# Effects of higher-order Casimir-Polder interactions on Rydberg atom
spectroscopy

B. Dutta  Laboratoire de Physique des Lasers, Université Sorbonne Paris Nord,
F-93430, Villetaneuse, France  CNRS, UMR 7538, LPL, 99 Avenue J.-B. Clément,
F-93430 Villetaneuse, France  J. C. de Aquino Carvalho  Laboratoire de
Physique des Lasers, Université Sorbonne Paris Nord, F-93430, Villetaneuse,
France  CNRS, UMR 7538, LPL, 99 Avenue J.-B. Clément, F-93430 Villetaneuse,
France  G. Garcia-Arellano  Laboratoire de Physique des Lasers, Université
Sorbonne Paris Nord, F-93430, Villetaneuse, France  CNRS, UMR 7538, LPL, 99
Avenue J.-B. Clément, F-93430 Villetaneuse, France  P. Pedri  Laboratoire de
Physique des Lasers, Université Sorbonne Paris Nord, F-93430, Villetaneuse,
France  CNRS, UMR 7538, LPL, 99 Avenue J.-B. Clément, F-93430 Villetaneuse,
France  A. Laliotis<EMAIL_ADDRESS>Laboratoire de Physique des
Lasers, Université Sorbonne Paris Nord, F-93430, Villetaneuse, France  CNRS,
UMR 7538, LPL, 99 Avenue J.-B. Clément, F-93430 Villetaneuse, France  C. Boldt
Institute for Physics, University of Rostock, Albert-Einstein-Straße 23-24,
D-18059 Rostock, Germany  J. Kaushal  Institute for Physics, University of
Rostock, Albert-Einstein-Straße 23-24, D-18059 Rostock, Germany  S. Scheel
Institute for Physics, University of Rostock, Albert-Einstein-Straße 23-24,
D-18059 Rostock, Germany

###### Abstract

In the extreme near-field, when the spatial extension of the atomic
wavefunction is no longer negligible compared to the atom-surface distance,
the dipole approximation is no longer sufficient to describe Casimir-Polder
interactions. Here we calculate the higher-order, quadrupole and octupole,
contributions to Casimir-Polder energy shifts of Rydberg atoms close to a
dielectric surface. We subsequently investigate the effects of these higher-
order terms in thin-cell and selective reflection spectroscopy. Beyond its
fundamental interest, this new regime of extremely small atom surface
separations is relevant for quantum technology applications with Rydberg or
surface-bound atoms interfacing with photonic platforms.

Highly excited (Rydberg) atoms have huge electric and magnetic transition
multipole moments that make them interact very strongly with their
environment. Thus, they are ideal candidates for studying dispersion forces
such as Casimir-Polder (atom-surface) [1, 2] or van der Waals (atom-atom)
interactions [3]. More recently, Rydberg atoms have attracted significant
attention for quantum technology applications. In particular, it was
demonstrated that probing Rydberg atoms inside thin vapour cells [4] presents
a simple way to fabricate room temperature single-photon sources based on the
Rydberg blockade effect [5], without having to resort to complex cold atom
manipulations. Moreover, Rydberg atoms in vapour cells are being used as
electric-field probes at THz or GHz frequencies [6, 7, 8]. However, the rapid
scaling of electric-dipole moment fluctuations makes Casimir-Polder (CP)
interactions a dominant spectroscopic contribution that impacts potential
hybrid systems such as tapered optical fibers [9, 10], hollow core fibers [11]
and thin cells [4, 5] that aim at interfacing atoms with photonic platforms.
Similarly, Rydberg interactions with van der Waals heterostructures have also
been investigated [12].

Theoretical studies of the interaction between dielectric surfaces and highly
excited atoms have exposed the limitations of the traditional Casimir-Polder
(CP) approach in which only the electric dipole interaction is taken into
account. Indeed, the dipole approximation breaks down and higher-order terms
need to be considered in the extreme near field [13] as Rydberg wavefunctions
can easily extend beyond 100nm (being proportional to ${n^{\star}}^{2}$, with
${n^{\star}}$ the effective quantum number). Moreover, perturbative approaches
have also been questioned when the expected energy shifts are comparable to
adjacent energy level spacings [14]. Although previous theoretical works have
presented the basic scaling laws governing quadrupole interactions [13],
detailed results of the higher order interaction coefficients have not yet
been presented for CP atom-surface interactions. Higher order effects have
nevertheless been studied in detail for the case of atom-atom or molecule-
molecule van der Waals interactions [15, 16].

Atom-surface experiments have similarly flourished during the past decades
[17, 18, 19, 20], shedding light on novel effects such as the temperature
dependence of CP interactions [19, 21], and atom- metamaterial interactions
[22] with an ever increasing precision in view of uncovering fundamental
forces beyond the standard model [20]. However, higher-order interactions
remain so far experimentally unexplored. A prominent and well developed
experimental method for performing such studies is vapour cell spectroscopy
[23, 24, 21, 25, 26]. Thin vapour cells allow for a controlled confinement of
atomic vapours within dielectric walls down to the nanometer regime, making
them excellent platforms for probing Rydberg atoms extremely close to
dielectric surfaces [4, 25]. Selective reflection in macroscopic cells has
also been used for atomic or molecular gases [23, 21, 27] close to a surface,
but provides no means of controlling the probing depth, typically defined by
the excitation wavelength.

Here, we investigate higher-order (quadrupole and octupole) CP interactions
between a Rydberg atom and a dielectric surface, providing explicit
calculations of the $C_{3}$ (dipole-dipole) and the $C_{5}$ (combined
quadrupole-quadrupole and dipole-octupole) coefficients for alkali Rydberg
atoms in the near field. We subsequently study the effects of higher-order
interactions on CP vapour cell spectroscopy, illuminating the conditions under
which higher-order interactions can be experimentally measurable. Multipole
contributions are of importance for experiments aiming at binding atoms to
surfaces and extending the optical control to the extreme near field [28].

We will conduct our studies in the electrostatic limit, where CP Rydberg-
surface interactions can be described as the interaction of the atomic charge
distribution (centred in $O$) with its surface-induced instantaneous image
(centred in $O^{\prime}$) in front of a perfectly conducting surface. We
assume that the Rydberg atom consists of a valence electron orbiting around a
positively charged core. The atom-surface interaction energy, $W$, is half the
electrostatic energy of the atomic charge distribution placed under the
external potential of its image, $\Phi^{\prime}(\vec{r^{\prime}})$, with the
corresponding field $\vec{E^{\prime}}(\vec{r^{\prime}})$,

$\begin{split}W=&-\frac{1}{2}\vec{p}\cdot\vec{E^{\prime}}(\vec{r}_{0})-\frac{1}{12}\sum_{i,j}Q_{ij}\frac{\partial{E^{\prime}_{i}}}{\partial{r_{j}^{\prime}}}(\vec{r}_{0})+\\\
&-\frac{1}{24}\sum_{i,j,k}T_{ijk}\frac{\partial^{2}{E^{\prime}_{i}}}{\partial{r_{j}^{\prime}}\partial{r_{k}^{\prime}}}(\vec{r}_{0})+\dots\end{split}$
(1)

Here, $\vec{r}_{0}$ is the position vector of the atom with coordinates
${r_{i}}$. The atomic multipole moments with respect to $O$ are denoted as
$\vec{p}$, $\bar{Q}$ and $\overline{\overline{T}}$ (dipole, quadrupole and
octupole moments, respectively). The potential created by the image can itself
be expanded into multipoles, giving

$\begin{split}\Phi^{\prime}(\vec{r}_{0})=&\frac{1}{4\pi\epsilon_{0}}\sum_{i}{p^{\prime}_{i}}\frac{{r}_{0i}}{r_{0}^{3}}+\frac{1}{8\pi\epsilon_{0}}\sum_{i,j}{Q^{\prime}_{ij}}\frac{r_{0i}r_{0j}}{r_{0}^{5}}+\\\
&+\frac{1}{24\pi\epsilon_{0}}\sum_{i,j,k}T^{\prime}_{ijk}\frac{r_{0i}r_{0j}r_{0k}}{r_{0}^{7}}+\dots\end{split}$
(2)

Here, $\vec{p^{\prime}}$, $\bar{Q^{\prime}}$, and
$\overline{\overline{T^{\prime}}}$ are the dipole, quadrupole and octupole
moments of the image with respect to $O^{\prime}$. Symmetry links the
components of the image moments to those of the atomic moments by
$p_{i}^{\prime}=(-1)^{\kappa+1}p_{i}$,
$Q_{ij}^{\prime}=(-1)^{\kappa+1}Q_{ij}$,
$T_{ijk}^{\prime}=(-1)^{\kappa+1}T_{ijk}$, where $\kappa$ is the number of
times $z$ appears as a tensor index.

The CP interaction energy can therefore be written as

$W=W_{pp}+W_{pQ}+W_{QQ}+W_{pT}.$ (3)

where $W_{pp}$, $W_{pQ}$, $W_{QQ}$, $W_{pT}$ are the dipole-dipole, dipole-
quadrupole, quadrupole- quadrupole and dipole-octupole contributions,
respectively. The dipole-dipole and quadrupole-quadrupole terms are given by

$\displaystyle
W_{pp}=-\frac{1}{4\pi\epsilon_{0}}\frac{p^{2}+p_{z}^{2}}{16z_{s}^{3}},$ (4)
$\displaystyle
W_{QQ}=-\frac{17Q_{zz}^{2}+16Q_{zy}^{2}+16Q_{zx}^{2}+2Q_{xx}^{2}+4Q_{yx}^{2}+2Q_{yy}^{2}}{4\pi\epsilon_{0}768z_{s}^{5}}.$
(5)

The cross terms such as the dipole-quadrupole and dipole-octupole
contributions with their $z_{s}^{-4}$ and $z_{s}^{-5}$ dependence,
respectively, are given in the Supplementary material. It should be noted that
$W_{pQ}$ includes both the energy of a dipole immersed in the field of a
quadrupole and vice-versa (the same applies to $W_{pT}$).

In the electrostatic limit, we can calculate the CP frequency shift, $\Delta
f$, of an atomic Rydberg state using first order perturbation theory as

$h\Delta f=\langle\psi_{n,l,J,M_{J}}|W|\psi_{n,l,J,M_{J}}\rangle\,.$ (6)

Here, $\psi_{n,l,J,M_{J}}$ is the wavefunction of the Rydberg electron, with
$n$ the principal quantum number, $l$, $J$ the orbital and total angular
momentum quantum numbers, and $M_{J}$ the projection of $J$ onto the z-axis.
For Rydberg atoms we can assume that the external electron is under the
influence of a central potential given by an effective Coulomb interaction
modified to include the polarisability of core electrons [29]. This allows us
to easily obtain the radial wavefunction by numerically solving Schrödinger’s
equation using the Numerov method. In our analysis, we ignore the hyperfine
structure of the atoms as it is usually very small compared to the CP energy
shifts for most alkali Rydberg atoms. The above analysis allows us to
calculate Rydberg-surface interactions at any multipole order.

Figure 1: $C_{3}$ and $C_{5}$ coefficients (blue and red colors, respectively)
for $S_{1/2}$ and $D_{3/2}$ states (squares and circles, respectively) of
cesium and rubidium (open and closed points, respectively) as a function of
the effective quantum number $n^{\star}=n-\delta$. The straight solid lines
correspond to the analytical expressions derived for a hydrogen $S$ state
[Eqs. (7) and (8)] providing an accurate estimate, to within 1$\%$, of the
interaction coefficients for all alkali Rydberg atoms.

The quantum mechanical averaging of the interaction energy gives the following
results: a) The dipole-dipole term ($\Delta f_{3}$) yields the well-known
near-field Casimir-Polder frequency shift $\Delta f_{3}=-C_{3}/z_{s}^{3}$,
where $C_{3}$ is a coefficient usually expressed in $\mathrm{MHz\,\mu m^{3}}$.
b) The dipole-quadrupole terms vanish for parity reasons. c) The quadrupole-
quadrupole and dipole-octupole terms give a frequency shift expressed as
$\Delta f_{5}=-C_{5}/z_{s}^{5}$, where $C_{5}$ is a coefficient expressed in
$\mathrm{MHz\,\mu m^{5}}$. We emphasize that the dipole-octupole contributions
do not necessarily vanish as both dipole and octupole interactions can act on
the same atomic transition ($\Delta l=\pm 1$ transitions can be both dipole
and octupole allowed). However, dipole-octupole terms only contribute to the
anisotropy of the atom-surface interaction, with an overall scalar component
(averaging over all $M_{J}$) that remains zero.

In Fig. 1 we plot the $C_{3}$ and $C_{5}$ coefficients for $S_{1/2}$ and
$D_{3/2}$ states of cesium and rubidium atoms as a function of the effective
quantum number ($n^{*}=n-\delta^{*}$), where $\delta^{*}$ is the quantum
defect. It shows that the interaction coefficients depend very little on the
core polarisability and on angular momentum. We have therefore calculated
analytical expressions for both $C_{3}$ and $C_{5}$ coefficients for a
hydrogen atom. For an $S_{1/2}$ state ($l=0$) the interaction coefficients
become

$\displaystyle
C_{3}=\frac{e^{2}a_{o}^{2}}{96\pi\epsilon_{0}h}{n^{\star}}^{2}\left(5{n^{\star}}^{2}+1\right)\approx\frac{5e^{2}a_{o}^{2}{n^{\star}}^{4}}{96\pi\epsilon_{0}h},$
(7) $\displaystyle
C_{5}=\frac{e^{2}a_{o}^{4}}{640\pi\epsilon_{0}h}{n^{\star}}^{4}\left(63{n^{\star}}^{4}+105{n^{\star}}^{2}+12\right)\approx\frac{63e^{2}a_{o}^{4}{n^{\star}}^{8}}{640\pi\epsilon_{0}h}.$
(8)

The full solution, for all states, is given in the Supplemental material.
Recall that for hydrogen ${n^{\star}}$ is simply the principal quantum number
and note that the leading term in the above polynomials is independent of
angular momentum.

Our assumption of a perfectly correlated image breaks down at distances
comparable to relevant transition wavelengths or when a frequency-dependent
dielectric constant is considered [30]. However, for Rydberg atoms, the
relevant multipole transitions are typically in the THz or GHz regime
suggesting that retardation can be ignored at micrometer distances. Moreover,
at these frequencies most materials do not possess surface resonances, and
their dielectric constant tends to their static ($\epsilon_{s}$) value.
Dielectric effects can therefore be accounted for by simply multiplying the
dispersion coefficients (Fig. 1) by the surface response
($S=\frac{\epsilon_{s}-1}{\epsilon_{s}+1}$), while temperature effects [31,
32, 21] are negligible and can be ignored [33]. The above arguments suggest
that the electrostatic limit is a good approximation for studying Rydberg-
surface interactions in the extreme near field. However, a QED treatment [34,
35, 36] would be necessary for treating the coupling of atoms with resonant
surfaces or meta-surfaces.

At this point, a word of caution is appropriate. Frequently, traceless
multipole tensors are used in the calculation. However, this is only justified
for quadrupole tensors, as its trace component does not contribute to the
energy. This is no longer true for the octupole tensor, where the trace does
in general contribute to the energy [37, 38]. Indeed, upon reducing the
quantity $r_{i}r_{j}r_{k}$ appearing in $\overline{\overline{T}}$ in terms of
spherical harmonics yields terms with $Y_{3m}(\Theta,\varphi)$ (traceless
part) as well as $Y_{1m}(\Theta,\varphi)$ (trace part). The latter can be
identified as a contribution to the dipole transition, and only further
symmetry considerations of the field distribution or atomic transition matrix
elements can cause them to vanish [39, 40] .

Having developed a theoretical framework that allows us to calculate both
$C_{3}$ and $C_{5}$ coefficients, we can estimate the effects of quadrupole
interactions on CP experiments. Atomic spectroscopy in vapour cells of
variable nanometric thickness (thin cells) is a well developed method for
measuring CP interactions with excited states including Rydberg atoms [25,
41]. Thin cells allow us to control the vapour confinement down to the
nanometer regime (thickness can be as small as 50nm) [25], giving them a
distinct advantage over other methods for probing higher order CP interactions
with Rydberg atoms.

Figure 2: (a) Schematic of the thin cell experiment analysed in our
simulations. A laser beam at 0.894$\mathrm{\mu m}$ or 0.852$\mathrm{\mu m}$
pumps cesium atoms to the 6$P_{1/2}$ or 6$P_{1/2}$ level, respectively.
Subsequently, a green laser at 0.514$\mathrm{\mu m}$ or 0.513$\mathrm{\mu m}$
probes Rydberg atoms at the 6$P_{1/2}$ $\rightarrow$ 16$S_{1/2}$ or the
6$P_{3/2}$ $\rightarrow$ 26$S_{1/2}$ transition, respectively. Higher lying
states can also be easily accessed via the same scheme. Due to interatomic
collisions, the population of the intermediate levels has a quasi-thermal
(Maxwell-Boltzmann) velocity distribution. (b) Relevant energy levels.

Typically, a two-step excitation technique is used to reach high-lying excited
states of alkali atoms [21, 24, 25]. For cesium (our atom of choice), a strong
pump excites atoms to their first excited state (6$P_{1/2}$ or 6$P_{3/2}$) and
a weak laser probes the 6$P_{1/2}$ $\rightarrow$ n$S$ or 6$P_{1/2}$
$\rightarrow$ n$D$ transitions. For states with $n\approx 20$, the transition
wavelength $\lambda_{probe}$ is about 510nm. In Fig. 2 we show the basic
principle of the experiment that will be analysed.

Figure 3: (a) Thin-cell transmission spectra for three different thicknesses,
calculated using Eq. (11) ($S_{T}$) with $C_{3}$=4.15 $\mathrm{MHz}\mathrm{\mu
m^{3}}$ and $C_{5}$=0.45 $\mathrm{kHz}\mathrm{\mu m^{5}}$, corresponding to Cs
16$S_{1/2}$ atoms. Dashed vertical line: position of the transition frequency
in the volume. Straight red lines: calculations including both dipole and
quadrupole interactions ($C_{3}$ and $C_{5}$). Black lines: dipole
interactions only. The transmission amplitude decreases with thickness by a
factor indicated in the figure. (b) Displacement of the transmission dip
$C_{3}^{eff}$ with respect to the volume transition frequency (dashed line),
multiplied by $L^{3}$, as a function of cell thickness. Red dots: calculations
with full CP potential. Black dots: dipole interactions only. Horizontal
dashed line: CP dipole shift in the centre of the cell multiplied by $L^{3}$.

Inter-atomic collisions and radiation trapping in the atomic vapour
redistribute the initially velocity selected 6$P$ population to many atomic
velocities and hyperfine states. This allows us to consider our atoms
essentially as two level systems in linear interaction with the probe
excitation field. Thin cells form a low finesse Fabry-Perot cavity that
eventually mixes the backward (reflection $I^{lin}_{R}$) and forward
(transmission $I^{lin}_{T}$) response of the polarised atomic vapour [42]
given by:

$\displaystyle
I^{lin}_{R}\propto\frac{N\mu^{2}}{\mathcal{F}}\int_{0}^{\infty}\frac{dv_{z}}{v_{z}}W(v_{z})\int_{0}^{L}dz\int_{0}^{z}dz^{\prime}e^{2ikz}e^{\frac{\mathcal{L}(z^{\prime})-\mathcal{L}(z)}{v_{z}}},$
(9) $\displaystyle
I^{lin}_{T}\propto\frac{N\mu^{2}}{\mathcal{F}}\int_{0}^{\infty}\frac{dv_{z}}{v_{z}}W(v_{z})\int_{0}^{L}dz\int_{0}^{z}dz^{\prime}e^{\frac{\mathcal{L}(z^{\prime})-\mathcal{L}(z)}{v_{z}}}\,.$
(10)

For a symmetric Fabry-Perot cavity with reflection coefficient $r$ for both
interfaces the transmission signal ($S_{T}$) that consists of the homodyne
beating between the atomic response with that of an empty cavity is given by

$S_{T}\propto\frac{2N\mu^{2}}{|\mathcal{F}|^{2}}\Re[I^{lin}_{T}+r^{2}e^{2ikL}I^{lin}_{T}-2rI^{lin}_{R}].$
(11)

Here, $\mathcal{F}=1-r^{2}e^{2ikL}$ with $r$ the reflection coefficient of the
windows and $k=2\pi/\lambda_{probe}$. The atomic velocity along the probe beam
propagation axis is denoted as $v_{z}$, the atomic vapour density inside the
cell is $N$ and the transition dipole moment is $\mu$. In the above equations,
the functions $\mathcal{L}(z^{\prime})-\mathcal{L}(z)$ are

$\mathcal{L}(z^{\prime})-\mathcal{L}(z)=\int_{z}^{z^{\prime}}\left[\Gamma/2-i(\delta+2\pi\Delta
f_{CP}(\xi)-kv_{z})\right]d\xi\,,$ (12)

where $\Delta f_{CP}(z)$ is the CP shift of the Rydberg state (the shift of
the 6$P$ state is negligible) inside the thin-cell cavity. The shift can be
separated into a dipole component and a quadrupole component that depend on
the $C_{3}$ and $C_{5}$ coefficients, respectively. As full calculation of the
CP shift inside a cavity requires a more elaborate theory, we will not
consider resonant effects due to surface polaritons and simply add the
potentials of both walls, neglecting the infinite series of multiple images.
In this case, the dipole and quadrupole shifts in the middle of the cell
become $-16C_{3}/L^{3}$ and $-64C_{5}/L^{5}$, respectively.

In Fig. 3(a) we show the transmission spectrum of a resonant
6$\mathrm{P_{1/2}}$ $\rightarrow$ 16$\mathrm{S_{1/2}}$ beam through a thin
cell of three different thicknesses. The atom-surface interaction coefficients
calculated for the 16$\mathrm{S_{1/2}}$ state of cesium (with a Bohr diameter
of $\approx$ 15nm) are $C_{3}$=4.15 $\mathrm{MHz}\mathrm{\mu m^{3}}$ and
$C_{5}$=0.45 $\mathrm{kHz}\mathrm{\mu m^{5}}$. The red curves represent the
calculated spectra using both dipole and quadrupole interactions, whereas for
black curves the quadrupole interactions are omitted. The effect of quadrupole
interactions becomes observable when the atomic vapour is confined at
thicknesses smaller than 200nm. At L=100nm the additional quadrupole shift
exceeds the predicted spectral width, suggesting that the $C_{5}$ coefficient
could be measurable with a thin-cell spectroscopy experiment. Figure 3(b)
shows the predicted displacement of the transmission dip $C_{3}^{\rm eff}$
with respect to the volume resonance away from the surface, multiplied by
$L^{3}$. When quadrupole interactions are ignored (black dots), $C_{3}^{\rm
eff}$ tends towards $16C_{3}$ (dashed horizontal line), suggesting that when
the cell is very thin, the dominant spectral contribution derives from atoms
at the centre of the cell. For increasing cell thicknesses, the contribution
of layers closer to the walls becomes more prominent leading to a larger
$C_{3}^{\rm eff}$. Red dots represent calculations including both dipole and
quadrupole interactions. From Fig. 3(b) we can see that quadrupole
interactions have no visible effect for thicknesses larger than 250nm. The
extent of the gray shaded area is $L^{3}w/5$, where $w$ is the width of the
transmission spectrum. The gray shaded area gives an indication of the
capability of the proposed experiment to discern between the two models (black
and red dots). Note that the discerning capability also depends on the signal
to noise ratio of the experiment. The reduction in signal amplitude is noted
in Fig. 3(a) next to the transmission curves. The effects of higher-order
interactions are evident for larger cell thicknesses when probing higher-lying
states such as $28\mathrm{S_{3/2}}$ via the 6$\mathrm{P_{1/2}}$ $\rightarrow$
28$\mathrm{S_{3/2}}$ transition (see Supplemental Material).

Our above analysis assumes that Rydberg atoms interact with surfaces only via
CP interactions. However, Stark shifts due to adsorbed atoms or parasitic
electric field in the surface of the dielectric windows are known to be an
important problem in precision atom-surface experiments [20, 43, 44, 45]. In
particular, high-lying states become extremely sensitive to electric fields
[46, 4] as their polarisability scales more rapidly ($\alpha\propto{n^{\star
7}}$) than the $C_{3}$ coefficient. Our analysis suggests that higher-order CP
effects can be measurable even with relatively low-lying Rydberg states with
${n^{\star}}\approx 10\ldots 15$, for which atom-surface interaction
experiments have already been demonstrated [1, 2]. This is a powerful
indication that vapour cell spectroscopic experiments could provide excellent
testbeds for further exploring Casimir-Polder physics.

###### Acknowledgements.

We acknowledge insightful discussions with Martial Ducloy and discussions with
Isabelle Maurin on the numerical modelling of thin cell spectra. This work was
financially supported by the ANR-DFG grant SQUAT (Grant Nos.
ANR-20-CE92-0006-01 and DFG SCHE 612/12-1), the DAAD and Campus France (via
the PHC-PROCOPE programme, grant No. 57513024), and the French Embassy in
Germany (via the Campus-France PHC-Procope project 44711VG and via PROCOPE
Mobilité, project DEU-22-0004 LG1).

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