# Photoexcitation of the 229Th nuclear clock transition using twisted light

Tobias Kirschbaum<EMAIL_ADDRESS>University of Würzburg,
Institute of Theoretical Physics and Astrophysics, Am Hubland, 97074 Würzburg,
Germany  Thorsten Schumm  Institute for Atomic and Subatomic Physics, TU Wien,
Stadionallee 2, 1020 Vienna, Austria  Adriana Pálffy  adriana.palffy-buss@uni-
wuerzburg.de University of Würzburg, Institute of Theoretical Physics and
Astrophysics, Am Hubland, 97074 Würzburg, Germany

###### Abstract

The 229Th nucleus has a unique transition at only 8 eV which could be used for
a novel nuclear clock. We investigate theoretically the prospects of driving
this transition with vortex light beams carrying orbital angular momentum.
Numerical results are presented for two experimental configurations which are
promising for the design of the planned nuclear clock: a trapped ion setup and
a large ensemble of nuclei doped into CaF2 crystals which are transparent in
the frequency range of the nuclear transition. We discuss the feasibility of
the vortex beam nuclear excitation and compare the excitation features with
the case of plane wave beams.

##  I Introduction

Twisted light or optical vortex beams refer to light beams with engineered
wave fronts that carry orbital angular momentum along their direction of
propagation Allen et al. (1992). These beams are qualitatively different from
plane waves, being characterized by a helical wave front, and spatial
inhomogeneous intensity patterns and momentum distributions Matula et al.
(2013); Peshkov et al. (2017); Yao and Padgett (2011). Their unique properties
have rendered them a trending topic in the past decade, in particular in
atomic physics, where vortex beams are a versatile tool to study photo-
absorption in atomic shell transitions. For example, these beams can be used
to address and separate quantum transitions with multipolarities higher than
electric dipole Afanasev et al. (2016, 2018a), to generate rich polarization
patterns in microscopic$/$macroscopic samples Afanasev et al. (2013); Scholz-
Marggraf et al. (2014); Surzhykov et al. (2015); Peshkov et al. (2018); Schulz
et al. (2019); Schmidt et al. (2023), or to suppress the unwanted light shift
in atomic clock transitions Schmiegelow et al. (2016); Lange et al. (2022);
Afanasev et al. (2018b). A few works have gone further to address
theoretically the interaction of vortex beams of higher frequency with nuclei,
for instance the manipulation of giant dipole resonances Lu et al. (2023), the
excitation of multipolar nuclear transitions Kazinski and Sokolov (2023),
Delta baryon photo-production Afanasev and Carlson (2022) and deuteron photo-
disintegration Afanasev et al. (2018c).

In this work we investigate theoretically the prospects of using vortex beams
to drive the 8 eV nuclear clock transition in 229Th. This is a unique nuclear
transition accessible by vacuum ultraviolet (VUV) light which renders possible
a nuclear frequency standard Peik and Tamm (2003) or a first nuclear laser
Tkalya (2011); Xu et al. (2023). The nuclear clock is a compelling alternative
to atomic clocks, potentially with superior accuracy and insensitive to some
of the shifts which affect atomic transitions Peik et al. (2021). As frequency
standard based on a nuclear transition, it would also have a significant
impact on other fields, for instance the improvement of satellite navigation
Thirolf et al. (2019), the detection of dark matter Peik et al. (2021); Fadeev
et al. (2020); Tsai et al. (2023) or investigating temporal variations of
fundamental constants Fadeev et al. (2020); Flambaum (2006).

The 8 eV nuclear transition in 229Th is a magnetic dipole ($\mathcal{M}1$) and
electric quadrupole ($\mathcal{E}2$) multipole mixture, for which beams with
orbital angular momentum might be advantageous. Until very recently, the 8 eV
isomeric (i.e., metastable) state could only be accessed indirectly, e.g. via
$\alpha$-decay of 233U Beck et al. (2007); Von Der Wense et al. (2016);
Seiferle et al. (2019); Sikorsky et al. (2020), x-ray excitation of the next
nuclear excited state at 29 keV Masuda et al. (2019) or $\beta$-decay of 229Ac
Kraemer et al. (2023). The first direct VUV laser excitation of the 229Th
isomer was reported just recently Tiedau et al. (2024), using plane wave
beams. Vortex beams can be generated in optical transmission from regular
plane wave beams Phua et al. (2007), and this technology is scalable to the
VUV frequency range. It is therefore timely to address scenarios in which the
nuclear clock transition is driven by twisted light beams.

Our study follows the two approaches pursued at the moment to build a nuclear
clock Peik and Okhapkin (2015). The first approach involves a single-ion
nuclear clock in an ion trap Campbell et al. (2012); Peik and Tamm (2003).
This setup is particularly clean and promises great accuracy; however, its
realization is technically demanding because of the very low excitation
probability per single nucleus. The second approach involves a solid-state
nuclear clock using Th-doped VUV transparent crystals Kazakov et al. (2012);
Rellergert et al. (2010); Thirolf et al. (2024); Dessovic et al. (2014);
Elwell et al. (2024). Here, a large number of doped nuclei
($N\approx${10}^{14}$$ to ${10}^{16}$) can be simultaneously interrogated,
leading to a superior signal to noise ratio and thus a higher stability. The
crystal approach leads to nuclear level shifts, splittings and broadenings due
to the interactions with the intrinsic electric and magnetic fields of the
host crystals Dessovic et al. (2014). The interaction of the nuclear
quadrupole moments with the electric field gradient generated by the crystal
lattice leads to the emergence of a nuclear quadrupole splitting on the order
$100\text{\,}\mathrm{MHz}$.

Our theoretical approach commences with the calculation of the twisted light
interaction matrix element, which we derive for Bessel beams using semi-
classical theory and express in terms of the nuclear reduced transition
probabilities $B(\mathcal{M}1)$ and $B(\mathcal{E}2)$ and the intensity of the
driving field. We then present and discuss nuclear excitation probabilities
obtained with the density matrix formalism for single Th ions for both on-axis
and off-axis geometry. Compared to plane wave excitation, the interaction with
vortex beams leads to a qualitatively different excitation pattern in space.
We note here that as it has been shown for atomic systems, the total
excitation achieved by vortex beams averaging over the impact parameter is not
higher than with plane wave beams Afanasev et al. (2013); Scholz-Marggraf et
al. (2014).

For the second nuclear clock approach, we present our results for vortex beams
interacting with an ensemble of nuclei doped in a VUV-transparent crystal. We
focus on the quadrupole splitting level scheme of 229Th doped in a large band
gap crystal CaF2 and calculate the nuclear excitation probability induced by a
circular polarized twisted light beam and a superposition of two such beams
with opposite helicity. The occupation of the nuclear quadrupole levels is
represented using an analytical solution of the multilevel Bloch equations. We
consider the cubic symmetry of the crystal and the average over the three
possible quantization axes entering the solution of the multilevel Bloch
equations. Our simulations show that the vortex excitation in a macroscopic
sample can have different features for single quantization axes, but taken
into account all crystal orientations, the overall result turns out to be
similar to plane wave excitation.

The paper is structured as follows. A short introduction to twisted light and
the corresponding derivation of the nuclear interaction matrix element is
presented in Sec. II. Then, our numerical results for the two nuclear clock
approaches are presented and discussed in Sec. III. The paper concludes with a
brief discussion in Sec. IV.

##  II Theoretical Background

This section introduces the theoretical background for the interaction between
twisted light and nuclei. Though developed independently, our approach
resembles the theory work in Ref. Kazinski and Sokolov (2023). Throughout this
paper, we consider Bessel modes in order to describe the twisted light field.
While Bessel-Gauss and Laguerre-Gauss modes are also often used in the
literature, Bessel modes are most convenient to be treated analytically and
numerically and theoretical predictions have proven so far to match
experimental observations quite well Afanasev et al. (2018a); Afanasev and
Carlson (2022). For this reason, the section starts with a short introduction
to twisted light based on Bessel modes. For a more detailed review on twisted
light, we refer to Refs. Matula et al. (2013); Peshkov et al. (2017). We then
proceed to address the underlying nuclear interaction matrix element derived
in the semiclassical theory, i.e., treating the light field classically, and
the nucleus quantum mechanically.

###  II.1 Bessel beams

A twisted light beam propagating in the $z$-direction with well defined
transverse momentum $\zeta=\left|k_{\perp}\right|$, longitudinal momentum
$k_{z}$, total angular momentum (TAM) projection $m_{\gamma}$ and helicity
$\Lambda$ is characterized by the following vector potential Schulz et al.
(2020); Peshkov et al. (2023a):

$\bm{A}^{\text{(tw)}}(\bm{r})=A_{0}\int\bm{e}_{\bm{k}\Lambda}e^{i\bm{k}\cdot\bm{r}}a_{\zeta
m_{\gamma}}k_{\perp}\frac{dk_{\perp}}{2\pi}\frac{d\alpha_{k}}{2\pi}$ (1)

with

$a_{\zeta
m_{\gamma}}=\left(-i\right)^{m_{\gamma}}e^{im_{\gamma}\alpha_{k}}\frac{2\pi}{\zeta}\delta(k_{\perp}-\zeta)$
(2)

being the Fourier amplitude. Here, $A_{0}$ is the field amplitude, $\bm{k}$
the wave vector, which is related to the energy of the radiation via
$E=\hbar\omega=\hbar c\left|\bm{k}\right|$, $\bm{e}_{\bm{k}\Lambda}$ the
polarization vector pointing in the directions determined by
$\left\\{\theta_{k},\alpha_{k}\right\\}$ and $\Lambda=\pm 1$ the helicity,
respectively.

In this representation, the Bessel light illustrates a coherent superposition
of circularly polarized plane waves where the different $\bm{k}$ vectors span
the surface of a cone. This cone has a constant opening angle
$\theta_{k}=\arctan{(\zeta/k_{z})}$, the so-called pitch angle, illustrated in
Fig. 1.

Figure 1: Twisted light beam in k-space. The superposition of the wave vectors
$\bm{k}$ form the surface of a cone with constant opening angle
$\theta_{k}=\arctan{(\zeta/k_{z})}$.

The real space representation of the vector potential can be determined upon
evaluating the integral in Eq. (1). Thereby, in order to integrate over
$\alpha_{k}$, one first needs to expand the polarization vector according to
Matula et al. (2013)

$\bm{e}_{\bm{k}\Lambda}=\sum_{m_{s}}c_{m_{s}}e^{-im_{s}\alpha_{k}}\bm{\eta}_{m_{s}}$
(3)

where the sum runs over all possible eigenvalues of the spin angular momentum
operator $m_{s}$. Here, $\bm{\eta}_{\pm 1}=(1,\pm i,0)/\sqrt{2}$ and
$\bm{\eta}_{0}=(0,0,1)$ are the eigenvectors of the spin angular momentum
operator. Moreover, the coefficients $c$ explicitly read

$c_{\pm
1}=\frac{1}{2}\left(1\pm\Lambda\cos{(\theta_{k}})\right),\,\,\,\,c_{0}=\frac{\Lambda}{\sqrt{2}}\sin{(\theta_{k})}.$
(4)

Rewriting the scalar product

$\bm{k}\cdot\bm{r}=\bm{k}_{\perp}\cdot\bm{r}_{\perp}+k_{z}z=\zeta
r_{\perp}\cos{(\alpha_{k}-\phi_{r})}+k_{z}z$ (5)

and using the integral representation of the Bessel function Scholz-Marggraf
et al. (2014); Schulz et al. (2020)

$\int_{0}^{2\pi}\frac{d\alpha_{k}}{2\pi}e^{in\alpha_{k}\pm
iz\cos{(\alpha_{k})}}=(\pm i)^{n}J_{n}(z)$ (6)

yields the real space vector potential Matula et al. (2013); Schulz et al.
(2020)

$\displaystyle\bm{A}^{\text{(tw)}}(\bm{r})$
$\displaystyle=A_{0}e^{ik_{z}z}\sum_{m_{s}}i^{-m_{s}}c_{m_{s}}\bm{\eta}_{m_{s}}$
(7) $\displaystyle\times J_{m_{\gamma}-m_{s}}(\zeta
r_{\perp})e^{i(m_{\gamma}-m_{s})\phi_{r}}.$

In the equations above, $J_{n}$ is the Bessel function of first kind and order
$n$. The expression of the real space vector potential shows that Bessel beams
have a spatially inhomogeneous intensity pattern in the transverse plane, in
contrast to plane waves Matula et al. (2013). Note that usually only paraxial
twisted light fields, i.e. $\theta_{k}<<1$, are available in experiments. As
such the sum (7) is solely restricted to a single leading term where
$m_{s}=\Lambda$ holds. Furthermore, in the paraxial regime the Bessel beam has
a well defined orbital angular momentum projection $m_{l}$, i.e. the TAM
projection decouples to $m_{\gamma}=m_{l}+\Lambda$. This means that the beam
is not uniquely characterized by $m_{\gamma}$ but also by certain values for
the pair $m_{l}$ and $\Lambda$. Apart from that, one can easily show that for
$\theta_{k}=0$ and $m_{\gamma}=\Lambda$ the twisted light vector potential
recovers the vector potential of a circularly polarized plane wave with an
insignificant phase factor $i^{-\Lambda}$. However, for the sake of
generality, we will address also the case of non-paraxial Bessel beams with
larger $\theta_{k}$ values.

###  II.2 Nuclear interaction matrix element

As a starting point to derive the nuclear interaction matrix element, we
consider the semi-classical interaction Hamiltonian for a nucleus in an
electromagnetic field given by Pálffy et al. (2008)

$\mathcal{H}_{I}=-\frac{1}{c}\int\bm{j}(\bm{r})\cdot\bm{A}(\bm{r})d^{3}\bm{r}.$
(8)

Here, $c$ is the speed of light, $\bm{j}$ the nuclear current density and
$\bm{A}$ the vector potential of the impinging electromagnetic field, so far
treated as a plane wave.

In order to take the case of twisted-light-matter interaction into account, we
have to implement two modifications in Eq. (8). First, we replace the plane
wave vector potential by the twisted wave vector potential (1). Second, a
(two-dimensional) impact parameter $\bm{b}$ is introduced via
$e^{-i\bm{k}_{\perp}\bm{b}}$ in order to specify the position of the nucleus
within the spatially inhomogeneous wave front. Thereby, the impact parameter
$\bm{b}=b(\cos{(\phi_{b})},\sin{(\phi_{b})},0)$ is defined with respect to the
beam center at $r_{\perp}=0$. Introducing the expression in Eq. (1) in Eq. (8)
and evaluating the integral over $k_{\perp}$, the interaction Hamiltonian can
be written as

$\displaystyle\mathcal{H}^{\text{(tw)}}_{I}$
$\displaystyle=-\frac{\left(-i\right)^{m_{\gamma}}A_{0}}{c}$ (9)
$\displaystyle\times\int\bm{j}(\bm{r})\cdot\bm{e}_{\bm{k}\Lambda}e^{i\bm{k}\bm{r}}e^{-i\bm{k}_{\perp}\bm{b}}e^{im_{\gamma}\alpha_{k}}\frac{d\alpha_{k}}{2\pi}d^{3}\bm{r}.$

Note that here the perpendicular component of $\bm{k}$ is now fixed by the
Dirac delta function in Eq. (2). We can further evaluate Eq. (9) by expanding
$\bm{e}_{\bm{k}\Lambda}e^{i\bm{k}\bm{r}}$ into spherical multipoles according
to

$\displaystyle\bm{e}_{\bm{k}\Lambda}e^{i\bm{k}\bm{r}}$
$\displaystyle=\sqrt{2\pi}\sum_{L,M}i^{L}\sqrt{2L+1}\,D_{M\Lambda}^{L}(\alpha_{k},\theta_{k},0)$
(10)
$\displaystyle\times\left(\bm{\mathcal{A}}^{\mathcal{M}}_{LM}+i\Lambda\bm{\mathcal{A}}^{\mathcal{E}}_{LM}\right).$

Here, $D_{M\Lambda}^{L}(\alpha_{k},\theta_{k},0)$ is the Wigner rotation
matrix accounting for the different quantization axes of the light field and
of the nucleus. This matrix can be expanded as
$D_{M\Lambda}^{L}(\alpha_{k},\theta_{k},0)=e^{-iM\alpha_{k}}d_{M\Lambda}^{L}(\theta_{k})$
where $d_{M\Lambda}^{L}(\theta_{k})$ is the Wigner small $d$-function.
Moreover, $\bm{\mathcal{A}}^{\mathcal{M}}_{LM}$ is the magnetic amplitude of
multipolarity $L$ and projection $M$, and
$\bm{\mathcal{A}}^{\mathcal{E}}_{LM}$ the electric amplitude, respectively.
Using the integral representation of the Bessel function (6)

$\displaystyle\int_{0}^{2\pi}e^{i(m_{\gamma}-M)\alpha_{k}}e^{-i\zeta
b\cos{(\alpha_{k}-\phi_{b})}}\,\frac{d\alpha_{k}}{2\pi}$ (11)
$\displaystyle=(-i)^{m_{\gamma}-M}e^{i(m_{\gamma}-M)\phi_{b}}J_{m_{\gamma}-M}(\zeta
b)\,,$

the interaction Hamiltonian can be written as

$\displaystyle\mathcal{H}^{\text{(tw)}}_{I}$
$\displaystyle=-\frac{A_{0}\sqrt{2\pi}}{c}\sum_{L,M,\mu}(i\Lambda)^{\delta_{\mu,\mathcal{E}}}i^{L}(-i)^{2m_{\gamma}-M}$
(12) $\displaystyle\times
e^{i\phi_{b}(m_{\gamma}-M)}\sqrt{2L+1}d_{M\Lambda}^{L}(\theta_{k})J_{m_{\gamma}-M}(\zeta
b)$
$\displaystyle\times\int\bm{j}(\bm{r})\cdot\bm{\mathcal{A}}_{LM}^{\mu}\,d^{3}\bm{r}\,,$

where $\delta_{\mu,\mathcal{E}}$ is the Kronecker delta and
$\mu\in\left\\{\mathcal{E},\mathcal{M}\right\\}$. Again, by setting
$\theta_{k}=0$, the interaction Hamiltonian recovers Eq. (8) as long as
$m_{\gamma}=\Lambda$. The integral in Eq. (12) can then be evaluated with aid
of Pálffy et al. (2008); Ring and Schuck (2004)

$\int\bm{j}(\bm{r})\cdot\bm{\mathcal{A}}_{LM}^{\mu}\,d^{3}\bm{r}=\frac{k^{L}c}{i(2L+1)!!}\sqrt{\frac{L+1}{L}}\mathbb{Q}^{\mu}_{LM}$
(13)

where $\mathbb{Q}^{\mu}_{LM}$ denotes the multipole moment operator of type
$\mu$. With that a general expression of the Hamiltonian describing the
twisted light nucleus interaction is found as

$\displaystyle\mathcal{H}^{\text{(tw)}}$
$\displaystyle=-\sqrt{2\pi}E\sum_{L,M,\mu}i^{L}(-i)^{2m_{\gamma}-M+1}e^{i\phi_{b}(m_{\gamma}-M)}$
(14)
$\displaystyle\times(i\Lambda)^{\delta_{\mu,\mathcal{E}}}\sqrt{\frac{(2L+1)(L+1)}{L}}\frac{k^{L-1}}{(2L+1)!!}$
$\displaystyle\times d_{M\Lambda}^{L}(\theta_{k})J_{m\gamma-M}(\zeta
b)\mathbb{Q}^{\mu}_{LM}$

where we used $A_{0}=E/k$ with $E$ being the amplitude of the electric field.
Then, the interaction matrix element can then be simply evaluated as

$\displaystyle M^{(\text{tw})}_{fi}$
$\displaystyle=\bra{I_{e}m_{e}}\mathcal{H}^{(\text{tw})}_{I}\ket{I_{g}m_{g}}$
(15)
$\displaystyle=-\sqrt{\frac{4\pi\mathcal{I}}{c\varepsilon_{0}}}\sum_{L,M,\mu}i^{L}(-i)^{2m_{\gamma}-M+1}e^{i\phi_{b}(m_{\gamma}-M)}$
$\displaystyle\times(i\Lambda)^{\delta_{\mu,\mathcal{E}}}\sqrt{\frac{(2L+1)(L+1)}{L}}\frac{k^{L-1}}{(2L+1)!!}$
$\displaystyle\times d_{M\Lambda}^{L}(\theta_{k})J_{m\gamma-M}(\zeta
b)\bra{I_{e}m_{e}}\mathbb{Q}^{\mu}_{LM}\ket{I_{g}m_{g}}$

where $I_{g/e}$ denotes the angular momentum of the nuclear ground$/$excited
state and $m_{g/e}$ the respective projections on the nuclear quantization
axis. Here, $E=\sqrt{\frac{2\mathcal{I}}{c\varepsilon_{0}}}$ was expressed in
terms of the intensity $\mathcal{I}$ of the driving field where
$\varepsilon_{0}$ denotes the vacuum permittivity.

By applying the Wigner-Eckhart theorem Edmonds (1996)

$\displaystyle\bra{I_{e}m_{e}}\mathbb{Q}^{\mu}_{LM}\ket{I_{g}m_{g}}=\frac{(-1)^{I_{g}-m_{g}}}{\sqrt{2L+1}}$
(16) $\displaystyle\times\braket{I_{e}m_{e}I_{g}-m_{g}}{LM}\langle
I_{e}||\mathbb{Q}^{\mu}_{LM}||I_{g}\rangle$

and using the expression of the reduced transition probability Ring and Schuck
(2004)

$B(\mu L,I_{g}\rightarrow I_{e})=\frac{1}{2I_{g}+1}\left|\langle
I_{e}||\mathbb{Q}^{\mu}_{LM}||I_{g}\rangle\right|^{2}$ (17)

the final form of the twisted nuclear interaction matrix element reads

$\displaystyle M_{fi}^{(\text{tw})}$
$\displaystyle=-\sum_{L,\mu}i^{L}(-i)^{2m_{\gamma}-\Delta
m+1}e^{i\phi_{b}(m_{\gamma}-\Delta m)}$ (18)
$\displaystyle\times(i\Lambda)^{\delta_{\mu,\mathcal{E}}}\sqrt{\frac{4\pi\mathcal{I}}{c\varepsilon_{0}}}\sqrt{\frac{(L+1)(2I_{g}+1)}{L}}$
$\displaystyle\times(-1)^{I_{g}-m_{g}}\frac{k^{L-1}}{(2L+1)!!}J_{m\gamma-\Delta
m}(\zeta b)d_{\Delta m\Lambda}^{L}(\theta_{k})$
$\displaystyle\times\braket{I_{e}m_{e}I_{g}-m_{g}}{L\Delta m}\sqrt{B(\mu
L,I_{g}\rightarrow I_{e})}$ $\displaystyle=\sum_{L,\mu}M_{fi,\mu
L}^{(\text{tw})}$

where the sum over $M$ was dropped due to the selection rule $\Delta
m=m_{e}-m_{g}$ embodied by the Clebsch-Gordan coefficient. From this, we can
define our Rabi frequency as

$\Omega^{(\text{tw})}_{fi}=\frac{\left|M_{fi}^{\text{(tw)}}\right|}{\hbar}.$
(19)

In general, this matrix element describes the interaction of a vortex beam
with any nucleus located at position $b$ with respect to the beam center whose
nuclear transitions is described by the reduced transition probability $B(\mu
L)$. We observe that for each multipole order $M_{fi,\mu L}^{(\text{tw})}$,
the interaction matrix element factorizes as a product of the standard plane
wave matrix element, multiplied by the Bessel function and the Wigner small
$d$-function. The latter induces selection rule modifications. A similar
factorization has been deduced also in Refs. Kazinski and Sokolov (2023); Lu
et al. (2023) and is known also from the interaction of vortex beams with
atomic transitions, see, for instance, Ref. Afanasev et al. (2018a).

In the case of 229Th, the transition from the ground to the excited state
proceeds via the $\mathcal{M}1$ channel with a small $\mathcal{E}2$ mixture.
Consequently, we can restrict the summation in Eq. (18) to these two multipole
orders. We calculate the square of the Rabi frequency defined in Eq. (19)
normalized to the Clebsch-Gordan coefficient of the respective hyperfine
transition and to the incoming intensity corresponding to the field amplitude
$A_{0}$. The results are presented in Fig. 2 for each multipole channel of the
229Th transition individually as a function of impact parameter for different
TAM projections $m_{\gamma}$. The corresponding input parameters used in the
calculation are listed in the figure caption.

Figure 2: Squared normalized Rabi frequency in units of
${\mathrm{m}}^{2}\text{\,}{\mathrm{s}}^{-2}\text{\,}{\mathrm{W}}^{-1}$ as a
function of impact parameter $b$ (in units of transition wavelength $\lambda$)
for the a) $\mathcal{M}1$ channel and the b) $\mathcal{E}2$ channel of the
229Th nuclear clock transition. The Rabi frequency is normalized according to
$\tilde{\Omega}^{2}=\left|\Omega_{fi,\mu
L}^{(\mathrm{tw})}\right|^{2}/(\left|\braket{I_{e}m_{e}I_{g}-m_{g}}{L\Delta
m}\right|^{2}\mathcal{I})$. For a), we use $B(\mathcal{M}1,I_{g}\rightarrow
I_{e})=0.017$ W.u. derived with the experimental values in Ref. Kraemer et al.
(2023), $m_{\gamma}=1$, $\Lambda=1$ and
$\theta_{k}=$45\text{\,}\mathrm{\SIUnitSymbolDegree}$$. For b), we use
$B(\mathcal{E}2,I_{g}\rightarrow I_{e})=27.04$ W.u. taken from the theoretical
predictions in Ref. Minkov and Pálffy (2017), $m_{\gamma}=2$, $\Lambda=1$ and
$\theta_{k}=$45\text{\,}\mathrm{\SIUnitSymbolDegree}$$.

The interaction with a twisted light beam leads to the emergence of $2L+1$
transition amplitudes which are position dependent. Due to the spatial
modulation by the Bessel function, certain transitions can be enhanced or
suppressed relative to each other. Of particular interest is the impact
parameter value $b=0$. In this case, the vortex beam’s TAM is completely
transferred to the internal degrees of freedom which leads to the selection
rule $\Delta m=m_{\gamma}$. For this reason it becomes possible to address
higher order multipole transitions and separate them from lower order
contributions. Concretely, for the case of 229Th, it is possible to
disentangle the $\mathcal{M}1$ \+ $\mathcal{E}2$ multipole mixing by
transferring $\Delta m=2$ units of angular momentum. Then, the transition can
only proceed via the $\mathcal{E}2$ channel although the respective radiative
rate is almost $10$ orders of magnitude suppressed compared to the
$\mathcal{M}1$ channel.

##  III Numerical examples

In this Section, we present numerical results for nuclear photo-absorption in
229Th using twisted light. Two experimental configurations based on the two
nuclear clock approaches are considered. First, we investigate the interaction
of a single trapped ion with a vortex beam, taking into account both the
nuclear hyperfine interaction and Zeeman splitting for the ion level scheme.
Thereby, the temporal dynamics induced by resonant driving is investigated for
the impact parameter configurations on-axis ($b=0$) and off-axis ($b\neq 0$)
for different transitions in the ion. Second, we investigate the interaction
of twisted light with a macroscopic sample in which the nuclei are distributed
homogeneously over the entire whole beam as would be the case of a Th-doped
VUV transparent crystal. In particular, we consider the case of the 229Th:CaF2
crystal including nuclear quadrupole splitting induced by the crystal fields.
We calculate the distribution of the magnetic sublevel population for a
circularly polarized Bessel beam and a superposition of two Bessel beams with
the same TAM $m_{\gamma}$, but opposite helicities $\Lambda$.

###  III.1 Single nucleus

Figure 3: 229Thq+ ion placed at impact parameter $b$ interacting with a
resonant vortex beam. a) Level scheme of 229Thq+ in an external magnetic
field. b) Effective level system modelling the resonant driving.

The scenario under investigation is illustrated in Fig. 3. A single trapped
229Thq+ ion is placed at impact parameter $b$ in a vortex beam with photon
energy resonant to one nuclear transition between the ground and excited state
manifolds. We consider an external magnetic field applied in $z$-direction
coinciding with the propagation direction of the beam. In the semiclassical
approach, the center of mass motion is treated via the well-defined impact
parameter. As such, the coupling to rotational degrees of freedom is
neglected, see Peshkov et al. (2023b) for a recent treatment of this effect.
The charge state $q$ can in principle be arbitrarily chosen, albeit with some
restrictions related to nuclear decay channels that involve the electronic
shell and can significantly shorten the nuclear excited state lifetime.
Theoretical calculations have shown that H-, Li- or B-like 229Th ions display
a strong nuclear hyperfine mixing which affects the lifetime of the nuclear
excited state Shabaev et al. (2022). We do not consider these effects here.

As indicated in panel a) of Fig. 3, the nuclear energy levels experience
hyperfine splitting described by the quantum number $F_{g/e}$. The emergence
of this splitting is related to the coupling of nuclear and electronic degrees
of freedom. Here, the quantum number $F_{g/e}$ runs from
$\left|J-I_{g/e}\right|\leq F_{g/e}\leq J+I_{g/e}$ where $J$ is the angular
momentum quantum number of the electronic shell and $I_{g/e}$ the angular
momentum quantum number of the nuclear ground$/$excited state. In order to
specify the quantization axis of the system, an external weak magnetic field
is applied. This leads to a further splitting of the energy levels $F_{e/g}$
where each level splits into $2F_{e/g}+1$ Zeeman sublevels running from
$-F_{e/g}\leq m_{e/g}\leq F_{e/g}$. We will consider resonant driving with a
twisted light field for different Zeeman $\Delta m=m_{e}-m_{g}$ transitions.

The multilevel system is effectively modelled as a three level system
described by the density matrix $\rho=\sum_{e,g,\tilde{g}}\rho_{ij}$ where the
diagonal elements correspond to the level populations and the off-diagonal
elements to the coherences, respectively. Here, the resonant driving occurs
between a fixed ground $\ket{g}$ and excited $\ket{e}$ state. We model the
relaxation processes proceeding via a quasi degenerate ground state
$\ket{\tilde{g}}$ which accounts for the nuclear excited state population
decaying to a variety of other magnetic or hyperfine substates.

The dynamics of the system are governed by the Master equation Scully and
Zubairy (1999)

$\dot{\rho}=\frac{1}{i\hbar}\left[\mathcal{H},\rho\right]+\mathcal{L}[\rho]$
(20)

where $\mathcal{H}$ is the Hamiltonian describing the light-matter interaction
and $\mathcal{L}[\rho]$ the Lindblad operator accounting for relaxation
processes, respectively. The concrete form of the Hamiltonian (on resonance)
is given by

$\mathcal{H}=-\hbar\left(\Omega_{eg}^{(\text{tw})}\ket{e}\bra{g}+\,\text{h.c.}\right)$
(21)

where $\Omega_{eg}^{(\text{tw})}$ is the twisted light Rabi frequency driving
the transition $\ket{g}\rightarrow\ket{e}$ whose general expression is given
in Eq. (19). For a proper description of the system including hyperfine
splitting, the Rabi frequency must be modified by replacing the Clebsch-Gordan
coefficient in Eq. (18) with von der Wense et al. (2020)

$\displaystyle\braket{I_{e}m_{e}I_{g}-m_{g}}{L\Delta
m}\rightarrow\sqrt{2F_{e}+1}\sqrt{2F_{g}+1}$ (22)
$\displaystyle\times\braket{F_{e}m_{e}F_{g}-m_{g}}{L\Delta
m}\begin{Bmatrix}I_{g}&L&I_{e}\\\ F_{e}&J&F_{g}\end{Bmatrix}\,,$

where $\\{...\\}$ denotes the Wigner-$6J$ symbol. Note that this
transformation only holds if the electronic shell remains in its state $J$
during the radiative coupling.

Moreover, the Lindblad operator reads

$\displaystyle\mathcal{L}[\rho]$
$\displaystyle=\Gamma\rho_{ee}\left(\ket{\tilde{g}}\bra{\tilde{g}}-\ket{e}\bra{e}\right)$
(23)
$\displaystyle-\left(\frac{\Gamma+\Gamma_{\ell}}{2}\right)\left(\rho_{eg}\ket{e}\bra{g}+\,\mathrm{h.c.}\right),$

where we have neglected relaxation between sublevels of the same manifold and
relaxation of the coherences involving $\ket{\tilde{g}}$, since these are not
affected by resonant driving. The total relaxation rate $\Gamma$ is thereby
given by $\Gamma=\Gamma_{\gamma}+\Gamma_{\text{el}}$ with $\Gamma_{\gamma}$
the radiative decay rate and $\Gamma_{\text{el}}$ the nuclear decay rate via
energy transfer to the electronic shell, respectively. The nucleus can
transfer its excitation energy to the electronic shell via internal conversion
or electronic bridge (EB) processes. Internal conversion refers to the process
in which the nuclear excited state decays by transferring its energy to an
shell electron which is then expelled into the continuum. Since the second
ionization potential of Th lies at 12 eV, this channel should be energetically
forbidden already for the first ionized state 229Th+. In turn, EB describes
the energy transfer between nucleus and electronic shell without a change in
the ionic charge and with an accompanying photon emission to account for the
energy mismatch of available electronic and nuclear states Porsev and Flambaum
(2010a, b).

Moreover, $\Gamma_{\ell}$ in Eq. (23) denotes the laser bandwidth which leads
to a faster dissipation of the coherences. For the radiative decay rate we
take $\Gamma_{\gamma}=$3.14\text{\times}{10}^{-4}\text{\,}{\mathrm{s}}^{-1}$$
Kraemer et al. (2023), while the value $\Gamma_{\text{el}}$ depends on the ion
species under investigation. Upon solving Eq. (20) numerically with the
initial condition $\rho_{gg}(0)=1$, we can determine the temporal dynamics of
the nuclear excited state population as a function of impact parameter. For
all upcoming calculations, we consider a vortex beam with TAM projection
$m_{\gamma}=2$ and $\Lambda=1$ interacting with a well localized 229Thq+
target for different opening angles.

####  III.1.1 On-axis ($b=0$)

For zero impact parameter, the selection rule $\Delta m=m_{\gamma}$ comes into
play. Thus, by choosing $\Delta m=2$, only the $\mathcal{E}2$ channel can be
driven on-axis, such that the two multipolarities of the nuclear clock
transition can be completely spatially separated. For the isomeric transition
in 229Th, the radiative rate for the $\mathcal{E}2$ channel is almost 10
orders of magnitude suppressed compared to the $\mathcal{M}1$ one. This means
that while in principle one could drive pure $\mathcal{E}2$ Rabi oscillations,
this requires either a very large Rabi frequency or a moderate Rabi frequency
and extremely long interrogation times. In both cases, ion loss from the trap
due to multi-photon ionization or chemical reactions with background gases may
occur. Moreover, a long interrogation time is limited by the relaxation of the
excited state population. A way to reduce unwanted ionization is to choose a
higher ion charge state for this case, for instance 229Th35+, discussed in the
context of electron beam ion trap generation via electron-atom collisions in
Ref. Bilous et al. (2020). We consider in the following this charge state with
the electron shell in its $J=15/2$ ground state. Based on the analysis in Ref.
Bilous et al. (2020) and the lack of electronic states with energies matching
the nuclear transition energy, we neglect the spontaneous EB decay rate in the
calculation and set $\Gamma_{\text{el}}=0$.

For nuclear excitation, we envisage the
$\ket{F_{g}=5\,m_{g}=5}\rightarrow\ket{F_{e}=7\,m_{e}=7}$ transition, since
the corresponding product of Clebsch-Gordan coefficient and Wigner $6$-$J$
symbol in Eq. (22) is largest. Although the transition is per se of
$\mathcal{E}2$ multipolarity, due to the selection rule $\Delta F=2$, resonant
driving is optimized with the choice of $\Delta m=2$. We note that for the on-
axis case ($b=0$), regardless of the choice of $\Delta F$, only the
$\mathcal{E}2$ transition will be driven once we choose $m_{\gamma}=2$.

Figure 4: Calculated excited state population as a function of irradiation
time for the resonant $\mathcal{E}2$ driving considering a single 229Th35+ ion
in the center of a vortex beam with $m_{\gamma}=2$ and $\Lambda=+1$ for
different laser bandwidths. The pitch angles of the beams are a)
$\theta_{k}=$1\text{\,}\mathrm{\SIUnitSymbolDegree}$$, b)
$\theta_{k}=$5\text{\,}\mathrm{\SIUnitSymbolDegree}$$, c)
$\theta_{k}=$10\text{\,}\mathrm{\SIUnitSymbolDegree}$$ and d)
$\theta_{k}=$60\text{\,}\mathrm{\SIUnitSymbolDegree}$$.

In Fig. 4 we present our numerical results for resonant excitation considering
different opening angles and laser bandwidths. We choose a fixed set of ground
and excited nuclear states and consider a fictitious laser intensity value of
$\mathcal{I}=$15\text{\,}\mathrm{W}\text{\,}{\mathrm{cm}}^{-2}$$. The excited
state population as a function of the irradiation time for small $\theta_{k}$
angles, also known as the paraxial regime, is presented in Fig. 4a). For a
perfectly coherent cw laser $(\Gamma_{\ell}=0\cdot$\mathrm{Hz}$)$, the
excitation probability reaches a maximum of only
$\approx$27\text{\,}\mathrm{\char 37\relax}$$ and drops down back to zero
completing a Rabi cycle in a few hours of interrogation time. The situation is
similar for a laser with a bandwidth of the order of the radiative decay rate
$(\Gamma_{\ell}=\Gamma_{\gamma})$ with the main difference that the amplitude
is slightly suppressed. For a more realistic
$\Gamma_{\ell}=2\pi\cdot$\mathrm{Hz}$$ corresponding to the bandwidth of a
mode of narrowband frequency comb with high repetition rate von der Wense et
al. (2020); von der Wense and Zhang (2020), no excitation via the
$\mathcal{E}2$ channel is achieved.

The driving becomes more efficient for larger pitch angles. We consider the
vicinity of the paraxial regime in Figs. 4b)-c). Several Rabi cycles with a
maximal excitation probability of $\approx$70\text{\,}\mathrm{\char
37\relax}$$ to $84\text{\,}\mathrm{\char 37\relax}$ can be passed within a few
hours time span for $\Gamma_{\ell}=0\cdot$\mathrm{Hz}$$ and
$\Gamma_{\ell}=\Gamma_{\gamma}$. Compared to the paraxial regime results, this
is due to the larger Rabi frequency which results from the Wigner small
$d$-function

$d_{21}^{2}(\theta_{k})=2\cos{\left(\frac{\theta_{k}}{2}\right)}^{3}\sin{\left(\frac{\theta_{k}}{2}\right)}$
(24)

in Eq. (18). The Wigner small $d$-function is increasing in the angular
interval $[0^{\circ},60^{\circ}]$, such that a higher pitch angle increases
the driving efficiency. We therefore present in Fig. 4d) the excitation
probability for $\theta_{k}=60^{\circ}$. In this case, several Rabi cycles can
be passed for $\Gamma_{\ell}=0\cdot$\mathrm{Hz}$$ and
$\Gamma_{\ell}=\Gamma_{\gamma}$ already within one hour with excitation
probabilities approaching a maximum probability of $\approx$
$95\text{\,}\mathrm{\char 37\relax}$. Only at this pitch angle can also a
narrowband frequency comb produce a non-negligible excitation probability,
albeit only the small value of $\approx$ $5\text{\,}\mathrm{\char 37\relax}$
reached after approx. 2.5 hours.

Overall, pure Rabi oscillations of the $\mathcal{E}2$ channel alone seem to be
very challenging experimentally. We note that the nuclear reduced transition
probability $B(\mathcal{E}2)$ has not been determined experimentally so far.
Thus, the prospect of an experiment which would lead to its measured value is
appealing. However, practical considerations show that unrealistically large
intensities and long interrogation times would be required. In addition, one
requires an accurate confinement of the particle at zero impact parameter over
a long period of time. Small deviation from $b=0$ might open the
$\mathcal{M}1$ channel which would immediately dominate the excitation process
due to its much stronger nuclear reduced transition probability. We thus
conclude that the on-axis case with its challenges is of rather academic
interest and turn to the next case of off-axis driving, i.e., at non-zero
impact parameter.

####  III.1.2 Off-axis ($b\neq 0$)

Once off-axis, the special selection rule $\Delta m=m_{\gamma}$ no longer acts
and we will have $\mathcal{M}1$ \+ $\mathcal{E}2$ multipole mixing. Due to the
much stronger $\mathcal{M}1$ radiative channel, we will neglect for this case
the $\mathcal{E}2$ multipole mixing. We therefore consider transitions with
$\Delta m\in\left\\{-1,0,1\right\\}$. The required laser intensities for
achieving a reasonable fraction of excited nuclei is less dramatic and the
corresponding interrogation time shorter. Therefore, one can envisage also
lower ionic charge states. In the following, we consider 229Th3+ which is the
most promising ion species for a single ion nuclear clock Campbell et al.
(2012); Peik and Tamm (2003). Similar to assumptions made in the on-axis case,
the electronic shell of 229Th3+ remains in its $J=5/2$ ground state during the
radiative driving. We consider three $\mathcal{M}1$ transitions between
sublevels of the ground and excited nuclear levels of 229Th3+ with hyperfine
splitting

$\displaystyle\ket{F_{g}=5\,m_{g}=\pm 5}\rightarrow\ket{F_{e}=4\,m_{e}=\pm
4}\,,$ (25)
$\displaystyle\ket{F_{g}=5\,m_{g}=0}\rightarrow\ket{F_{e}=4\,m_{e}=0}\,,$

where the former two were proposed as clock transitions Campbell et al.
(2012). Since the electronic shell remains in its ground state, the
spontaneous EB decay is on the order of
$\Gamma_{\mathrm{eb}}\approx\mathcal{O}(${10}^{-5}\text{\,}{\mathrm{s}}^{-1}$)$
Porsev and Flambaum (2010b), about one order of magnitude smaller than the
radiative decay rate. We therefore consider $\Gamma\approx\Gamma_{\gamma}$.

Figure 5: Population dynamics as a function of impact parameter $b$ (in units
of the transition wave length $\lambda$) and time $t$ for three $\mathcal{M}1$
transitions in a 229Th3+ ion. Here, $m_{\gamma}=2$, $\Lambda=1$,
$\theta_{k}=$10\text{\,}\mathrm{\SIUnitSymbolDegree}$$ and
$\mathcal{I}=$0.5\text{\,}\mathrm{W}\text{\,}{\mathrm{cm}}^{-2}$$ for all
calculations. a) $\ket{F_{g}=5\,m_{g}=-5}\rightarrow\ket{F_{e}=4\,m_{e}=-4}$
for $\Delta m=1$ and $\Gamma_{\ell}=0\cdot$\mathrm{Hz}$$. b)
$\ket{F_{g}=5\,m_{g}=0}\rightarrow\ket{F_{e}=4\,m_{e}=0}$ for $\Delta m=0$ and
$\Gamma_{\ell}=0\cdot$\mathrm{Hz}$$. c)
$\ket{F_{g}=5\,m_{g}=5}\rightarrow\ket{F_{e}=4\,m_{e}=4}$ for $\Delta m=-1$
and $\Gamma_{\ell}=0\cdot$\mathrm{Hz}$$. d)-f) same as a)-c), with the
difference $\Gamma_{\ell}=2\pi\cdot$\mathrm{Hz}$$.

We calculate the excited state population $\rho_{ee}$ for nuclear transitions
driven resonantly with a tuneable vortex beam with parameters
$\theta_{k}=$10\text{\,}\mathrm{\SIUnitSymbolDegree}$$, $m_{\gamma}=2$,
$\Lambda=1$ and
$\mathcal{I}=$0.25\text{\,}\mathrm{W}\text{\,}{\mathrm{cm}}^{-2}$$. The
population dynamics of the excited state $\rho_{ee}$ as a function of position
and time for two laser band widths $\Gamma_{\ell}=0\cdot$\mathrm{Hz}$$ (cw)
and $\Gamma_{\ell}=2\pi\cdot$\mathrm{Hz}$$ are presented in Fig. 5. The case
of a completely coherent cw laser ($\Gamma_{\ell}=0\cdot$\mathrm{Hz}$$) is
presented in the colormaps in Figs. 5a) for $\Delta m=1$, b) for $\Delta m=0$
and c) for $\Delta m=-1$, respectively. The excited state population is
presented as a function of impact parameter and time. Bright regions in the
map display high excitation probabilities while dark regions display low
probabilities. In Fig. 5a) we observe that strong driving of the
$\mathcal{M}1$ transition for $\Delta m=1$ occurs already at small impact
parameters, only slightly off-axis. Since the ion impact parameter cannot be
kept precisely fixed in the trap, this result supports our conclusion of the
previous subsection that pure resonant $\mathcal{E}2$ driving at $b=0$ is
rather unrealistic with current experimental capabilities.

Along the abscissa, the excited state population exhibits Rabi oscillations
resulting from resonant driving. Along the ordinate, $\rho_{ee}$ is modulated
according to $J_{m_{\gamma}-\Delta m}^{2}(\zeta b)$. For a fixed time
indicated by the vertical dashed line, we present on the panels left of the
colormaps the corresponding projected impact-parameter dependence of the
excited state population. For a better understanding of the displayed minima,
we plot in each projection also the corresponding Bessel function squared
(normalized to unity) to distinguish between the zeros of
$J_{m_{\gamma}-\Delta m}^{2}(\zeta b)$ and Rabi oscillation minima due to the
chosen time point.

For a fixed impact parameter value (indicated by the horizontal dashed line),
we present projections in the panels below the colormaps to show the excited
state population as a function of time. Since the Wigner small-$d$ functions
entering the expression of the Rabi frequency obey the relation

$d_{11}^{1}(\theta_{k})>d_{01}^{1}(\theta_{k})>d_{-11}^{1}(\theta_{k})$ (26)

for the considered pitch angle
$\theta_{k}=$10\text{\,}\mathrm{\SIUnitSymbolDegree}$$, the transition
$\ket{F_{g}=5\,m_{g}=-5}\rightarrow\ket{F_{e}=4\,m_{e}=-4}$ with $\Delta m=1$
is driven on the fastest timescale. We note that the order of the Wigner
functions in Eq. (26) is reversed if we choose $\Lambda=-1$. Correspondingly,
the timescales of the Rabi cycles for the three $\Delta m=0,\pm 1$ cases is
also reversed for $\Lambda=-1$: for the same pitch angle, the transition
$\Delta m=-1$ is fastest while $\Delta m=1$ is slowest. The times required for
a Rabi cycle however follow $J_{m_{\gamma}-\Delta m}^{2}(\zeta b)$ and will
therefore not have the same values as for the case $\Lambda=1$. We note that
tuning the pitch angle can also manipulate the duration of a Rabi cycle.
However, by means of varying $\theta_{k}$ also the vertical modulation via
$J_{m_{\gamma}-\Delta m}^{2}(\zeta b)$ is influenced such that the zeros of
the Bessel function shift.

In case of $\Gamma_{\ell}=2\pi\cdot$\mathrm{Hz}$$, the oscillation patterns of
the excited state population $\rho_{ee}$ become more washed out. Although the
vertical Bessel modulation in impact parameter is conserved, the temporal
dynamics in terms of Rabi oscillations differ. The smallest modifications
occur in the case of $\Delta m=1$, where the Rabi oscillations present a
slightly decaying envelope. The dynamics for $\Delta m=0$ and $\Delta m=-1$
are mediated by incoherent pumping resulting from a faster relaxation of the
coherences in Eq. (23). This leads to the suppression or even absence of Rabi
oscillations together with a reduced amplitude as it is shown in the
projection of the colormaps. However, coherent dynamics can be recovered by
either increasing the pitch angle or the intensity provided that the condition
$\Omega>>\Gamma_{\ell}/2$ is fulfilled von der Wense et al. (2020).

The off-axis setup can be used to probe different $\mathcal{M}1$ transitions
in 229Th3+ with a single tunable laser beam. The selection rules for vortex
beam excitation allows driving of all three transitions; however, a narrow-
band tunable laser will selectively drive just the transition to which its
frequency is resonant. This can be interesting for metrology applications, for
instance comparing clock transitions. The excitation strength can be tuned
with respect to the position of the nucleus in the wave front according to the
projection in Fig. 5 or by varying the pitch angle. We note that in comparison
with the on-axis case discussed in the previous Subsection, this scenario
seems to be more feasible, since shorter interrogation times and lower laser
intensities are required. According to Refs. von der Wense and Zhang (2020);
Peik et al. (2021); Zhang et al. (2022) a high repetition frequency comb with
comb mode bandwidth $\Gamma_{\ell}=2\pi\cdot$\mathrm{Hz}$$ and
$\mathcal{I}=\mathcal{O}($1\text{\,}\mathrm{W}\text{\,}{\mathrm{cm}}^{-2}$)$
should be available in the near future.

The Bessel modes used here represent an idealization, similar to the
frequently used idealized plane wave picture. Modelling a real physical
scenario requires the usage of Laguerre-Gaussian modes. However, close to the
beam center and in the paraxial approximation we expect Laguerre-Gauss and
Bessel modes to predict a similar behaviour Afanasev et al. (2018b). When
considering Laguerre-Gaussian beams, we expect a temporally finite modulation
with slightly modified spatial pattern of the excitation. This is however
beyond the scope of the present work.

###  III.2 Ensembles of nuclei

In the second part of this Section, we consider the interaction of twisted
light with a macroscopic 229Th target where the nuclei are distributed
homogeneously over the entire beam. Here, our model system is 229Th:CaF2. The
host crystal CaF2 is an ideal environment for doping 229Th. The band gap of
$\approx$ $11\text{\,}\mathrm{eV}$-$12\text{\,}\mathrm{eV}$ Rubloff (1972);
Barth et al. (1990); Tsujibayashi et al. (2002) renders the crystal
transparent at the wavelength of the nuclear clock transition. In addition,
high doping densities upwards $n\approx${10}^{17}\text{\,}{\mathrm{cm}}^{-3}$$
can be achieved Beeks et al. (2023). As a consequence, a larger number of
nuclei can be interrogated at the same time leading to an improved clock
stability.

The underlying cubic lattice hosts 229Th in a charge state $4+$ leading to a
suppression of the internal conversion decay channel. Within the lattice, one
of the calcium ions is replaced by the thorium ion and two more fluorine
interstitial ions emerge for charge compensation. Ab initio DFT studies
predict several preferred doping configurations Dessovic et al. (2014), where
two are of particular interest Nickerson et al. (2018). The fluorine ions are
either in a $90\text{\,}\mathrm{\SIUnitSymbolDegree}$ or
$180\text{\,}\mathrm{\SIUnitSymbolDegree}$ configuration as illustrated in
Fig. 6a). In contrast to the free ionic system, thorium doped CaF2 experiences
a quadrupole splitting according to Collins and Travis (1967)

$\hat{\mathcal{H}}_{\mathcal{E}2}=\frac{eQV_{zz}}{4I(2I-1)}\left[3\hat{I}_{z}-\hat{I}+\frac{\eta}{2}(\hat{I}_{+}^{2}+\hat{I}_{-}^{2})\right].$
(27)

Here, $e$ denotes the electric charge, $Q$ are the nuclear quadrupole moments,
$V_{zz}$ is the dominant component of the electric field gradient (EFG) at the
thorium nucleus and $\eta=(V_{xx}-V_{yy})/V_{zz}$ is the asymmetry parameter
of the EFG. The operators $\hat{I}$ and $\hat{I}_{z}$ denote the angular
momentum and angular momentum projection operators, while $\hat{I}_{+/-}$ are
the raising$/$lowering operators. For the sake of simplicity, we consider
throughout this section the case of the
$180\text{\,}\mathrm{\SIUnitSymbolDegree}$ configuration where $\eta=0$. In
that case, the nuclear level scheme splits into $(2I_{g}+1)$ ground and
$(2I_{e}+1)$ excited states of different energies as shown in Fig. 6b).
Furthermore, for this configuration the quantization axis is not determined by
an external magnetic field but instead by the orientation of the EFG along the
F--229Th4+-F- bond.

Figure 6:  a)229Th:CaF2 crystal structure with preferred charge compensation
directions. Adapted from Ref. Nickerson et al. (2018). b)229Th:CaF2 crystal
interacting with twisted light. The nucleus experiences electric quadrupole
hyperfine splitting due to the coupling to the electronic surrounding. This
leads to the emergence of $6$ ground states and $4$ excited states. A
broadband pulse, much broader than the splitting of a hyperfine level
$\Gamma_{\ell}>>\mathcal{O}($\mathrm{MHz}$)$, drives all 12 available
$\mathcal{M}1$ transitions simultaneously.

For monitoring the excitation of the isomeric state in the crystal without
relying on fluorescence, radiofrequency nuclear quadrupole resonance
spectroscopy (NQRS) of the excited state hyperfine splitting manifold has been
proposed Peik et al. (2021). To detect a NQRS signal, a certain population
difference, i.e., a polarization, between the involved levels needs to be
implemented. NQRS cannot discern between sublevels with the same absolute
value of the spin projection $|m_{e}|$. Thus, the polarization of excited
state populations should occur between
$\rho_{\mathrm{ex},1}+\rho_{\mathrm{ex},4}$ and
$\rho_{\mathrm{ex},2}+\rho_{\mathrm{ex},3}$. This seems extremely challenging
thermally, so in the following we investigate how the interaction with vortex
beams can produce such a polarization.

For our calculations, we assume that a weak, broadband vortex field with
arbitrary TAM projection $m_{\gamma}$ interacts with nuclei embedded in the
host crystal. For a macroscopic sample, the TAM projection $m_{\gamma}$ and
the spatially inhomogeneous wave front are not controlling the driven
transitions; only the pitch angle plays a role Scholz-Marggraf et al. (2014).
We consider at first a fixed helicity value $\Lambda$. Towards the end of the
Section, we present also results for a superposition or two circularly
polarized beams with different $\Lambda$ values. All $12$ $\mathcal{M}1$
transitions, $4$ of each allowed $\Delta m$, are driven at once. Again, the
$\mathcal{E}2$ channel of the nuclear transition is neglected due to its much
smaller radiative coupling. In addition, in the crystal all dopant
configurations are equally probable, leading to an equal distribution over the
three possible EFG orientations along the
$180\text{\,}\mathrm{\SIUnitSymbolDegree}$ bonds. This is related to the
underlying cubic lattice structure as illustrated in Fig. 7. As such, the
system has in total three mutually perpendicular quantization axes which are
shared each by $1/3$ of the total nuclear population. We therefore investigate
the excited state population $\rho_{\mathrm{ex},j}$ in a system with multiple
quantization axes.

The excited substate populations $\rho_{\mathrm{ex},j}$ are determined via the
multilevel Bloch equations in the low saturation limit. In the first place, we
calculate a general expression for the matrix element involving an arbitrary
orientation of the quantization axis. Then, we solve the multilevel Bloch
equations. Taking into account all three quantization axes, the occupation of
a certain magnetic substate can be written as

$\rho_{\mathrm{ex},j}=\frac{\rho^{q_{1}}_{\mathrm{ex},j}+\rho^{q_{2}}_{\mathrm{ex},j}+\rho^{q_{3}}_{\mathrm{ex},j}}{3}$
(28)

where the superscript corresponds to the different orientations of the
quantization axis (see Fig. 7).

In order to account simultaneously for all three orientations of the
quantization axis, we need to set first our laboratory frame. We choose the
latter as depicted in Fig. 7, with the $z$ axis parallel to $k_{z}$, i.e., to
the propagation direction of the twisted field. Additionally, we require a
rotation of the nuclear states according to Varshalovich et al. (1988)

$\displaystyle\ket{I_{g}m_{g}}_{n}$
$\displaystyle=\sum_{m_{g}^{\prime}}D^{I_{g}}_{m_{g},m_{g}^{\prime}}(\alpha,\beta,\gamma)\ket{I_{g}m_{g}^{\prime}}_{\ell}$
(29)
$\displaystyle=e^{-im_{g}\alpha}\sum_{m_{g}^{\prime}}d^{I_{g}}_{m_{g},m_{g}^{\prime}}(\beta)e^{-im_{g}^{\prime}\gamma}\ket{I_{g}m_{g}^{\prime}}_{\ell}$

where $\alpha=\beta=\gamma=0$ corresponds to the case where $k_{z}$ and the
orientation of the EFG are parallel, see Eq. (18). Here, the subscript $n$
corresponds to the angular momentum eigenstates in the nuclear reference frame
and $\ell$ to the light field reference frame, respectively. The
transformation of the sets $\ket{I_{e}m_{e}}$ works in a similar fashion. By
following the steps in Sec. II for $L=1$ and $\mu=\mathcal{M}$ with the given
state convention and applying some angular momentum algebra, we arrive at

$\displaystyle M_{fi}^{(\mathrm{tw})}$
$\displaystyle=-\sqrt{\frac{8\pi\mathcal{I}}{9c\varepsilon_{0}}}\sqrt{2I_{g}+1}\sqrt{B(\mathcal{M}1,I_{g}\rightarrow
I_{e})}$ (30)
$\displaystyle\times(-1)^{I_{g}-m_{g}}\braket{I_{e}m_{e}I_{g}-m_{g}}{1\Delta
m}\sum_{M}(-i)^{2m_{\gamma}-M}$ $\displaystyle\times
e^{i(m_{\gamma}-M)\phi_{b}}J_{m_{\gamma}-M}(\zeta
b)d^{1}_{M\Lambda}(\theta_{k})D^{1}_{M\Delta m}(\alpha,\beta,\gamma).$

The main difference to Eq. (18) is the additional sum over $M$ and as well as
the additional Wigner function $D^{1}_{M\Delta m}(\alpha,\beta,\gamma)$. This
is in correspondence with the results in Schulz et al. (2020) where a similar
geometry was investigated for atomic photo-absorption.

In case of a macroscopic sample in which the nuclei are distributed
homogeneously over the entire beam, the matrix element modulus squared
(related to the observable) must be averaged over a disk of radius $R$ in
order to determine the average rate of excitation. This is done via

$\left|\tilde{M}_{fi}^{\mathrm{(tw)}}\right|^{2}=\frac{1}{R^{2}\pi}\int\left|M_{fi}^{(\mathrm{tw})}\right|^{2}d^{2}\bm{b}.$
(31)

By using the integral relations Afanasev et al. (2013)

$\lim\limits_{R\to\infty}\int_{0}^{R}J_{m_{\gamma}-\Delta m}^{2}(\zeta
b)b\,db=\frac{R}{\pi\zeta}$ (32)

and

$\int_{0}^{2\pi}e^{i(M^{\prime}-M)\phi_{b}}\,d\phi_{b}=2\pi\delta_{M,M^{\prime}}$
(33)

we obtain for the averaged matrix element modulus squared

$\displaystyle\left|\tilde{M}_{fi}^{\mathrm{(tw)}}\right|^{2}$
$\displaystyle=\left|\braket{I_{e}m_{e}I_{g}-m_{g}}{1\Delta
m}\right|^{2}B(\mathcal{M}1,I_{g}\rightarrow I_{e})$ (34)
$\displaystyle\times\frac{16\mathcal{I}(2I_{g}+1)}{9c\varepsilon_{0}R\zeta}\sum_{M}d^{1}_{M\Lambda}(\theta_{k})^{2}d^{1}_{M\Delta
m}(\beta)^{2}$

which is independent of the impact parameter $b$ as well as the Euler angles
$\alpha$, $\gamma$ and hence only sensitive to $\beta$ and $\theta_{k}$.

The optical Bloch equations in the low saturation limit are given by von der
Wense et al. (2020)

$\dot{\rho}_{ee}=-i\sum_{g}\Omega_{eg}\left(\rho_{ge}-\rho_{eg}\right)-\rho_{ee}\Gamma$
(35)

for the excited state population and

$\dot{\rho}_{ge}=-i\Omega_{eg}\left(\rho_{ee}-\rho_{gg}\right)-\rho_{ge}\tilde{\Gamma}$
(36)

for the coherences. Here, $\Gamma=\Gamma_{\gamma}$, since spontaneous EB decay
is on the order of
$\Gamma_{\text{eb}}\approx\mathcal{O}(${10}^{-8}\text{\,}{\mathrm{s}}^{-1}$)$
in 229Th:CaF2 Nickerson et al. (2020, 2021) and can be neglected. Once more we
neglect relaxation between sublevels of the same manifold. In the Bloch
equation (36), $\tilde{\Gamma}\approx\Gamma_{\ell}/2$ corresponds to the decay
rate of the coherences which is dominated by the large linewidth of the laser.
A suitable laser source for this procedure can be for instance the system
described in Ref. Thielking et al. (2023). The large linewidth of the laser
leads to a fast dissipation of the coherences which in turn leads to a rapidly
evolving steady state. In this case, one can set $\dot{\rho}_{ge}=0$. In
combination with the weak driving field, the ground state population remains
basically unaffected such that $\rho_{gg}-\rho_{ee}\approx 1/(2I_{g}+1)$ holds
for an initial ground state quadrupole structure. The sum in Eq. (35) thereby
runs over all initial magnetic substates from which the final state can be
reached via a $\mathcal{M}1$ transition, i.e. $m_{g}=m_{e}-\Delta m$ due to
the fact that a vortex beam has $2L+1$ amplitudes. Thus, we can replace
$\Omega_{eg}$ by $\Omega_{eg}^{(\text{tw})}$.

With these assumptions, solving Eq. (35) with $\rho_{ee}(0)=0$ provides for
the nuclear ensemble excitation in the wavefront

$\displaystyle\rho_{ee}^{(\mathrm{tw})}$
$\displaystyle=\overbrace{\frac{2}{\Gamma\tilde{\Gamma}}\frac{1-e^{-\Gamma
t}}{2I_{g}+1}}^{\alpha(t)}\sum_{g}\left(\Omega_{eg}^{(\mathrm{tw})}\right)^{2}$
(37)
$\displaystyle=\frac{\alpha(t)}{R\zeta}\frac{16\mathcal{I}(2I_{g}+1)}{9c\varepsilon_{0}\hbar^{2}}B(\mathcal{M}1,I_{g}\rightarrow
I_{e})$
$\displaystyle\times\sum_{m_{g},M}\left|\braket{I_{e}m_{e}I_{g}-m_{g}}{1\Delta
m}\right|^{2}d^{1}_{M\Lambda}(\theta_{k})^{2}d^{1}_{M\Delta m}(\beta)^{2}.$

In the following, we will refer to the excited magnetic substates as
$\rho_{\mathrm{ex},j}^{(\mathrm{tw})}$ instead of $\rho_{ee}^{(\mathrm{tw})}$
as introduced in the labels in Fig. 6.

Figure 7: 229Th:CaF2 crystal structure with
$180\text{\,}\mathrm{\SIUnitSymbolDegree}$ fluoride interstitials. Here, all
three possible orientations of the EFG and the corresponding Euler angles with
respect to the laboratory reference frame (left) are illustrated.

Summing over all four excited substates we obtain

$\displaystyle{\rho}_{\text{ex}}^{(\mathrm{tw})}$
$\displaystyle=\sum_{j}{\rho}_{\mathrm{ex},j}^{(\mathrm{tw})}$ (38)
$\displaystyle=\frac{\alpha(t)}{R\zeta}\frac{16\mathcal{I}(2I_{g}+1)}{9c\varepsilon_{0}\hbar^{2}}B(\mathcal{M}1,I_{g}\rightarrow
I_{e})\,,$

where the Clebsch-Gordan coefficients and the Wigner small-$d$ functions
vanish due to their unitarity. Then, the fraction of an excited magnetic
substate driven by a circularly polarized vortex field is given in the low
saturation limit by

$\displaystyle\frac{\rho_{\mathrm{ex},j}^{(\mathrm{tw})}}{\rho_{\text{ex}}^{(\mathrm{tw})}}=$
(39) $\displaystyle\sum_{m_{g},M}\left|\braket{I_{e}m_{e}I_{g}-m_{g}}{1\Delta
m}\right|^{2}d^{1}_{M\Lambda}(\theta_{k})^{2}d^{1}_{M\Delta m}(\beta)^{2}.$

Apart from circularly polarized Bessel beams, it has also been discussed in
literature that different beam polarizations can be generated via
superpositions of twisted light beams Schulz et al. (2020); Ramakrishna et al.
(2022). In the following we also consider the superposition of two Bessel
beams with the same TAM projection $m_{\gamma}$ and opposite helicities
$\Lambda$ of the form

$\bm{A}^{(\text{tw})}_{\text{sup}}=\frac{1}{\sqrt{2}}\left(\bm{A}_{m_{\gamma},\Lambda=+1}^{(\text{tw})}+e^{i\varphi}\bm{A}_{m_{\gamma},\Lambda=-1}^{(\text{tw})}\right)$
(40)

where $\varphi$ is an arbitrary phase. In the paraxial regime and the special
case $m_{\gamma}=0$, this superposition would correspond to a radially
polarized vortex field for $\varphi=0$ and an azimuthally polarized vortex
field for $\varphi=\pi$ Schulz et al. (2020).

Following the aforementioned steps in the derivation of the matrix element and
sublevel population, the partial excited states for such a superposition can
be written as

$\displaystyle\frac{\rho_{\mathrm{ex},j}^{(\mathrm{tw})}}{\rho_{\text{ex}}^{(\mathrm{tw})}}$
$\displaystyle=\frac{1}{2}\sum_{m_{g},M}\left|\braket{I_{e}m_{e}I_{g}-m_{g}}{1\Delta
m}\right|^{2}$ (41) $\displaystyle\times d^{1}_{M\Delta
m}(\beta)^{2}\Big{[}d^{1}_{M\Lambda}(\theta_{k})^{2}+d^{1}_{M-\Lambda}(\theta_{k})^{2}$
$\displaystyle+2\cos{(\varphi)}d^{1}_{M-\Lambda}(\theta_{k})d^{1}_{M\Lambda}(\theta_{k})\Big{]}.$

Since both expressions (LABEL:eq:twcrosscirc) and (41) are independent of
$\alpha$ and $\gamma$, the contribution of the misaligned axes are equal such
that Eq. (28) simplifies to

$\rho_{\mathrm{ex},j}=\frac{\rho_{\mathrm{ex},j}(\beta=0^{\circ})+2\rho_{\mathrm{ex},j}(\beta=90^{\circ})}{3}.$
(42)

In Fig. 8, the partial populations
$\rho_{\mathrm{ex},j}^{(\mathrm{tw})}/\rho_{\text{ex}}^{(\mathrm{tw})}$ as
function of the pitch angle are presented for a circularly polarized Bessel
beam and for the two-beam superpositions following Eq. (41) with $\varphi=0$
and $\varphi=\pi$. The scenarios $\beta=0^{\circ}$, $\beta=90^{\circ}$ and
their combination in Eq. (42) are discussed. We choose the pitch angle range
$[0^{\circ},90^{\circ}]$. In practice, the value
$\theta_{k}=$90\text{\,}\mathrm{\SIUnitSymbolDegree}$$ is not feasible
experimentally, since this corresponds to a propagating wave without a
longitudinal component $k_{z}$ which is unphysical. Furthermore, most current
experiments with twisted light beams are in the paraxial regime such that an
excitation pattern significantly different from plane wave excitation is
challenging at the moment. Nevertheless, according to Ref. Scholz-Marggraf et
al. (2014), pitch angles in the range
$20\text{\,}\mathrm{\SIUnitSymbolDegree}$ to
$60\text{\,}\mathrm{\SIUnitSymbolDegree}$ seem to be experimentally feasible.

Figure 8: Distribution of magnetic sublevel population as a function of pitch
angle for different vortex beam polarizations and crystal quantization axes.
a)-c) correspond to a circularly polarized Bessel beam. a) $\beta=0$, b)
$\beta=$90\text{\,}\mathrm{\SIUnitSymbolDegree}$$ and c) the sum (28). d)-f)
correspond to a beam superposition (40) with $\varphi=0$ and g)-i) correspond
to a beam superposition (40) with $\varphi=\pi$. The quantization axes are
column-wise the same as in a)-c).

Figures 8a)-c) represent the results for a circularly polarized Bessel beam
with $\Lambda=+1$, starting with a), which corresponds to $\beta=0^{\circ}$.
At small opening angles the distribution hardly deviates from the plane wave
result for a circularly polarized beam. Here, the occupation probability is
dominated by the underlying Clebsch-Gordan coefficient
$|\braket{I_{e}m_{e}I_{g}-m_{g}}{11}|^{2}$ of the transition. With increasing
$\theta_{k}$, a change in the sublevel distribution becomes visible. The two
magnetic substates $\rho_{\mathrm{ex},1}$ and $\rho_{\mathrm{ex},2}$ with the
largest occupation probability become less populated, while the populations of
$\rho_{\mathrm{ex},3}$ and $\rho_{\mathrm{ex},4}$ start to increase. This is
related to the fact that the Wigner functions in Eq. (LABEL:eq:twcrosscirc)
for $\Delta m=0$ and $\Delta m=-1$ are increasing with $\theta_{k}$. This
progression takes place until the occupation of magnetic substates with the
same projection quantum number $\left|m_{e}\right|$ becomes equal and
approaches the value

$\frac{\rho_{\mathrm{ex},j}^{(\mathrm{tw})}}{\rho_{\text{ex}}^{(\mathrm{tw})}}=\frac{3}{16}+\frac{1}{4}\left|\braket{I_{e}m_{e}I_{g}-m_{g}}{10}\right|^{2}$
(43)

at $\theta_{k}=$90\text{\,}\mathrm{\SIUnitSymbolDegree}$$.

In Fig. 8b), we present our results once the quantization axis is rotated by
$90\text{\,}\mathrm{\SIUnitSymbolDegree}$. For small values of $\theta_{k}$,
the occupation follows Eq. (43). With increasing pitch angle, the distribution
changes only slightly. The corresponding results accounting for all three
quantization axis via Eq. (28) are displayed in Fig. 8c). For small opening
angles, the occupation probability follows the pattern visible in part a),
however with a different magnitude. One can observe that states with the same
absolute value $\left|m_{e}\right|$ are symmetrically distributed around $1/4$
corresponding to a homogeneous sublevel population distribution throughout the
entire range of $\theta_{k}$. Thus, in the context of NQRS, the case of most
interest would be the small pitch angle range for $\beta=0^{\circ}$
quantization, where the relevant populations
$\rho_{\mathrm{ex},2}+\rho_{\mathrm{ex},3}$ and
$\rho_{\mathrm{ex},1}+\rho_{\mathrm{ex},4}$ differ by $\approx 0.1$. However,
it is not clear whether any means to favor this particular quantization axis
in the sample preparation process could be developed.

Figures 8d)-f) show the results for a vortex beam superposition with relative
phase $\varphi=0$. For all quantization axes, the excitation probabilities are
independent of the pitch angle $\theta_{k}$ and are the same as plane wave
excitation. This is due to the unitarity relation of the Wigner small-$d$
functions. For $\beta=0^{\circ}$ the excitation probability illustrated in
Fig. 8d) is given by

$\displaystyle\frac{\rho_{\mathrm{ex},j}^{(\mathrm{tw})}}{\rho_{\text{ex}}^{(\mathrm{tw})}}$
$\displaystyle=\frac{1}{2}\Big{(}\left|\braket{I_{e}m_{e_{j}}I_{g}-m_{g_{j}}}{11}\right|^{2}$
(44)
$\displaystyle+\left|\braket{I_{e}m_{e_{k}}I_{g}-m_{g_{k}}}{1-1}\right|^{2}\Big{)}$

throughout the entire range of $\theta_{k}$. For the
$\beta=$90\text{\,}\mathrm{\SIUnitSymbolDegree}$$ case illustrated in Fig.
8e), the situation is similar, however with the difference that the sublevel
distribution follows Eq. (43) throughout the entire range of $\theta_{k}$.
This is again related to the unitarity relation of the Wigner function for
$L=1$ and $M\neq 0$ and additionally through the misaligned quantization axis
leading to an admixture of $\Delta m=0$ transitions. When including all three
axes in Fig. 8f), all magnetic substates are equally populated over the entire
range of $\theta_{k}$. This could also be achieved by a plane wave field with
a linearly polarized beam von der Wense et al. (2020).

Finally, Figs. 8g)-i) present the calculated excitation probabilities for a
superposition of vortex beams with relative phase $\varphi=\pi$. For the case
with $\beta=0$ in Fig. 8g) the excited state population is determined for
small opening angles by Eq. (44) while later on for
$\theta_{k}=$90\text{\,}\mathrm{\SIUnitSymbolDegree}$$ it is proportional to
the remaining Clebsch-Gordan coefficient
$|\braket{I_{e}m_{e}I_{g}-m_{g}}{10}|^{2}$. Similarly, in Fig. 8h) the
population is proportional to the expression in Eq. (43) for small angles, and
follows Eq. (44) for large pitch angles. Taking into account all three
quantization axes the magnetic substates are equally occupied throughout the
entire range of $\theta_{k}$.

Summarizing, depending on the twisted light field mode under consideration the
polarization of a 229Th:CaF2 crystal can be slightly modified compared to the
plane wave case by varying the transverse momentum $\zeta$ of a broadband
vortex beam. This effect is most pronounced along a single quantization axis
in the sample, while taking into account all three intrinsic crystal axes
leads to a polarization pattern rather similar to the plane wave result. The
most interesting result is obtained for a circularly polarized vortex beam.
The largest excited state polarization is achieved for small pitch angles,
thus closer to the plane wave result, however only for the single-quantization
axis aligned with the direction of pulse propagation with $\beta=0^{\circ}$;
averaged over all three quantization axis the polarization vanishes. We note
that it is unclear how the case of a single quantization axis could be
implemented in practice.

Our results for 229Th:CaF2 are limited to the $180^{\circ}$ interstitial
fluorine ions configuration. The $90^{\circ}$ one is more complicated and
could offer other possibilities for excited state polarization. Also, other
VUV-transparent crystals with a more complicated structure than CaF2 might
also offer the possibility to work with a single quantization axis and
different quadrupole splitting level schemes.

##  IV Conclusion & Outlook

Nuclear photoabsorption in 229Th using twisted light was investigated
theoretically within a semiclassical approach, employing a classical
description of the electromagnetic field. Two scenarios were investigated.
First, we focused on the temporal and spatial excitation dynamics of a single
229Th ion interacting with a resonant vortex beam. Thereby, it was shown that
driving of the $\mathcal{E}2$ transition can be optimized in the center of the
vortex beam, yet under at present rather challenging experimental parameters.
Adjacent to the beam center, the $\mathcal{E}2$ channel becomes negligible and
$\mathcal{M}1$ transitions dominate. We have presented numerical results for
nuclear clock transitions between the hyperfine-split levels of 229Th3+ driven
by narrow-band vortex beams and discussed their spatial and temporal
excitation patterns.

Second, we have studied the interaction of broad-band twisted light with a
macroscopic 229Th:CaF2 crystal. Here all $\mathcal{M}1$ transitions were
addressed by the field at once, and the excited state polarization of the
crystal was deduced taking into account all three orientations of the
quadrupole splitting quantization axis. We have investigated the sublevel
population as a function of the beam pitch angle. Our results show that for
practical NQRS applications one would require a single aligned EFG
quantization axis, in the paraxial limit of small $\theta_{k}$. As the
quantization axis plays an important role for the excited state population,
our study should be extended to the second relevant EFG configuration in
229Th:CaF2 and perhaps also to different VUV-transparent crystals which could
offer completely different hyperfine splitting features, such as LiCAF and
LiSAF. On a different front, our results could be extended to Laguerre-
Gaussian modes to better predict experimentally available vortex beams, and to
take into account the coherent pulse propagation through macroscopic samples.
This will be the subject of future work.

###### Acknowledgements.

AP gratefully acknowledges support from the Deutsche Forschungsgemeinschaft
(DFG, German Science Foundation) in the framework of the Heisenberg Program
(PA 2508/3-1). The research was supported by the Austrian Science Fund (FWF)
[grant DOI:10.55776/F1004] (COMB.AT) together with the DFG (PA 2508/5-1). This
work is also part of the ThoriumNuclearClock project that has received funding
from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation programme (Grant Agreement No. 856415).

## References

  * Allen et al. (1992) L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992), URL https://link.aps.org/doi/10.1103/PhysRevA.45.8185.
  * Matula et al. (2013) O. Matula, A. G. Hayrapetyan, V. G. Serbo, A. Surzhykov, and S. Fritzsche, Journal of Physics B: Atomic, Molecular and Optical Physics 46, 205002 (2013).
  * Peshkov et al. (2017) A. Peshkov, D. Seipt, A. Surzhykov, and S. Fritzsche, Physical Review A 96, 023407 (2017).
  * Yao and Padgett (2011) A. M. Yao and M. J. Padgett, Advances in optics and photonics 3, 161 (2011).
  * Afanasev et al. (2016) A. Afanasev, C. E. Carlson, and A. Mukherjee, Journal of Optics 18, 074013 (2016).
  * Afanasev et al. (2018a) A. Afanasev, C. E. Carlson, and M. Solyanik, Physical Review A 97, 023422 (2018a).
  * Afanasev et al. (2013) A. Afanasev, C. E. Carlson, and A. Mukherjee, Physical Review A 88, 033841 (2013).
  * Scholz-Marggraf et al. (2014) H. M. Scholz-Marggraf, S. Fritzsche, V. G. Serbo, A. Afanasev, and A. Surzhykov, Phys. Rev. A 90, 013425 (2014), URL https://link.aps.org/doi/10.1103/PhysRevA.90.013425.
  * Surzhykov et al. (2015) A. Surzhykov, D. Seipt, V. G. Serbo, and S. Fritzsche, Phys. Rev. A 91, 013403 (2015), URL https://link.aps.org/doi/10.1103/PhysRevA.91.013403.
  * Peshkov et al. (2018) A. Peshkov, A. Volotka, A. Surzhykov, and S. Fritzsche, Physical Review A 97, 023802 (2018).
  * Schulz et al. (2019) S.-L. Schulz, S. Fritzsche, R. A. Müller, and A. Surzhykov, Physical Review A 100, 043416 (2019).
  * Schmidt et al. (2023) R. Schmidt, S. Ramakrishna, A. Peshkov, N. Huntemann, E. Peik, S. Fritzsche, and A. Surzhykov, arXiv preprint arXiv:2310.10197 (2023).
  * Schmiegelow et al. (2016) C. T. Schmiegelow, J. Schulz, H. Kaufmann, T. Ruster, U. G. Poschinger, and F. Schmidt-Kaler, Nature communications 7, 12998 (2016).
  * Lange et al. (2022) R. Lange, N. Huntemann, A. Peshkov, A. Surzhykov, and E. Peik, Physical Review Letters 129, 253901 (2022).
  * Afanasev et al. (2018b) A. Afanasev, C. E. Carlson, C. T. Schmiegelow, J. Schulz, F. Schmidt-Kaler, and M. Solyanik, New Journal of Physics 20, 023032 (2018b).
  * Lu et al. (2023) Z.-W. Lu, L. Guo, Z.-Z. Li, M. Ababekri, F.-Q. Chen, C. Fu, C. Lv, R. Xu, X. Kong, Y.-F. Niu, et al., Phys. Rev. Lett. 131, 202502 (2023), URL https://link.aps.org/doi/10.1103/PhysRevLett.131.202502.
  * Kazinski and Sokolov (2023) P. Kazinski and A. Sokolov, arXiv preprint arXiv:2312.03803 (2023).
  * Afanasev and Carlson (2022) A. Afanasev and C. E. Carlson, Annalen der Physik 534, 2100228 (2022).
  * Afanasev et al. (2018c) A. Afanasev, V. G. Serbo, and M. Solyanik, Journal of Physics G: Nuclear and Particle Physics 45, 055102 (2018c).
  * Peik and Tamm (2003) E. Peik and C. Tamm, Europhysics Letters 61, 181 (2003).
  * Tkalya (2011) E. V. Tkalya, Phys. Rev. Lett. 106, 162501 (2011), URL https://link.aps.org/doi/10.1103/PhysRevLett.106.162501.
  * Xu et al. (2023) H. Xu, H. Tang, G. Wang, C. Li, B. Li, P. Cappellaro, and J. Li, Phys. Rev. A 108, L021502 (2023), URL https://link.aps.org/doi/10.1103/PhysRevA.108.L021502.
  * Peik et al. (2021) E. Peik, T. Schumm, M. Safronova, A. Palffy, J. Weitenberg, and P. G. Thirolf, Quantum Science and Technology 6, 034002 (2021).
  * Thirolf et al. (2019) P. G. Thirolf, B. Seiferle, and L. von der Wense, Annalen der Physik 531, 1800381 (2019).
  * Fadeev et al. (2020) P. Fadeev, J. C. Berengut, and V. V. Flambaum, Phys. Rev. A 102, 052833 (2020), URL https://link.aps.org/doi/10.1103/PhysRevA.102.052833.
  * Tsai et al. (2023) Y.-D. Tsai, J. Eby, and M. S. Safronova, Nature Astronomy 7, 113 (2023).
  * Flambaum (2006) V. V. Flambaum, Phys. Rev. Lett. 97, 092502 (2006), URL https://link.aps.org/doi/10.1103/PhysRevLett.97.092502.
  * Beck et al. (2007) B. R. Beck, J. A. Becker, P. Beiersdorfer, G. V. Brown, K. J. Moody, J. B. Wilhelmy, F. S. Porter, C. A. Kilbourne, and R. L. Kelley, Phys. Rev. Lett. 98, 142501 (2007), URL https://link.aps.org/doi/10.1103/PhysRevLett.98.142501.
  * Von Der Wense et al. (2016) L. Von Der Wense, B. Seiferle, M. Laatiaoui, J. B. Neumayr, H.-J. Maier, H.-F. Wirth, C. Mokry, J. Runke, K. Eberhardt, C. E. Düllmann, et al., Nature 533, 47 (2016).
  * Seiferle et al. (2019) B. Seiferle et al., Nature (London) 573, 243 (2019).
  * Sikorsky et al. (2020) T. Sikorsky, J. Geist, D. Hengstler, S. Kempf, L. Gastaldo, C. Enss, C. Mokry, J. Runke, C. E. Düllmann, P. Wobrauschek, et al., Phys. Rev. Lett. 125, 142503 (2020), URL https://link.aps.org/doi/10.1103/PhysRevLett.125.142503.
  * Masuda et al. (2019) T. Masuda, A. Yoshimi, A. Fujieda, and al., Nature 573, 238 (2019), URL https://www.nature.com/articles/s41586-019-1542-3#citeas.
  * Kraemer et al. (2023) S. Kraemer, J. Moens, M. Athanasakis-Kaklamanakis, S. Bara, K. Beeks, P. Chhetri, K. Chrysalidis, A. Claessens, T. E. Cocolios, J. G. Correia, et al., Nature 617, 706 (2023).
  * Tiedau et al. (2024) J. Tiedau, M. V. Okhapkin, K. Zhang, J. Thielking, G. Zitzer, E. Peik, F. Schaden, T. Pronebner, I. Morawetz, L. Toscani De Col, et al., Physical Review Letters in press (2024).
  * Phua et al. (2007) P. Phua, W. Lai, Y. L. Lim, K. Tiaw, B. Lim, H. Teo, and M. Hong, Optics letters 32, 376 (2007).
  * Peik and Okhapkin (2015) E. Peik and M. Okhapkin, Comptes Rendus Physique 16, 516 (2015).
  * Campbell et al. (2012) C. J. Campbell, A. G. Radnaev, A. Kuzmich, V. A. Dzuba, V. V. Flambaum, and A. Derevianko, Physical review letters 108, 120802 (2012).
  * Kazakov et al. (2012) G. Kazakov, A. Litvinov, V. Romanenko, L. Yatsenko, A. Romanenko, M. Schreitl, G. Winkler, and T. Schumm, New Journal of physics 14, 083019 (2012).
  * Rellergert et al. (2010) W. G. Rellergert, D. DeMille, R. R. Greco, M. P. Hehlen, J. R. Torgerson, and E. R. Hudson, Phys. Rev. Lett. 104, 200802 (2010), URL https://link.aps.org/doi/10.1103/PhysRevLett.104.200802.
  * Thirolf et al. (2024) P. G. Thirolf, S. Kraemer, D. Moritz, and K. Scharl, The European Physical Journal Special Topics pp. 1–19 (2024).
  * Dessovic et al. (2014) P. Dessovic, P. Mohn, R. Jackson, G. Winkler, M. Schreitl, G. Kazakov, and T. Schumm, Journal of Physics: Condensed Matter 26, 105402 (2014).
  * Elwell et al. (2024) R. Elwell, C. Schneider, J. Jeet, J. E. S. Terhune, H. W. T. Morgan, A. N. Alexandrova, H. B. T. Tan, A. Derevianko, and E. R. Hudson, _Laser excitation of the 229th nuclear isomeric transition in a solid-state host_ (2024), eprint 2404.12311.
  * Schulz et al. (2020) S.-L. Schulz, A. Peshkov, R. Müller, R. Lange, N. Huntemann, C. Tamm, E. Peik, and A. Surzhykov, Physical Review A 102, 012812 (2020).
  * Peshkov et al. (2023a) A. A. Peshkov, E. Jordan, M. Kromrey, K. K. Mehta, T. E. Mehlstäubler, and A. Surzhykov, Annalen der Physik 535, 2300204 (2023a), eprint https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.202300204, URL https://onlinelibrary.wiley.com/doi/abs/10.1002/andp.202300204.
  * Pálffy et al. (2008) A. Pálffy, J. Evers, and C. H. Keitel, Phys. Rev. C 77, 044602 (2008), URL https://link.aps.org/doi/10.1103/PhysRevC.77.044602.
  * Ring and Schuck (2004) P. Ring and P. Schuck, _The nuclear many-body problem_ (Springer Science & Business Media, 2004).
  * Edmonds (1996) A. R. Edmonds, _Angular momentum in quantum mechanics_ (Princeton university press, 1996).
  * Minkov and Pálffy (2017) N. Minkov and A. Pálffy, Physical review letters 118, 212501 (2017).
  * Peshkov et al. (2023b) A. Peshkov, Y. Bidasyuk, R. Lange, N. Huntemann, E. Peik, and A. Surzhykov, Physical Review A 107, 023106 (2023b).
  * Shabaev et al. (2022) V. Shabaev, D. Glazov, A. Ryzhkov, C. Brandau, G. Plunien, W. Quint, A. Volchkova, and D. Zinenko, Physical Review Letters 128, 043001 (2022).
  * Scully and Zubairy (1999) M. O. Scully and M. S. Zubairy, _Quantum optics_ (1999).
  * von der Wense et al. (2020) L. von der Wense, P. V. Bilous, B. Seiferle, S. Stellmer, J. Weitenberg, P. G. Thirolf, A. Pálffy, and G. Kazakov, The European Physical Journal A 56, 1 (2020).
  * Porsev and Flambaum (2010a) S. G. Porsev and V. V. Flambaum, Phys. Rev. A 81, 042516 (2010a), URL https://link.aps.org/doi/10.1103/PhysRevA.81.042516.
  * Porsev and Flambaum (2010b) S. G. Porsev and V. V. Flambaum, Phys. Rev. A 81, 032504 (2010b), URL https://link.aps.org/doi/10.1103/PhysRevA.81.032504.
  * Bilous et al. (2020) P. V. Bilous, H. Bekker, J. C. Berengut, B. Seiferle, L. von der Wense, P. G. Thirolf, T. Pfeifer, J. R. C. López-Urrutia, and A. Pálffy, Phys. Rev. Lett. 124, 192502 (2020), URL https://link.aps.org/doi/10.1103/PhysRevLett.124.192502.
  * von der Wense and Zhang (2020) L. von der Wense and C. Zhang, The European Physical Journal D 74, 1 (2020).
  * Zhang et al. (2022) C. Zhang, P. Li, J. Jiang, L. von der Wense, J. F. Doyle, M. E. Fermann, and J. Ye, Optics Letters 47, 5591 (2022).
  * Rubloff (1972) G. W. Rubloff, Physical review B 5, 662 (1972).
  * Barth et al. (1990) J. Barth, R. Johnson, M. Cardona, D. Fuchs, and A. M. Bradshaw, Physical Review B 41, 3291 (1990).
  * Tsujibayashi et al. (2002) T. Tsujibayashi, K. Toyoda, S. Sakuragi, M. Kamada, and M. Itoh, Applied physics letters 80, 2883 (2002).
  * Beeks et al. (2023) K. Beeks, T. Sikorsky, F. Schaden, M. Pressler, F. Schneider, B. N. Koch, T. Pronebner, D. Werban, N. Hosseini, G. Kazakov, et al., arXiv preprint arXiv:2312.13713 (2023).
  * Nickerson et al. (2018) B. S. Nickerson, W.-T. Liao, and A. Pálffy, Physical Review A 98, 062520 (2018).
  * Collins and Travis (1967) R. Collins and J. C. Travis, in _Mössbauer Effect Methodology: Volume 3 Proceedings of the Third Symposium on Mössbauer Effect Methodology New York City, January 29, 1967_ (Springer, 1967), pp. 123–161.
  * Varshalovich et al. (1988) D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, _Quantum theory of angular momentum_ (World Scientific, 1988).
  * Nickerson et al. (2020) B. S. Nickerson, M. Pimon, P. V. Bilous, J. Gugler, K. Beeks, T. Sikorsky, P. Mohn, T. Schumm, and A. Pálffy, Physical Review Letters 125, 032501 (2020).
  * Nickerson et al. (2021) B. S. Nickerson, M. Pimon, P. V. Bilous, J. Gugler, G. A. Kazakov, T. Sikorsky, K. Beeks, A. Grüneis, T. Schumm, and A. Pálffy, Phys. Rev. A 103, 053120 (2021), URL https://link.aps.org/doi/10.1103/PhysRevA.103.053120.
  * Thielking et al. (2023) J. Thielking, K. Zhang, J. Tiedau, J. Zander, G. Zitzer, M. Okhapkin, and E. Peik, New Journal of Physics 25, 083026 (2023).
  * Ramakrishna et al. (2022) S. Ramakrishna, J. Hofbrucker, and S. Fritzsche, Physical Review A 105, 033103 (2022).