda_{\mathrm{res}}}\int_{-L}^{0}\differential
x^{\prime}e^{ik_{0}\nu_{m}\absolutevalue{x-x^{\prime}}}\sigma_{ge}(x^{\prime},t-\absolutevalue{x-x^{\prime}}\nu_{m}/c).$
(3)

Here, $\Lambda_{\mathrm{res}}$ ($47\text{\,}\mathrm{nm}$ for ^57Fe) is the on-
resonance attenuation length; $\nu_{m}$ is the complex effective refractive
index of the mode, and $\zeta_{m}$ the dimensionless coupling-coefficient of
the WG mode. The latter effectively rescales $\Lambda_{\mathrm{res}}$ and is
given by $\zeta_{m}=du_{m}(z_{0})^{2}$, where $d$ and $z_{0}$ are thickness
and position of the ^57Fe-layer, respectively and $u_{m}(z)$ is the complex
transversal mode profile, described in Ref. [30], which is bi-normalized to
$\int[u_{m}(z)]^{2}\differential z=1$. The field produced by the exciton is
obtained from the oscillating magnetization,
$\mathbf{M}=\mathbf{m}\rho\sigma_{ge}\exp(-i\omega_{0}t)+\text{h.c.}$. Note
that (3) not only describes planar WGs (after integrating out $y$) as
discussed here, but also 1d channel WGs, and furthermore homogeneous slabs
(foils) with translational symmetry in $y$-$z$ [36], with different values for
$\nu_{m}$ ($=n$ for slab) and $\zeta_{m}$ ($=1$ for slab).

Equation (1) is often treated as an eigenvalue problem, decomposing
$\sigma_{ge}$ into radiative eigenmodes [6, 18, 19]. Instead, we employ a
_forward-scattering approximation_ (FSA) that allows us to solve (3)
analytically for the two experimental settings sketched in Fig. 1. The rapidly
oscillating phase factor in (3) strongly suppresses the back-scattered field
[29]. Hence, we change the upper limit of integration in (3) to $x$.

In the FC geometry [Fig. 1(a)], the only surviving component of the excitation
pulse is the fundamental guided mode, so that
$\Omega(x,t)=\Omega_{0}\exp(ik_{0}\nu_{m}x)\Pi(t_{\mathrm{ret}})$, with some
slowly varying envelope $\Pi(t)$ where $t_{\mathrm{ret}}=t-k_{0}\nu_{m}x/c$.
Note that the excitation pulse decays with $x$, as
$\imaginary\\{\nu_{m}\\}>0$. For finite $L$, (3) becomes a Volterra integral
equation due to the FSA and hence has _no_ radiative eigenmodes. Its exact
analytical solution for the given $\Omega(x,t)$ is

$\sigma_{ge}(x,t)=iAe^{ik_{0}\nu_{m}x-\gamma
t_{\mathrm{ret}}/2}J_{0}\left[\sqrt{\gamma
t_{\mathrm{ret}}x\zeta_{m}\Lambda^{-1}_{\mathrm{res}}}\right],$ (4)

for $t_{\mathrm{ret}}\geq 0$, where $J_{0}$ is a Bessel function of the first
kind and $A=\Omega_{0}\int\Pi(t)\differential t$ the pulse area. This is the
well known solution for NFS [23] with $\Lambda_{\mathrm{res}}$ rescaled by
$\zeta_{m}$. The exciton emits directionally and purely into the WG mode (see
Supplemental Material [32]). The emission undergoes the dynamical beats, well
known from coherent pulse propagation through resonant media [37, 13] – and
clearly observed in our experiment. From the experimental data, we obtain an
estimate for the coupling coefficient
$\absolutevalue{\zeta_{m}}=L_{\mathrm{foil}}/L\approx$0.029$$, slightly
smaller than the theoretical value (from design parameters) of $0.038$.

In the GI geometry [Fig. 1(b)], the incidence angle $\theta_{\mathrm{in}}$
fixes the in-plane wavevector to
$k_{\mathrm{in}}=k_{0}\cos\theta_{\mathrm{in}}$. The excitation pulse can be
written as
$\Omega(x,t)=\Omega_{0}\exp(ik_{\mathrm{in}}x)\Pi(t_{\mathrm{ret}})$, with
here $t_{\mathrm{ret}}=t-k_{\mathrm{in}}x/\omega_{0}$. Assuming that $L$ is
significantly larger than the off-resonance attenuation length of the mode,
$\Lambda_{m}=(2k_{0}\imaginary\\{\nu_{m}\\})^{-1}$, (typically a few
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$) we can extend the lower boundary
of integration in (3) to $-\infty$. This eliminates the boundary effect in the
forward direction. The linear spatial phase in $\Omega(x,t)$ corresponds to a
radiative eigenmode of the exciton, so that the exciton evolves harmonically
with shifted frequency and enhanced decay. We obtain (see Supplemental
Material [32])

$\displaystyle\sigma_{ge}(x,t)$
$\displaystyle=iA\exp\left[ik_{0}x\cos\theta_{\mathrm{in}}-\left(\frac{\gamma}{2}-i\eta\right)t_{\mathrm{ret}}\right],$
(5) $\displaystyle\eta(\theta_{\mathrm{in}})$
$\displaystyle=\frac{\zeta_{m}\Lambda_{m}}{\Lambda_{\mathrm{res}}}\frac{1}{2\Lambda_{m}q_{\mathrm{\theta}}-i}\frac{\gamma}{2},$
(6)

with
$q_{\mathrm{\theta}}=k_{0}\cos\theta_{\mathrm{in}}-k_{0}\real\\{\nu_{m}\\}\approx-
k_{0}(\theta_{\mathrm{in}}-\theta_{m})\sin\theta_{m}$ and pulse area $A$ as
previously defined. The mode angle is defined via
$\cos\theta_{m}=\real\\{\nu_{m}\\}$. The exciton decays superradiantly with
peak speedup of $1+\zeta_{m}\Lambda_{m}/\Lambda_{\mathrm{res}}$ depending on
ratio between off- and on-resonant attenuation lengths, as well as the mode
coupling coefficient $\zeta_{m}$. Note that
$\Lambda_{m}/\Lambda_{\mathrm{res}}=\rho\Lambda_{m}\sigma_{\mathrm{res}}$
($\sigma_{\mathrm{res}}$: resonant cross section, $\rho$: number density of
nuclei) gives the number of nuclei within the off-resonant absorption length
(compare Ref. [22]), clearly demonstrating the collective superradiant nature
of the speedup, contrasting earlier interpretations based on an increase of
the photon density of states [14]. Figure 4(b) shows the speedup from
calculations with (6) using the design parameters as well as from the
experimental data. This simple mono-modal model already explains the observed
peak speedup. It does not fully account for the increased angular resonance
width, which can be attributed to the residual hyperfine splitting and angular
divergence observed in the experiment [32]. Thus, we see that (3)
qualitatively reproduces both experiments.

In summary, we have investigated the evolution of nuclear excitons in x-ray
WGs for two different modes of excitation. In the FC geometry, we have
observed dynamical beats emitted by WGs with a $20\text{\,}\mathrm{nm}$-thick
guiding core and varying lengths from about $0.1\text{\,}\mathrm{mm}$ to
$2\text{\,}\mathrm{mm}$ into the fundamental WG mode. In the GI geometry, we
have observed exponential decay into the resonantly excited WG mode with
tuneable superradiant speedup. Our theory consistently describes the
observations of both experiments in a one-dimensional waveguide picture. When
a single mode dominates, the equation of motion is formally identical to that
of a homogeneous slab with length (foil thickness) rescaled by a nucleus-mode
coupling constant $\zeta_{m}$. Depending on the _off-resonant_ optical
thickness of the WG, two regimes are realized. For optically thin WGs, the
sharp jump in nuclear density at the WG entrance gives rise to dynamical
beats. The grazing incidence geometry allows to study optically thick WGs,
where off-resonant absorption suppresses the boundary effect and the exciton
evolves harmonically with resonant dispersion relation.

Going beyond the single-mode WGs discussed here will enable to observe a wider
range of phenomena. The front coupling geometry in particular can support
multiple guiding modes, which can be used to implement effective two-beam
control techniques for the x-ray frequency regime, and opens the door for a
vast possibility of new methods of narrowband x-ray control via the
engineering of collective radiation patterns with extreme transversal
coherence. Exploiting interference between two resonant modes in a
longitudinally microstructured layer, a phenomenon similar to selective sub-
radiance is predicted [29]. Even a gravitational sensor has been proposed that
uses similar WG structures and the nuclear resonance of scandium [38]. Thus,
Mössbauer nuclei in x-ray WGs are a viable platform to study the collective
interaction of localized two-level systems with photons propagating in a one-
dimensional WG, realizing the low-saturation regime with a very high number of
emitters, and allowing to control the temporal evolution of the collective
decay via the excitation and coupling geometry.

###### Acknowledgements.

Warm thanks go to Mike Kanbach for preparing the WG assemblies, to Olaf
Leupold for his support and fruitful discussions during the first beamtime,
and to Dieter Lott and Helmholtz-Zentrum Hereon for providing and supporting
the reflectometer to calibrate the thin film samples. We acknowledge DESY
(Hamburg, Germany), a member of the Helmholtz Association HGF, for the
provision of experimental facilities. Beamtime was allocated for proposals
I-20211677; I-20221193. We acknowledge partial funding by Max Planck School of
Photonics; Deutsche Forschungsgemeinschaft (DFG) (432680300 SFB 1456/C03,
429529648 TRR 306/C04, 390858490 EXC 2147). A. Pálffy acknowledges support
from the DFG in the framework of the Heisenberg Program. LML, MV, TS, and RR
are part of the Max Planck School of Photonics.

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