# Nuclear coherent population transfer to the 229mTh isomer using x-ray pulses

Tobias Kirschbaum<EMAIL_ADDRESS>Department of Physics, Friedrich-
Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany  Nikolay
Minkov  Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of
Sciences, Tzarigrad Road 72, BG-1784 Sofia, Bulgaria  Max-Planck-Institut für
Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  Adriana Pálffy
<EMAIL_ADDRESS>Department of Physics, Friedrich-Alexander-
Universität Erlangen-Nürnberg, 91058 Erlangen, Germany  Max-Planck-Institut
für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany

###### Abstract

Population of the 8 eV 229mTh isomer via the second nuclear excited state at
29.19 keV by means of coherent x-ray pulses is investigated theoretically. We
focus on two nuclear coherent population transfer schemes using partially
overlapping x-ray pulses known from quantum optics: stimulated Raman adiabatic
passage (STIRAP), and successive $\pi$ pulses. Numerical results are presented
for three possible experimental setups. Our results identify the Gamma Factory
as the most promising scenario, where two ultraviolet pulses combined with
relativistically accelerated ions deliver the required intensities for
efficient isomer population. Our simulations require knowledge of the in-band
and cross-band nuclear transition probabilities. We give theoretically
predicted values for the latter and discuss them in the context of recent
experiments.

##  I Introduction

Among the entire nuclear chart, the 229Th isotope presents a unique isomer
which is the lowest known metastable first excited state at energy
$\approx$8\text{\,}\mathrm{eV}$$ von der Wense et al. (2016); Bec (2009);
Sikorsky et al. (2020). This very low energy is an exception among otherwise
typically much higher nuclear level energies, and has the peculiarity that it
could be in principle addressed by a narrow-band vacuum ultraviolet (VUV)
laser. This renders 229Th an ideal candidate for a nuclear clock exceeding the
accuracy of present atomic clocks E. Peik and Chr. Tamm (2003); Campbell et
al. (2012); Peik and Okhapkin (2015); Peik et al. (2021). In addition to a
very precise and novel nuclear frequency standard, a nuclear clock would have
significant impact also for other applications, for instance the detection of
dark matter Peik et al. (2021); Fadeev et al. (2020a), investigating the
temporal variation of fundamental constants Flambaum (2006); Fadeev et al.
(2020b), the construction of the first nuclear laser Tkalya (2011), or
improving the global positioning system Thirolf et al. (2019).

A major hurdle towards the nuclear clock experimental implementation is the
relatively large uncertainty on the isomeric state energy. Recently, the
energy of the isomeric state was reported to be
$E_{\text{iso}}=$8.28(17)\text{\,}\mathrm{eV}$$ Seiferle et al. (2019) using
the direct measurement of internal conversion electrons,
$E_{\text{iso}}=$8.30(92)\text{\,}\mathrm{eV}$$Yamaguchi et al. (2019) from
the determination of transition rates and energies from the second excited
state at $29.19\text{\,}\mathrm{keV}$ or
$E_{\text{iso}}=$8.10(17)\text{\,}\mathrm{eV}$$ Sikorsky et al. (2020) from
$\gamma$-spectroscopic measurements of the $\alpha$-decay of the parent
nucleus 233U. Up to this point, direct laser excitation failed, and the
radiative decay of the isomer could not be observed. The isomer energy
measurements are based on two indirect excitation methods. The first method
populates the isomeric state with a probability of
$\approx$2\text{\,}\mathrm{\char 37\relax}$$ during the $\alpha$-decay of 233U
Thielking et al. (2018). However, the nuclear decay is a purely statistical
phenomenon and therefore the population transfer is not controllable. The
second method incoherently populates the isomeric level via x-ray pumping of
the second excited state at $29.19\text{\,}\mathrm{keV}$. In the measurements
in Ref. Masuda et al. (2019), this state is populated by synchrotron radiation
pulses and partially decays to the 8 eV isomeric state.

In this work we theoretically investigate and optimize excitation schemes for
the isomer via the 29.190 keV level taking advantage of coherence-based
schemes known from quantum optics and adapted for x-ray quantum optics Adams
et al. (2013). We consider the 229Th three-level system of $\Lambda$-type
(named so because of the level scheme reminiscent of the upper case Greek
letter $\Lambda$) comprising the ground ($|1\rangle$), isomeric ($|2\rangle$)
and second excited ($|3\rangle$) states as illustrated in Fig. 1. We study
efficient nuclear coherent population transfer (NCPT) from the ground state to
the 8.19 eV isomeric state via two quantum optical transfer schemes: (i)
Stimulated Raman adiabatic passage (STIRAP)Bergmann et al. (1998, 2015);
Vitanov et al. (2017); Bergmann et al. (2019), and (ii) two subsequent
$\pi$-pulses Scully and Zubairy (1997) which pump the entire population first
from $|1\rangle\rightarrow|3\rangle$ and subsequently from
$|3\rangle\rightarrow|2\rangle$. Provided sufficient x-ray intensity, these
transfer schemes allow to place the entire population from the ground state
into the isomeric state in a coherent and controlled manner. Prior theoretical
studies have discussed the possibility of nuclear STIRAP in the context of
high-energy gamma-ray transitions Liao et al. (2011, 2013); Nedaee-Shakarab et
al. (2016, 2017). The peculiarities of the investigated 229Th three-level
system are that the two transitions to the upper state have low energy and
narrow resonances, are almost degenerate in energy and have dominating
internal conversion decay channels, with non-negligible multipole mixing.

Figure 1: Three-level $\Lambda$ system comprising of the ground and two first
excited states of 229Th. Each state is labelled with its angular momentum,
parity, Nilsson quantum numbers Nilsson (1955) and energy. The letters P/S
correspond to the pulse type (pump/Stokes) coupling to the respective levels.
Energies are not to scale.

We model the three-level-$\Lambda$-system within the density matrix approach
and solve the master equation to determine the population transfer to the
isomeric state. As the experimental setup involving two coherent x-ray pulses
is not trivial, we envisage three different scenarios for NCPT. The first
scenario involves the interaction of highly charged 229Th ions circulating at
relativistic speed in a storage ring, with two UV laser fields driving the two
x-ray transitions in the nuclear rest frame Budker et al. (2022). This
scenario was proposed at the future Gamma Factory (GF) facility at CERN Krasny
(2015); Budker et al. (2020, 2022). Similarly, the second scenario discusses
the interaction with coherent x-rays from an x-ray free electron laser (XFEL)
Huang et al. (2021a); Pellegrini et al. (2016); Altarelli (2011); Nam et al.
(2021); Galayda (2018) combined with moderately relativistic 229Th ions in a
storage ring. The third scenario investigates the interaction of a 229Th
solid-state target with coherent and resonant x-rays
($E_{\lambda}\geq$29.19\text{\,}\mathrm{keV}$$) from a cavity-based x-ray
source, for instance the x-ray free electron laser oscillator (XFELO) Kim et
al. (2008); Adams et al. (2019); Qin et al. (2018a); Lindberg et al. (2011).
Our results identify the most feasible scenario from the three and investigate
the specific requirements for each NCPT mechanism, in particular the required
laser intensities for efficient population of the isomer.

The NCPT theoretical calculations require knowledge of nuclear transition
parameters which are not experimentally available at the moment. In
particular, the nuclear reduced transition probabilities $B(M1)$ for the
magnetic dipole channel of both $|3\rangle\rightarrow|1\rangle$ and
$|3\rangle\rightarrow|2\rangle$ transitions are an input for the calculation.
We use our recently introduced nuclear structure model Minkov and Pálffy
(2017, 2019, 2021) to calculate these values and discuss the choice of our
model parameters. Furthermore, we compare the newly obtained values for
$B(M1)$ and related quantities with experimental data available from two
recent experiments Masuda et al. (2019); Sikorsky et al. (2020). While theory
and experiment agree reasonably well on the $B(M1)$ values, our analysis
reveals that the three experimentally available quantities are not completely
consistent. We discuss the possible values for the radiative and total
branching ratios which are of interest also for other experiments.

The paper is structured as follows. The theoretical model for the quantum
optical transfer schemes to achieve NCPT is introduced in Sec. II. Section III
introduces the nuclear transition properties of the three level-$\Lambda$
system and briefly discusses the nuclear model input used to obtain them.
Furthermore, our analysis on the available experimental data is presented
here. Our numerical results are given and discussed in Sec. IV. The paper
concludes with a brief discussion in Sec. V.

##  II Quantum optical transfer schemes

This Section introduces the theoretical models for the STIRAP and $\pi$-pulses
quantum optical transfer schemes for our $\Lambda$ three-level system in 229Th
illustrated in Fig. 1. We investigate the interaction of two resonant fields
driving our nuclear $\Lambda$ three-level system and the resulting change in
population. NCPT can be achieved by means of STIRAP and two $\pi$-pulses. What
distinguishes the two schemes is the arrival sequence of the two pulses. For
STIRAP, the first arriving pulse is the one driving the transition
$\left|2\right\rangle\leftrightarrow\left|3\right\rangle$, called the Stokes
pulse. The pump pulse follows, partially overlapping with the Stokes pulse,
and drives the second transition
$\left|1\right\rangle\leftrightarrow\left|3\right\rangle$. In the two
$\pi$-pulses scheme, the pump pulse arrives first, followed by the Stokes
pulse. The two pulse sequences are illustrated for generic Gaussian-shaped
pulses in Fig. 2.

Figure 2: a) Generic pulse sequence for STIRAP. $\Delta\tau=\tau_{p}-\tau_{s}$
indicates the delay between the pulses. b) Generic pulse sequence for NCPT via
$\pi$-pulses.

STIRAP coherently transfers population between the two ground states of a
$\Lambda$-type system independently of the branching ratio or decay channels
of the upper state Bergmann et al. (1998, 2015). The preceding Stokes pulse
couples to the unoccupied states $\left|2\right\rangle$ and
$\left|3\right\rangle$ and creates a coherent superposition thereof.
Consequently, the overlapping pump pulse couples to the fully occupied ground
state $\left|1\right\rangle$ and the pre-built coherence of
$\left|3\right\rangle$ and $\left|2\right\rangle$. In this process a dark
state Bergmann et al. (1998)

$\left|\text{D}\right\rangle=\frac{\Omega_{s}\left(t\right)}{\sqrt{\Omega_{s}\left(t\right)^{2}+\Omega_{p}\left(t\right)^{2}}}\left|1\right\rangle-\frac{\Omega_{p}\left(t\right)}{\sqrt{\Omega_{s}\left(t\right)^{2}+\Omega_{p}\left(t\right)^{2}}}\left|2\right\rangle,$
(1)

is formed. Here, $\Omega_{s/p}$ are the time dependent Rabi frequencies Pálffy
et al. (2008); Liao et al. (2011) of the Stokes and Pump pulse, respectively.
By optimization of the laser parameters such as intensity or delay time
$\Delta\tau$ between the Stokes and pump pulses, one is able to transfer the
entire population from $\left|1\right\rangle$ to $\left|2\right\rangle$ via
the dark state, without occupying the intermediate state
$\left|3\right\rangle$. A rather empirical condition for each pulse, namely
the adiabatic condition Vitanov et al. (2001); Kuhn et al. (1992)

$\Omega_{0}^{2}\geq\frac{2\ln{2}\cdot 100}{t_{\text{pul}}^{2}}$ (2)

provides a guideline for the optimization. In this equation $\Omega_{0}$
denotes the Rabi frequency amplitude of the respective pulse and
$t_{\text{pul}}$ the pulse duration at FHWM, respectively. Note that this
formula only holds if the pulses are fully temporally coherent and have no
phase fluctuations. For simplicity, we will assume throughout this paper that
no (relative) phase fluctuations occur.

For the $\pi$-pulse technique, the pulse sequence is opposite to STIRAP.
Thereby, the population is initially transferred from $\left|1\right\rangle$
to $\left|3\right\rangle$ by the pump pulse and from there immediately to
$\left|2\right\rangle$ by the Stokes pulse. To transfer the entire population
into the desired state each pulse should satisfy Scully and Zubairy (1997)

$\int_{-\infty}^{\infty}\Omega\left(t\right)\text{d}t=\pi\,.$ (3)

By solving the integral, we arrive to an expression for the pulse requirements
to achieve transfer rates of unity. For Gaussian pulses we obtain the
expression

$\Omega_{0}^{2}=2\ln{2}\frac{\pi}{t_{\text{pul}}^{2}}\,.$ (4)

Both methods are a challenge to the experiment, since the transfer criteria in
Eqs. (2) and (3) require very large laser intensities for the available pulse
durations. However, implementing such coherence-based schemes has the
advantage that the population transfer is no longer dependent on incoherent
processes like spontaneous decay. If the lifetime of the intermediate state
$\left|3\right\rangle$ is shorter than the radiative coupling, then STIRAP is
the method of choice. Apart from that, the $\pi$-pulse method is applicable.
An advantage of the $\pi$-pulse method is that one has a larger window for the
pulse delay time, which is only constrained by the lifetime of the
intermediate state $\left|3\right\rangle$. Furthermore, the $\pi$-pulse method
requires smaller intensities than STIRAP. However, STIRAP is more robust than
the $\pi$-pulses method as far as parameter variations are concerned Bergmann
et al. (1998), as long as the temporal coherence of the pulses is secured.

###  II.1 Density matrix approach

We model the population of the three-level-$\Lambda$ system within the density
matrix approach. The density matrix is defined as

$\rho\left(t\right)=\sum_{i,j}\rho_{ij}\left(t\right)\left|i\right\rangle\left\langle
j\right|$ (5)

where $\\{i,j\\}\in\\{1,2,3\\}$. The diagonal elements $\rho_{ii}$ denote the
level-population and the off-diagonal elements $\rho_{ij}$ the coherences,
respectively. Our starting point to compute the population transfer in 229Th
is the master equation Scully and Zubairy (1997); Pálffy et al. (2008)

$\partial_{t}\rho\left(t\right)=\frac{1}{i\hbar}\left[\mathcal{H}\left(t\right),\rho\left(t\right)\right]+\rho_{\text{relax}}\left(t\right),$
(6)

where $\mathcal{H}\left(t\right)$ denotes the interaction Hamiltonian and
$\rho_{\text{relax}}\left(t\right)$ the relaxation term, respectively.
Furthermore, $\hbar$ denotes the reduced Planck constant. The Hamiltonian
reads Liao et al. (2011, 2013)

$\mathcal{H}\left(t\right)=-\frac{\hbar}{2}\begin{pmatrix}0&0&\Omega_{p}^{\ast}\left(t\right)\\\
0&-2\left(\Delta_{p}-\Delta_{s}\right)&\Omega_{s}^{\ast}\left(t\right)\\\
\Omega_{p}\left(t\right)&\Omega_{s}\left(t\right)&2\Delta_{p}\end{pmatrix}.$
(7)

Here, $\Omega_{p/s}$ is the time dependent Rabi-frequency and $\Delta_{p/s}$
the detuning of the pump$/$Stokes field, respectively. The expressions for the
Rabi frequencies are given below. The relaxation matrix embodying several
decay channels has the form

$\rho_{\text{relax}}\left(t\right)=\frac{1}{\hbar}\begin{pmatrix}\text{BR}_{31}\Gamma\rho_{33}&0&-0.5\Gamma\rho_{13}\\\
0&\text{BR}_{32}\Gamma\rho_{33}&-0.5\Gamma\rho_{23}\\\
-0.5\Gamma\rho_{31}&-0.5\Gamma\rho_{32}&-\rho_{33}\Gamma\end{pmatrix}.$ (8)

where $\text{BR}_{3j}$ denotes the branching ratio for the transition
$\left|3\right\rangle\rightarrow\left|j\right\rangle$ where $j\in\\{1,2\\}$
and $\Gamma$ is the total decay rate of $\left|3\right\rangle$, respectively.
For simplicity we assume throughout this work that both laser fields are fully
temporally coherent during the radiative coupling. In addition, we neglect the
decay of the isomeric level, since the lifetime of $\left|2\right\rangle$ is
much larger than the lifetime of $\left|3\right\rangle$. If not mentioned
otherwise, initially the nuclear population is in the ground state such that
$\rho_{11}\left(0\right)=1$ and
$\rho_{22}\left(0\right)=\rho_{33}\left(0\right)=0$.

####  II.1.1 Lab frame

The time dependent Rabi frequencies in the laboratory frame are assumed to be
Gaussian, such that

$\Omega_{p\left(s\right)}=\Omega_{0,p\left(s\right)}\exp\left(-2\ln{2}\frac{\left(t-\tau_{p\left(s\right)}\right)^{2}}{t_{\text{pul}^{2}}}\right)$
(9)

where $\Omega_{0,p\left(s\right)}$ denotes the Rabi amplitude and
$\tau_{p\left(s\right)}$ the temporal peak position of the pump$/$Stokes
field, respectively. We can express the Rabi amplitude for the nuclear
transition $\left|i\right\rangle\rightarrow\left|j\right\rangle$ as Liao et
al. (2011, 2013)

$\begin{split}\Omega_{0,ij}=\sqrt{\frac{16\pi
I_{0}}{c\epsilon_{0}\hbar^{2}}}\sqrt{\frac{\left(2I_{i}+1\right)\left(L_{ij}+1\right)}{L_{ij}}}\\\
\times\frac{k_{ij}^{L_{ij}-1}}{\left(2L_{ij}+1\right)!!}\sqrt{B_{ij}\left(\mu
L_{ij}\right)}=\sqrt{I_{0}}\xi_{ij}\end{split}$ (10)

where $I_{0}$ indicates the peak intensity of the radiation field, $c$ the
speed of light, $\epsilon_{0}$ the vacuum permittivity, $I_{i}$ the nuclear
spin of level $\left|i\right\rangle$, $L_{ij}$ the multipolarity, $k_{ij}$ the
wave number and $B_{ij}\left(\mu L_{ij}\right)$ the reduced transition
probability of the transition, respectively. The index $\mu$ corresponds to
the radiation multipole type electric or magnetic $\mu\in\\{E,M\\}$. For the
purpose of our calculation of radiative couplings, we only consider the first
dominant multipole order in our Rabi frequency, which is the $M1$ channel.

With this concrete expression for the Rabi frequency, we can rewrite the
transfer criteria in Eq. (2) and Eq. (3) in terms of radiation intensity. We
then obtain

$I_{0}\geq\frac{2\ln{2}\cdot 100}{t_{\text{pul}}^{2}\xi_{ij}^{2}}$ (11)

for STIRAP and

$I_{0}=\frac{2\ln{2}\cdot\pi}{t_{\text{pul}}^{2}\xi_{ij}^{2}}$ (12)

for the $\pi$-pulses, respectively. We use these expressions to compute the
ideal intensity required to transfer the entire population from the ground
state to the desired isomeric state with hardly any losses.

####  II.1.2 Ion rest frame

In case of the relativistic acceleration of the nuclei, the laser photon
parameters are relativistically boosted in the rest frame of the ion according
to

$\omega_{3}-\omega_{1(2)}=\gamma\left(1+\beta\cos{\theta}\right)\omega_{P(S)}\,,$
(13)

where $\gamma$ is the relativistic Lorentz factor, $\beta=v/c$, with $v$ the
ion velocity, and $\theta$ denotes the impact angle between radiation and
nuclei, respectively. Thus, the pulse width and the peak intensity change in
the nuclear rest frame as Bürvenich et al. (2006)

$\displaystyle
t_{\text{pul}}\rightarrow\frac{t_{\text{pul}}}{\gamma\left(1+\beta\right)}\,,$
(14) $\displaystyle I_{0}\rightarrow I_{0}\gamma^{2}\left(1+\beta\right)^{2}.$
(15)

Here, we have assumed that the interaction between the ion bunch and the laser
pulses is collinear ($\theta\approx 0$). Furthermore, as the ions approach the
speed of light, the relativistic factor $\gamma\left(1+\beta\right)\rightarrow
2\gamma$. With the above considerations the time dependent Rabi frequency in
the nuclear rest frame becomes

$\Omega_{p\left(s\right)}^{\text{Rest}}=2\gamma\Omega_{0,p\left(s\right)}^{\text{Lab}}\exp\left(-2\ln{2}\frac{4\gamma^{2}\left(t-\tau_{p\left(s\right)}\right)^{2}}{t_{\text{pul}}^{2}}\right)\,.$
(16)

Also in the ion rest frame we can rewrite the adiabaticity criterion for the
boosted intensities in analogy with Eqs. (11) and (12).

##  III The 229Th Three-Level-$\Lambda$ system

In this Section we discuss our present knowledge on the nuclear transition
properties of the 229Th three-level-$\Lambda$ system under investigation. From
the nuclear structure point of view, the 29.19 keV state with angular momentum
$I=5/2^{+}$ and the isomeric state with $I=3/2^{+}$ belong to the excited
(non-yrast) band built on the $K^{\pi}[Nn_{z}\Lambda]=3/2^{+}[631]$ single-
neutron orbital, with $K$ the projection of the angular momentum on the
intrinsic nuclear symmetry axis, $\pi$ the parity and $N$, $n_{z}$ and
$\Lambda$ the Nilsson asymptotic quantum numbers Nilsson and I.Ragnarsson
(1995), respectively. Thus, the transition
$\left|1\right\rangle\leftrightarrow\left|3\right\rangle$ corresponds to the
cross-band (cr) transition, while the transition
$\left|2\right\rangle\leftrightarrow\left|3\right\rangle$ corresponds to the
in-band (in) transition. For laser excitation, the radiative channels of the
two transitions are of interest, with the reduced transition probabilities
$B(M1)$ entering the Rabi frequency amplitudes in Eqs. (10). However, the
relaxation matrix (8) contains also other quantities, such as incoherent decay
rates, which could include also the internal conversion (IC) channel, and
several types of branching ratios. Thus, we are interested in $B(M1)$ and
$B(E2)$ values for both in-band and cross-band transitions, as well as the
related IC coefficients, and the resulting branching ratios $\mathrm{BR}_{31}$
and $\mathrm{BR}_{32}$. The corresponding transition rates are only partially
known from experiments. Alternatively, we can obtain these quantities from a
nuclear structure model which was so far successfully used to predict the low-
lying level structure of 229Th Minkov and Pálffy (2017, 2019, 2021). In the
following we present our nuclear structure results on the transition
properties of the in-band and cross-band transition. We discuss these values
in comparison with available experimental data and check the consistency of
the available parameter sets.

###  III.1 Theoretical predictions

The actinide nuclei and in particular the even-odd isotopes among them present
a rich nuclear structure. A model approach capable to incorporate the shape-
dynamic properties together with the intrinsic structure characteristics
typical for the actinide nuclei has been under development in the last decade
Minkov et al. (2006, 2007); Cwiok et al. (1987); Walker and Minkov (2010);
Minkov et al. (2009, 2010); Minkov (2013). It considers a collective
quadrupole-octupole (QO) vibration-rotation motion of the nucleus which in the
particular case of odd-mass nuclei is coupled to the motion of the single
(odd) nucleon within a reflection-asymmetric deformed potential. The
collective motion is described through the so-called coherent QO mode (CQOM)
giving raise to the quasi parity-doublet structure of the spectrum Minkov et
al. (2006, 2007), whereas the single-particle (s.p.) one is determined by the
deformed shell model (DSM) with reflection-asymmetric Woods-Saxon potential
Cwiok et al. (1987) and pairing correlations of Bardeen-Cooper-Schrieffer
(BCS) type included as in Ref. Walker and Minkov (2010). The Coriolis
interaction between CQOM and the odd nucleon was originally considered in
Minkov et al. (2009, 2010), whereas the effect of Coriolis decoupling and
$K$-mixing on the rotation-vibration was taken into account in Ref. Minkov
(2013).

All the model aspects outlined above have been assembled together in Ref.
Minkov and Pálffy (2017) in a detailed nuclear-structure-model description of
the low-lying positive- and negative-parity excited levels and transition
probabilities observed in 229Th. This allowed the description of several
experimentally known $B(M1)$ and $B(E2)$ reduced probabilities and prediction
of the two unknown isomeric ones, responsible for the radiative decay of the
8.19 eV $K^{\pi}=3/2^{+}$-isomer to the $K^{\pi}=5/2^{+}$ ground state,
corresponding to the cross-band transition
$\left|1\right\rangle\leftrightarrow\left|2\right\rangle$ in the scheme of
Fig. 1.

The two states are considered as almost degenerate quasi-particle bandheads
with a superposed collective QO vibration-rotation mode giving raise to yrast
$K^{\pi}=5/2^{+}$ and non-yrast $K^{\pi}=3/2^{+}$ quasi parity-doublet
structures. The isomer decay is obtained as the result of a Coriolis mixing
emerging from a remarkably fine interplay between the coherent QO motion of
the core and the single-nucleon motion within the reflection-asymmetric
deformed potential. Despite earlier statements on the weakness of the Coriolis
mixing Dykhne and Tkalya (1998); Tkalya et al. (2015), we found that only
because of the Coriolis $K$-mixing interaction can we explain the presence of
the $K$-forbidden $M1$ and $E2$ transitions between the yrast and non-yrast
bands, otherwise forbidden due to the overall axial symmetry of the problem.
The same holds also for the cross-band $|3\rangle\rightarrow|1\rangle$
transition of the three-level system investigated here, which can only be
accounted for by the inclusion of Coriolis mixing in the model.

Within this model it is also clear that the two electromagnetic multipole
contributions have different origins. The $E2$ transition is mainly related to
the collective part, whereas the $M1$ component emerges from the single-
nucleon degree of freedom Ring and Schuck (1980). Nevertheless, the collective
QO mode has a strong indirect influence on the $M1$ transition via the s.p.
coupling to the nuclear core. Vice-versa, the collective part is decisive for
the $E2$ transition, however with indirect influence from the single nucleon
via the particle-core coupling. The very fine balance between the different
degrees of freedom and its role in the formation of the total dynamics of the
nucleus and its isomer properties including energy, electromagnetic transition
rates and magnetic moments were examined in detail in Minkov and Pálffy
(2021). It was confirmed that the reasoning for the existence and the decay
properties of the 229mTh state is strongly related to all nuclear structure
model ingredients originally considered in Ref. Minkov and Pálffy (2017),
namely, the collective core, the single-nucleon motion in the deformed
potential and the Coriolis interaction.

A specificity of the model in calculating the $M1$ reduced transition
probability is the use of an additional input related to the intrinsic-spin
and collective gyromagnetic factors, which are attenuated by multiplication
with the respective quenching factors $q_{s}$ and $q_{R}$, as explained in
Refs. Minkov and Pálffy (2019, 2021). The nuclear-structure grounds and
theoretical motivation for the introduction of the two quenching factors are
discussed in Ref. Minkov and Pálffy (2019) based on the original adjustment of
the model parameters to the 229Th energy levels and transition rates reported
in Minkov and Pálffy (2017). The dependence of the predicted 229mTh
electromagnetic properties on the $q_{s}$ and $q_{R}$ quenching factors and QO
deformations is examined in Ref. Minkov and Pálffy (2021) through different
fits including in addition the observed ground- and isomeric- state magnetic
dipole moments. Physically consistent model descriptions of energy, transition
rates and magnetic moments are obtained in a narrow QO-deformation region by
varying $q_{s}$ between 0.6 and 0.55 and $q_{R}$ between 0.6 and 0.45, with
corresponding minimal (and smooth) changes in the adjusted CQOM and Coriolis-
mixing parameters, and with the BCS parameters being fixed as in Refs. Minkov
and Pálffy (2017, 2019) (see Figs. 6-12 in Minkov and Pálffy (2021)).

So far the in-band and cross-band decay transitions of the 29.19 keV level
were not considered in our works Minkov and Pálffy (2017, 2019, 2021). Here we
present our (new) theoretical predictions for the $B(M1)$ and $B(E2)$ values
in this level obtained through the quenching factors and other model
parameters as explained above. Table 1 presents the $B(M1)$ values obtained
through the CQOM-DSM-BCS and Coriolis-mixing parameters used in Ref. Minkov
and Pálffy (2019) for $q_{s}=0.6$ and $q_{s}=0.55$ with several values of the
collective gyromagnetic quenching factor $q_{R}$ between 1 and 0.45. The
corresponding values of the $B(E2)$ reduced transition probabilities, which
are independent of the two quenching factors, are
$B(E2;|3\rangle\rightarrow|1\rangle)=27.11$ W.u. (Weisskopf units) and
$B(E2;|3\rangle\rightarrow|2\rangle)=239.18$ W.u.

Table 2 presents the $B(M1)$ and $B(E2)$ values obtained at few pairs of
$q_{s}$ and $q_{R}$ values through the corresponding sets of CQOM and
Coriolis-mixing parameters adjusted in Ref. Minkov and Pálffy (2021) taking
into account the experimental magnetic moments with the DSM and BCS parameters
kept the same as in Ref. Minkov and Pálffy (2019). We note that here the
reduced transition probabilities for the $E2$ channel slightly vary with
$q_{s}$ and $q_{R}$ due to the (slightly) different CQOM and Coriolis-mixing
parameter values obtained in the different fits. A comparison between Table 1
and Table 2 shows that for the same set of quenching factors, the reduced
transition probabilities present some variations depending on the chosen CQOM
and Coriolis mixing model parameters. Overall, the in-band transition
$|3\rangle\rightarrow|2\rangle$ presents much stronger reduced transition
probabilities for both multipolarities than the cross-band transition
$|3\rangle\rightarrow|1\rangle$.

$q_{S}$ | $q_{R}$ |  $B(M1)$ (W.u.)
---|---|---
$|3\rangle\rightarrow|1\rangle$ | $|3\rangle\rightarrow|2\rangle$
0.6 | 1 | 0.0012 | 0.0648
| 0.8 | 0.0020 | 0.0544
| 0.7 | 0.0025 | 0.0495
| 0.6 | 0.0030 | 0.0449
| 0.5 | 0.0036 | 0.0405
| 0.45 | 0.0039 | 0.0383
0.55 | 0.5 | 0.0028 | 0.0357
| 0.45 | 0.0031 | 0.0337
Table 1: Predicted $B(M1)$ values obtained for several spin and gyromagnetic quenching factors $q_{s}$ and $q_{R}$ with all other model parameters taken from Ref. Minkov and Pálffy (2019). See text for further explanations. $q_{S}$ | $q_{R}$ |  $B(M1)$ (W.u.) |  |  $B(E2)$ (W.u.)
---|---|---|---|---
$|3\rangle\rightarrow|1\rangle$ | $|3\rangle\rightarrow|2\rangle$ |  | $|3\rangle\rightarrow|1\rangle$ | $|3\rangle\rightarrow|2\rangle$
0.6 | 0.6 | 0.0043 | 0.0432 |  | 39.49 | 234.86
| 0.5 | 0.0050 | 0.0390 |  | 38.23 | 235.52
0.55 | 0.5 | 0.0035 | 0.0348 |  | 34.19 | 235.87
| 0.45 | 0.0035 | 0.0332 |  | 31.44 | 236.11
Table 2: Predicted $B(M1)$ and $B(E2)$ values obtained for several spin and
gyromagnetic quenching factors $q_{s}$ and $q_{R}$ with all other model
parameters taken from Ref. Minkov and Pálffy (2019). See text for further
explanations.

####  III.1.1 Radiative rates

The reduced transition probabilities give access to the radiative transition
rates which can be calculated according to Ring and Schuck (1980)

$T\left(M1\right)=$1.779\text{\times}{10}^{4}$\cdot E[$\mathrm{keV}$]^{3}\cdot
B\left(M1\right)$ (17)

for $M1$ transitions and

$T\left(E2\right)=$1.223\text{\times}{10}^{-6}$\cdot
E[$\mathrm{keV}$]^{5}\cdot B\left(E2\right)$ (18)

for $E2$ transitions, where the rates $T(\mu L)$ are given in s-1,
$E[$\mathrm{keV}$]$ is the transition energy in keV, $B(M1)$ in $\mu_{N}^{2}$
with $\mu_{N}$ the nuclear magneton and $B(E2)$ in e2fm2, respectively. The
total decay rate is obtained as a sum over the rates of the two
multipolarities $M1$ and $E2$, and the corresponding width for the
in-(cross-)band transitions is given by

$\Gamma_{\gamma}=\hbar\sum_{\mu L}T(\mu L)\,.$ (19)

We note that due to the suppressing factors in the expression (18), the $E2$
contribution for the radiative decay is approx. two orders of magnitude
smaller than the $M1$ one for both in- and cross-band transitions. Thus, the
$E2$ channel can be neglected in Eq. (19). However, this will not be the case
for the corresponding IC rates, as it will be discussed in the following.

####  III.1.2 IC rates

In the process of IC, the energy of the nuclear excited state is transferred
to an atomic electron, which is kicked out of its shell. The nuclear
excitation energy should therefore exceed the binding energy of the IC
electron. IC rates can be calculated taking into account the interplay between
atomic and nuclear degrees of freedom. However, since the nuclear part can be
related to the reduced transition probabilities $B(\mu L)$, the quantity of
interest is the IC coefficient $\alpha$, defined as the ratio of the IC and
radiative rates. The IC coefficient contains the remaining atomic structure
information and can be accurately calculated with existing codes such as BrIcc
Kibédi et al. (2008).

The IC coefficients are calculated according to the nuclear transition energy
and the transition multipolarity. For the two considered transitions
$|3\rangle\rightarrow|2\rangle$ and $|3\rangle\rightarrow|1\rangle$, the
transition energies differ by just 8 eV and the calculated IC coefficients are
for all practical purposes identical for each considered multipolarity. The
$E2$ IC coefficients are approx. a factor 30 higher than the $M1$
coefficients, compensating for the smaller radiative rates of the former.
Thus, the $E2$ IC channels are not negligible. Both $E2$ and $M1$ IC
coefficients are much larger than one, rendering IC the strongest decay
channel for the 29 keV excited state. The total IC rate for the
in-(cross-)band transition is given by the sum of the $M1$ and $E2$ channels,
with the corresponding IC width

$\Gamma_{\mathrm{I}C}=\hbar\sum_{\mu L}\alpha(\mu L)T(\mu L)\,.$ (20)

In the following we use $\alpha\left(M1\right)=151.07$ and
$\alpha\left(E2\right)=4401.61$ for both in-band and cross-band transition.
These values are interpolated from the IC coefficients tabulated in Ref. Rösel
et al. (1978).

###  III.2 Analysis of available experimental data

The experimental data on the transitions connecting the 29 keV level to the
ground and isomeric states is scarce. There are only three related quantities
that have been reported experimentally so far from two different experiments:
(i) the cross-band radiative transition width reported in Ref. Masuda et al.
(2019), $\Gamma_{\gamma}^{\mathrm{c}r}=1.70\pm 0.40\,$\mathrm{neV}$$, (ii)
from the same experiment, the total half-life of the state $|3\rangle$,
dominated by the two IC decays to the isomer and ground states,
$T_{1/2}=82.2\pm 4.0\,$\mathrm{ps}$$ and (iii) the radiative branching ratio
of the cross-band transition, $\mathrm{BR}_{31}^{\gamma}=9.3(6)~{}\%$ Sikorsky
et al. (2020).

From (i), we can deduce the corresponding $B(M1)$ for the cross-band
transition. Neglecting the $E2$ multipole mixing (which, as discussed above,
is a very good approximation), and using expression (17) we obtain
$B(M1;|3\rangle\rightarrow|1\rangle)=0.00326\pm 0.00076$ W.u. This value is in
good agreement with most of the theoretical predictions listed in Tables 1 and
2. Combining this value with the measured branching ratio
$\mathrm{BR}_{31}^{\gamma}$ reported in Ref. Sikorsky et al. (2020), we can
extract the in-band reduced transition probability. Indeed, neglecting the
multipole mixing in the radiative decay, we have

$\mathrm{BR}_{31}^{\gamma}=\frac{\Gamma_{\gamma}^{31}}{\Gamma_{\gamma}^{32}+\Gamma_{\gamma}^{31}}\,,$
(21)

leading to $B(M1;|3\rangle\rightarrow|2\rangle)=0.0318$ W.u., within the
uncertainty interval $[0.0227,0.0420]$ W.u. This value is slightly smaller
than the theoretical predictions in Tables 1 and 2.

The total half-life of the upper state $|3\rangle$ depends on both radiative
and IC decay channels of the two transitions, and here the multipole mixing
needs to be taken into account. As a result, we have

$\displaystyle T_{1/2}$ $\displaystyle=$
$\displaystyle\ln{2}\large\\{[1+\alpha(M1)][T_{31}(M1)+T_{32}(M1)]$ (22)
$\displaystyle+[1+\alpha(E2)][T_{31}(E2)+T_{32}(E2)]\\}^{-1}\,.$

Despite reliable theoretical values for the two IC coefficients (the numerical
values were given in Sec. III.1.2), this expression is not sufficient to
determine the two remaining unknowns $T_{32}(E2)$ and $T_{31}(E2)$ and the
corresponding reduced transition probabilities $B(E2)$. We note that mixing
ratios for the two transitions are not available from experiments. Thus, some
further assumption starting from our theory knowledge is required.

Inspection of the theoretically predicted $B(E2)$ values shows that regardless
of the used model parameters, the in-band reduced transition probability
$B(E2;|3\rangle\rightarrow|2\rangle)$ value is rather stable, with values
varying between 234.86 W.u. and 239.18 W.u. In the following we will use the
average value $B(E2;|3\rangle\rightarrow|2\rangle)=237.02$ W.u. to obtain
$T_{32}(E2)$ and derive the remaining unknown rate $T_{31}(E2)$ from Eq. (22).
For the $M1$ channels we use the values deduced above from the experimental
observations, dropping the error bars. We obtain $T_{31}(E2)=4.478\cdot
10^{5}$\mathrm{s}$^{-1}$ and correspondingly
$B(E2;|3\rangle\rightarrow|1\rangle)=207.65$ W.u. We note that this value is
unexpectedly large for the cross-band transition, being of the same order of
magnitude with the in-band reduced transition probability. In case we start
from a theoretical value for $B(E2;|3\rangle\rightarrow|1\rangle)$ in the
range shown in Table 2, we obtain for the in-band transition an unexpectedly
large reduced transition probability of
$B(E2;|3\rangle\rightarrow|2\rangle)>400$ W.u. These inconsistencies suggests
that the experimental values obtained from the two experiments do not match
exactly. We have checked that a consistent set of results can be obtained if
in expression (22) we use values close to or at the upper uncertainty limits
of the experimental values of the total half-life of state $|3\rangle$ and the
two $B(M1)$ transition rates, such as $T_{1/2}=85.6$ ps,
$B(M1;|3\rangle\rightarrow|1\rangle)=0.0040$ W.u. and
$B(M1;|3\rangle\rightarrow|2\rangle)=0.0420$ W.u. This may be considered as an
indication for the need of further more precise experimental determination of
these quantities.

We note that also Ref. Masuda et al. (2019) arrives at inconsistencies when
combining the reported experimental lifetime and cross-band transition width
with theoretical values from Refs. Kroger and Reich (1976); Barci et al.
(2003). The latter theoretical values are obtained from the standard
rotational model and are not expected to be particularly accurate for 229Th,
especially in view of the most recent insights into the nuclear structure
origins of the Th isomer Minkov and Pálffy (2017, 2019, 2021). By combining
the experimental $T_{1/2}$ and $\Gamma_{\gamma}^{\mathrm{c}r}$ with
theoretical predictions on the in-band radiative decay Barci et al. (2003),
the authors of Ref. Masuda et al. (2019) deduce a total cross-band IC
coefficient of 1,370. This value hints to an implausible multipole mixing for
the cross-band transition. Even by using the (equally implausible) value
$B(E2);|3\rangle\rightarrow|1\rangle)=207.65$ W.u. inferred above, the cross-
band IC coefficient would be a factor of two smaller. A closer inspection of
the theoretical values in Ref. Barci et al. (2003) shows that the predicted
cross-band transition width disagrees with the experimental value by a factor
2. The rotational model predictions are therefore not suitable for combining
experimental and theoretical results to deduce correct branching ratios and
related transition parameters. This inconsistency reflects in the unexpectedly
large IC coefficient value.

###  III.3 Choice of nuclear transition parameters

For our STIRAP calculations in Th ions or atoms we require the $M1$ reduced
transition probabilities for the in- and cross-band transitions, the total
decay rate of level $|3\rangle$ and the branching ratios $\mathrm{BR}_{31}$
and $\mathrm{BR}_{32}$ with and without including the IC channel. In the
following we list in Table 3 the used sets of nuclear transition parameters
and comment on the reliability of our choice.

|  | cross-band | in-band | total decay
---|---|---|---|---
|  | $|3\rangle\rightarrow|1\rangle$ | $|3\rangle\rightarrow|2\rangle$ | of $|3\rangle$
$B(M1)$ [W.u.] | theo111Values are obtained using quenching factors $q_{s}=q_{R}=0.6$ and the model parameters from Ref. Minkov and Pálffy (2019). | 0.0030 | 0.0449 |
exp222Values obtained combining the measured cross-band radiative width Masuda et al. (2019) and radiative branching ratio Sikorsky et al. (2020) values. | 0.00326(76) | $0.0318\begin{subarray}{c}+0.0102\\\ -0.0091\end{subarray}$ |
BRγ[%] | theo333Obtained with the theoretical $B(M1)$ values listed in this Table. | 6.3 | 93.7 |
exp444From Ref. Sikorsky et al. (2020). | 9.3(6) | 90.7(6) |
BR[%] | theo | 7.5 | 92.5 |
exp555Using the reduced transition probabilities extracted from the experimental data discussed above, i.e., $B(M1;|3\rangle\rightarrow|1\rangle)=0.00326$ W.u., $B(M1;|3\rangle\rightarrow|2\rangle)=0.0318$ W.u., $B(E2);|3\rangle\rightarrow|1\rangle)=207.65$ W.u. and the theoretical assumption $B(E2);|3\rangle\rightarrow|2\rangle)=237.02$ W.u. | 28.01 | 71.99 |
$\Gamma_{\gamma}$[neV] | theoc | 1.6 | 23.71 | 25.31
exp666the cross-band value was reported in Ref. Masuda et al. (2019); the other two values are obtained using in addition the experimental branching ratio reported in Ref. Sikorsky et al. (2020). | 1.7(4) | $16.56\begin{subarray}{c}+5.328\\\ -4.734\end{subarray}$ | $18.26\begin{subarray}{c}+5.728\\\ -5.134\end{subarray}$
$\Gamma$[$\mu$eV] | theo | 0.4 | 5.05 | 5.45
exp | 1.55 | 4.00 | 5.55777From the measured total halflife $T_{1/2}$ of the excited state Masuda et al. (2019).
Table 3: Reduced transition probabilities, branching ratios and radiative and
total decay widths for the cross-band and in-band transitions obtained from
theoretical predictions (theo), experimental data (exp) and a combination
thereof.

We note that the total decay rate of the upper state $|3\rangle$ obtained from
theoretical estimates is very close to the experimental value from Ref. Masuda
et al. (2019). The branching ratios and especially the total branching ratio
show the largest disagreement between theory and reconstruction from
experimental values. At first sight, this discrepancy could stem from the
different $B(M1)$ values appearing in the branching ratios. This holds true
for the radiative branching ratio BRγ, where the disagreement is however not
that critical. However, the much larger and most relevant disagreement for the
total branching ratio is caused by the already mentioned inconsistency in the
$B(E2)$ values obtained from partial reconstruction based on experimental
data.

##  IV Numerical Results

In this Section we present our numerical results for nuclear coherent
population transfer in the considered 229Th three-level-$\Lambda$ system. We
consider three different possible experimental scenarios and discuss their
feasibility. The first two scenarios investigate the interaction of
relativistic thorium ions accelerated in a storage ring considering (A) UV
radiation and strong acceleration or (B) x-ray radiation from an XFEL source
combined with moderate ion acceleration. The last scenario (C) addresses the
interaction of a generic 229Th-doped solid sample and highly energetic x-rays
from a cavity-based x-ray source.

###  IV.1 NCPT for highly accelerated nuclei

The first scenario investigates the radiative coupling of ultra relativistic
highly charged thorium ions with two UV laser fields. The experimental
implementation could be achieved at the GF, which is an ambitious research
tool for physics beyond colliders at CERN Krasny (2015); Budker et al. (2020,
2022). The ultra relativistic acceleration of Th ions appears to be feasible
given previous experiments at LHC with highly ionized Pb ions, which have a
similar mass to 229Th, where Lorentz factors up to $\gamma\approx 2950$ were
reached. We note that STIRAP in this scenario has been recently addressed in
Refs. Budker et al. (2022); Jin et al. (2022).

For our purpose, we consider highly charged 229Th ions (or even bare nuclei)
circulating at LHC with $v\approx c$ and a Lorentz factor $\gamma=2950$. We
assume the ions are distributed in a bunch with kinetic energy fluctuations in
the order of magnitude $\Delta\gamma/\gamma=${10}^{-4}$$. The transverse cross
section of the ion beam can be approximated as $\pi r_{\sigma}^{2}$, where
$r_{\sigma}=$16\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ is the $1$-$\sigma$
radius of the ion bunch Budker et al. (2022). The ion bunch has a revolution
frequency of $f_{b}=$11.2\text{\,}\mathrm{kHz}$$ and at the same time up to
1232 ion bunches can circulate in the storage ring. Each ion bunch carries up
to ${10}^{8}$ 229Th ions Budker et al. (2022). For resonant excitation we
require photon energies $E_{\lambda}=$29.19\text{\,}\mathrm{keV}$/2\cdot
2950\approx$4.95\text{\,}\mathrm{eV}$$ for the pump field. The Stokes field
photons are tuned such that a resonant coupling of isomer and second nuclear
excited state occurs. The initial pulse energy of each field is set to
$10\text{\,}\mathrm{\SIUnitSymbolMicro J}$, which is then increased by a
factor ${10}^{5}$ by means of cavity enhancement with a Fábry-Perot
interferometer to $1\text{\,}\mathrm{J}$. The pulse duration is chosen to be
$3.7\text{\,}\mathrm{ps}$ such that a broad spectral width is guaranteed. The
energy spread of the ion beam is accounted for approximatively by including a
detuning in the calculation, which reads in the nuclear rest frame

$\Delta_{P}=2\left(\gamma+\Delta\gamma\right)\omega_{P}-\omega_{\text{13}}=\frac{\Delta\gamma}{\gamma}\cdot\omega_{13}\approx\Delta_{S}.$
(23)

The peak intensity of each laser field in the lab frame is defined as
$I_{0}=\frac{E_{\text{pul}}}{t_{\text{pul}}\pi w^{2}}$, where $w$ is the beam
waist of the pulse. In addition, we assume the laser system has a repetition
rate $f_{\text{rep}}=1232\cdot f_{b}=$13.8\text{\,}\mathrm{MHz}$$ and it is
perfectly synchronized to the circulating ion bunches.

We start by estimating the required intensities for NCPT via the adiabaticity
and $\pi$-pulse criteria in Eqs. (11) and (12). These estimates require
knowledge of the nuclear reduced transition probabilities $B(M1)$ for the two
transitions. Our intensity values for the experimentally deduced and
theoretical $B(M1)$ sets in Table 3 are listed in Table 4. The ratio of the
pump and Stokes intensities is related to the different reduced transition
probabilities of the corresponding transitions, and is a factor $10$ for the
experimental values and a factor $15$ for the theoretical values.

Method | $B(M1)$ | $I_{0,P}[$\mathrm{W}\text{\,}{\mathrm{m}}^{-2}$]$ | $I_{0,S}[$\mathrm{W}\text{\,}{\mathrm{m}}^{-2}$]$
---|---|---|---
STIRAP | exp | $2.75\text{\times}{10}^{24}$ | $2.77\text{\times}{10}^{23}$
| theo | $2.93\text{\times}{10}^{24}$ | $1.96\text{\times}{10}^{23}$
$\pi$-pulses | exp | $8.64\text{\times}{10}^{22}$ | $8.70\text{\times}{10}^{21}$
| theo | $9.22\text{\times}{10}^{22}$ | $6.16\text{\times}{10}^{21}$
Table 4: Laser intensities required for NCPT via STIRAP and $\pi$-pulses. The
values were obtained from the adiabaticity and $\pi-$pulse criteria for the
given pulse duration $t_{\text{pul}}=$3.7\text{\,}\mathrm{ps}$$ for the
experimental (exp) and theoretical (theo) $B(M1)$ data sets.

We are interested in the fraction of nuclei which have reached the isomeric
state after one sequence of pump and Stokes pulses. Instead of the isomer
population at the end of the laser pulses, we consider the transfer rate which
also takes into account the ratio of ions unintentionally transferred to
$\left|3\right\rangle$ and their subsequent decay after one ion revolution in
the storage ring. The transfer rate for a single pulse sequence therefore
reads

$\eta\approx\rho_{22}+\text{BR}_{32}\rho_{33}\left(1-\exp{\left(-\frac{\ln{2}}{\gamma
T_{1/2}f_{b}}\right)}\right).$ (24)

Here, $\rho_{ii}$ are the density matrix elements giving the fraction of
population in state $\left|i\right\rangle$ and $\text{BR}_{32}$ the branching
ratio of the in-band decay, respectively. The latter term takes into account
the radiative decay of the intermediate state during one circulation in the
storage ring. With aid of the experimental and theoretical reduced transition
probabilitites $B(M1)$ we derive the radiative half life as
$T_{1/2}^{\text{exp}}=$25\text{\,}\mathrm{ns}$$ and
$T_{1/2}^{\text{th}}=$18\text{\,}\mathrm{ns}$$.

We start with NCPT via STIRAP considering the intensities in Table 4 estimated
for the experimental set of reduced transition probabilities $B(M1)$. Since
the adiabaticity criterion is rather empirical, we consider also smaller
intensities and vary the pulse delay to maximize the population transfer. To
this end, the peak intensity of both Stokes and pump fields is scaled down in
steps of 0.1. Our numerical results in Fig. 3(a) show that indeed, 100%
population transfer can be reached for a large range of delay times
$\Delta\tau$ for the intensity values $I_{0,P/S}$ of Table 4 (and, obviously,
for any higher intensities). However, for particular values of $\Delta\tau$,
STIRAP can be successful also for lower intensities. Figure 3(a) presents the
transfer rate to the state $|2\rangle$ for scaled pump and Stokes intensities
using three scaling factors 0.1, 0.2 and 0.3 where a trend change is observed.
For $\Delta\tau=$2.2\text{\,}\mathrm{ps}$$, 100% population transfer can be
achieved via STIRAP also for peak intensity values 0.3$I_{0,P/S}$. Our
numerical results for the populations of the three states using these
intensities are presented in Fig. 3(b). Also the scaling 0.2 leads to transfer
rates of almost unity. However, further lowering of the pulse intensities
leads to incomplete population transfer rates. Using the theoretical $B(M1)$
values we obtain very similar results, and the transfer rates deviate only by
a few percent.

Figure 3: (a) STIRAP transfer rate as a function of time delay (in the lab
frame) for different scaling factors of the laser intensities. (b) STIRAP
population transfer as a function of time for $0.3\cdot I_{0,P/S}$ and
$\Delta\tau=$2.2\text{\,}\mathrm{ps}$$ (lab frame). (c) $\pi$-pulse transfer
rate as a function of time delay between pulses $|\Delta\tau|$ (in the lab
frame) for different ion energy spreads. (d) Single $\pi$-pulse population
transfer sequence for (i) $\Delta\tau=$-0.7\text{\,}\mathrm{ps}$$, $6\cdot
I_{0,P/S}$ and $\Delta\gamma/\gamma=${10}^{-4}$$ and for (ii)
$\Delta\tau=$-5\text{\,}\mathrm{ps}$$, $I_{0,P/S}$ and
$\Delta\gamma/\gamma=${10}^{-5}$$ (in the lab frame).

Next, we turn to NCPT via $\pi$-pulses. Our numerical results in Figure 3(c)
show that transfer rates of unity cannot be reached due to the large detuning
in the ensemble. Only about $16\text{\,}\mathrm{\char 37\relax}$ of the
population are transferred to the isomeric state for a small delay window. The
transfer rates can be increased either by decreasing the ion beam energy
spread or by increasing the laser intensity. Indeed, as soon as we scale down
the ion energy spread, the transfer rates significantly increase. For
$\Delta\gamma/\gamma=${10}^{-5}$$ the transfer rates already increase to
$96\text{\,}\mathrm{\char 37\relax}$ and for $\Delta\gamma/\gamma=${10}^{-6}$$
transfer rates of unity are reached for a large time delay window. We note
that a better ion beam quality would be beneficial also for NCPT via STIRAP.
Keeping $\Delta\gamma/\gamma=${10}^{-4}$$ fixed, transfer rates of approx.
$100\text{\,}\mathrm{\char 37\relax}$ can be reached by increasing the
intensity. Considering the set of $B(M1)$ values deduced from experiments, for
the given pulse duration scaling by a factor 6 is required to obtain
$\eta=$100\text{\,}\mathrm{\char 37\relax}$$ for
$\Delta\tau=$-0.7\text{\,}\mathrm{ps}$$. This intensity corresponds to
$\approx$20\text{\,}\mathrm{\char 37\relax}$$ of the intensity required for
NCPT via STIRAP. Thereby, we observe that the sequence attains with the
preceding pump pulse slightly higher rates, though for both cases very close
to 100%.

This scenario allows only a small delay window for transfer rates approaching
unity, and a large pulse overlap is required. This is therefore less of the
traditional $\pi$-pulse scheme but rather a single train of coincident pulses,
which builds a bridge between both NCPT schemes Rangelov and Vitanov (2012).
This feature can be identified in the upper part of Fig. 3(d) which
illustrates a single population transfer event for
$\Delta\tau=$-0.7\text{\,}\mathrm{ps}$$, $6\cdot I_{0,P/S}$ and
$\Delta\gamma/\gamma=${10}^{-4}$$. In comparison a single event for
$\Delta\tau=$-5\text{\,}\mathrm{ps}$$, $I_{0,P/S}$ and
$\Delta\gamma/\gamma=${10}^{-5}$$ is shown below. The difference is clearly
visible. In (i) only a small fraction of the population is pumped to the
intermediate state compared to (ii).

Let us apply our theoretical considerations to the GF experimental setup and
its given parameters. Considering the laser beam waists to be equal to the
1-$\sigma$ radius of the ion bunch
$w_{\sigma}=$16\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ yields intensities
$I_{0,S}=I_{0,P}=$3.36\text{\times}{10}^{20}\text{\,}\mathrm{W}\text{\,}{\mathrm{m}}^{-2}$$
much smaller than the threshold deduced from the transfer criteria.
Correspondingly, we can expect only low population transfer rates of much less
than one percent. The solution is to increase the intensity by stronger
focusing and therefore smaller laser beam waist. The drawback is that less
nuclei in the ion beam are irradiated by the lasers and only a fraction
thereof can be promoted to the isomeric state. To investigate the trade-off
between intensity and number of addressed nuclei we start by approximating the
cross sections of the two laser beams as circular and concentric, and in
addition concentric to the ion beam cross section. Guided by the transfer
criteria, we assume that the two Rabi frequencies for pump and Stokes pulses
are equal for equal pulse durations. Considering the same laser energy for the
two beams, the smallest beam waist and therefore highest intensity is required
for the pump laser, since the $B(M1)$ value is smaller for the
$|1\rangle\rightarrow|3\rangle$ transition. The effective number of ions
interacting with the laser photons is then
$N_{\text{ion}}^{\text{eff}}=\left(\frac{w_{P}}{w_{\sigma}}\right)^{2}N_{\text{ion}}$
with $w_{P}$ the beam waist of the pump field. For a single NCPT process
during one revolution of the ion beam, up to
$N_{\text{iso}}=N_{\text{ion}}^{\text{eff}}\,\eta$ isomers can be populated.

Method | $\Delta\gamma/\gamma$ | Scaling | $w_{P}[$\mathrm{\SIUnitSymbolMicro m}$]$ | $\Delta\tau[$\mathrm{ps}$]$ | $\eta[$\mathrm{\char 37\relax}$]$ | $N_{\text{iso}}[${10}^{4}$]$ | $t_{ex}[$\mathrm{s}$]\gtrsim$
---|---|---|---|---|---|---|---
STIRAP | ${10}^{-4}$ | 1 | 0.18 | 2.9 | 100 | 1.3 | 1.38
| ${10}^{-4}$ | 0.3 | 0.32 | 2.2 | 100 | 4.0 | 0.42
| ${10}^{-4}$ | 0.2 | 0.40 | 1 | 99.1 | 6.2 | 0.32
| ${10}^{-4}$ | 0.1 | 0.56 | 0.1 | 69.6 | 8.5 | 0.20
$\pi$-pulses | ${10}^{-4}$ | 1 | 1 | -0.1 | 16.3 | 6.3 | 0.18
| ${10}^{-4}$ | 6 | 0.41 | -0.7 | 100 | 6.5 | 0.29
| ${10}^{-5}$ | 1 | 1 | -5 | 96 | 37.4 | 0.10
| ${10}^{-6}$ | 1 | 1 | -5 | 100 | 39.1 | 0.09
Table 5: Population transfer rate $\eta$, number of produced isomers
$N_{\text{iso}}$ and time $t_{ex}$ required to approximately reach saturation
for the two NCPT schemes in scenario A. The beam-waist of the Stokes field can
be determined via $w_{S}=\sqrt{I_{0,P}/I_{0,S}}\cdot w_{P}$. For an equal
focus ($w_{P}=w_{S}$) the pulse energy of the Stokes field should be lowered
according to $E_{S}=I_{0,S}/I_{0,P}\cdot E_{P}$. See text for further
explanations.

The results for both transfer schemes (for the experimental set of $B(M1)$
values) are summarized in Table 5. For STIRAP the number of produced isomers
is rather small, since the intensities $I_{0,S}/I_{0,P}$ corresponds to rather
small beam waists compared to the ion beam. Thus, only a small fraction of the
ion beam is addressed and just a few ten thousand ions are promoted to the
isomeric state. Scaling down the intensity values correspondingly leads to
lower population transfer rates, but due to the larger beam size more isomers
are excited. A more systematic study reveals that choosing the same pump and
Stokes beam waist (and therefore the same intensity for the two lasers) yields
the maximum of approx. $1.5\text{\times}{10}^{5}$ isomers for
$w_{P}=w_{S}=$1.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ for delay times
around $\Delta\tau=$0.1\text{\,}\mathrm{ps}$$. In comparison, this focus also
leads to better results than
$w_{P}=w_{S}=r_{\sigma}=$16\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, for which
the number of populated isomers is only about $3.3\text{\times}{10}^{4}$. For
NCPT with two $\pi$-pulses, our results show that the population transfer for
$\Delta\gamma/\gamma=${10}^{-4}$$ does not deviate much from the STIRAP
numbers. However, the implementation could be less challenging due to the
larger pump field waist. Provided it is possible to reduce the ion energy
spread of the beam, then high transfer rates and consequently high isomer
numbers can be produced as shown in the Table 5.

For applications that make use of the produced isomers, it is very unlikely
that a single pulse sequence is sufficient. We therefore investigate the
scenario of repeated coherent pumping, considering that Stokes and pump lasers
interact with the ion bunch with repetition rate $f_{b}$. We assume that for
each ion bunch revolution, the ions redistribute spatially in the beam. Thus,
naively one could approximate that after the time
$\frac{N_{\text{ion}}}{N_{\text{isomer}}}\cdot\frac{1}{f_{b}}$ (considered in
the lab frame), a large fraction of ions in the storage ring are transferred
to the isomeric state. However, this picture is not accurate since for
increasing numbers of excited isomers, the laser pulses will also drive
nuclear population from the isomeric state back to states $|1\rangle$ and
$|3\rangle$ leading to saturation.

To model this scenario, we consider the same beam waist for both pump and
Stokes pulses, such that both pulses address the same fraction of nuclei in
the beam. To compensate in the intensity, we consider a smaller Stokes pulse
energy. We model the spatial redistribution assuming that the ions with nuclei
in their respective states are distributed homogeneously within the bunch for
each pulse sequence/ion bunch revolution. The master equation is solved in a
loop for the input parameters in Tab. 5 where each iteration corresponds to a
pulse sequence. We obtain an isomer saturation of
$\approx$50\text{\,}\mathrm{\char 37\relax}$$ for all NCPT scenarios with
$\Delta\gamma/\gamma=${10}^{-4}$$ and $\approx$70\text{\,}\mathrm{\char
37\relax}$$ for all (non-overlapping) $\pi$-pulse scenarios with
$\Delta\gamma/\gamma\leq${10}^{-5}$$. Note that for a better ion energy spread
($\leq${10}^{-5}$$) also NCPT via STIRAP can reach a total isomer population
of $\approx$70\text{\,}\mathrm{\char 37\relax}$$. The excitation time $t_{ex}$
in Tab. 5 corresponds to the approximate time which is required for the isomer
population to reach the saturation limit. We see by means of $t_{ex}$ that it
is more advantageous for faster excitation to have a large beam waist. These
times are for most scenarios approximately a factor 2 larger than the naive
estimate $\frac{N_{\text{ion}}}{N_{\text{isomer}}}\cdot\frac{1}{f_{b}}$.

Incoherent pumping $\left|1\right\rangle\rightarrow\left|3\right\rangle$ with
the pump laser only and the subsequent spontaneous decay to
$\left|2\right\rangle$ provides an efficient alternative due to the
advantageous branching ratio $\text{BR}_{32}$ and the slow radiative decay of
the isomeric state. Thus, almost all ions in the bunch can reach the isomeric
state. Our numerical simulations of a sequence of pump pulses leads to almost
100% population of the isomeric state after a time which is approximately 5
times the $t_{ex}$ values in Table 5. However, it takes both STIRAP and
incoherent pumping approximately the same time to reach the STIRAP saturation
level of about 50%. Thus, the advantage of STIRAP to reduce the excitation
time (albeit with a more complicated experimental setup) only comes into play
with large if sufficient intensities are available for laser beams with waists
which approach the ion beam radius.

###  IV.2 NCPT with moderately accelerated nuclei

The second excitation scheme considers the radiative coupling of moderately
accelerated thorium ions with coherent x-ray pulses from an XFEL. In
comparison to optical laser systems, XFEL pulses may lack temporal coherence
and most currently operating facilities only have repetition rates of a few
tens up to a hundred Hz with a few exceptions Huang et al. (2021b). The photon
energies of XFELs range from hundred eV (soft x-rays) up to
$25\text{\,}\mathrm{keV}$ des (2022). Unfortunately, these energies are too
low to directly excite the 29 keV level in 229Th, so some acceleration of the
nuclear beam is required. For our purposes, we consider the pump field
operating at a photon energy of $3.5\text{\,}\mathrm{keV}$, while the Stokes
field is operating at a slightly smaller energy. For these energies, the ions
have to be accelerated to $\gamma\approx 4.2$ for resonance. This small
Lorentz factor is not available at CERN since there $\gamma_{\text{min}}=20$
Budker et al. (2022). Thus, the experiment would require a smaller storage
ring. For instance, the high energy storage ring (HESR) at GSI can deliver
sufficiently small Lorentz factors for heavy ions Sánchez et al. (2020).
Compared to LHC ($\approx$40\text{\,}\mathrm{h}$$)Budker et al. (2022), the
lifetime of U90+ (similar mass as Th) with $\gamma\approx 4$ is expected to be
$\approx$3000\text{\,}\mathrm{s}$$ Shevelko et al. (2018).

We assume the ion bunch carries up to ${10}^{8}$ ions with a transverse
1-$\sigma$ radius of $2.1\text{\,}\mathrm{mm}$ and the circulation frequency
of an ion bunch is $f_{b}=$522\text{\,}\mathrm{kHz}$$. The large ion beam size
is correlated with the relativistic Lorentz factor $\gamma$ since the
adiabatic damping requires larger particle momentum Lee (2018). If not
mentioned otherwise, we consider also for this case the ion energy spread
$\Delta\gamma/\gamma=${10}^{-4}$$. Again, this then leads to a rest frame
detuning
$\Delta_{P}=\Delta_{S}\approx\Delta\gamma/\gamma\cdot\omega_{\text{13}}$ in
the ensemble. The generic laser parameters we consider Huang et al. (2021a)
are pulse energy $1\text{\,}\mathrm{mJ}$, pulse duration
$30\text{\,}\mathrm{fs}$ and repetition rate
$f_{\text{rep}}=$60\text{\,}\mathrm{Hz}$$. We assume the pulses have full
temporal coherence, which is not far from what has been reported in Ref. Nam
et al. (2021). Since the repetition rate of the XFEL is small compared to the
circulation frequency, an ion bunch can circulate (in the HESR case)
$f_{b}/f_{\text{rep}}=8700$ times before another XFEL pulse sequence arrives.
Therefore, the transfer rate after one pulse sequence can be approximated as
$\eta\approx\rho_{22}+\text{BR}_{32}\rho_{33}$. The required theoretical and
experimental intensities for NCPT via STIRAP and $\pi$-pulses are listed in
Tab. 6. Compared to the figures in Table 4, it is clear that this setup
requires much larger laser intensities to be successful due to the fs XFEL
pulse durations.

Method | $B(M1)$ | $I_{0,P}[$\mathrm{W}\text{\,}{\mathrm{m}}^{-2}$]$ | $I_{0,S}[$\mathrm{W}\text{\,}{\mathrm{m}}^{-2}$]$
---|---|---|---
STIRAP | exp | $4.18\text{\times}{10}^{28}$ | $4.21\text{\times}{10}^{27}$
| theo | $4.46\text{\times}{10}^{28}$ | $2.98\text{\times}{10}^{27}$
$\pi$-pulses | exp | $1.31\text{\times}{10}^{27}$ | $1.32\text{\times}{10}^{26}$
| theo | $1.40\text{\times}{10}^{27}$ | $9.38\text{\times}{10}^{25}$
Table 6: Laser intensities required for NCPT via STIRAP and $\pi$-pulses for
scenario B and a pulse duration of $\approx$30\text{\,}\mathrm{fs}$$. See
caption of Table 4 for further explanations.

In the following we present our results for the case of the $B(M1)$ values
obtained from experimental data. Following the discussion in Sec. IV.1, at
first NCPT via STIRAP is discussed. Starting from the intensities delivered by
the adiabaticity criterion in Table 6, we investigate in Fig. 4(a) the
transfer rate as a function of delay time between XFEL pulses for down-scaled
intensity values. Transfer rates of unity are reached until $0.8\cdot
I_{0,P/S}$. The population transfer after a single STIRAP sequence for this
case is shown in Fig. 4(b). From a scaling of 0.7 the transfer rate slightly
decreases to $\approx$98.7\text{\,}\mathrm{\char 37\relax}$$, while a scaling
of 0.6 leads to a drop of $\eta$ to $\approx$91\text{\,}\mathrm{\char
37\relax}$$. In contrast to the results in Sec. IV.1, the broader plateaus of
the transfer rate as a function of delay time appear not for the reference
intensity $I_{0,P/S}$, but for the down-scaled (and also for upscaled) values.
Thus, also smaller intensities allow for a more feasible and robust STIRAP
implementation.

Figure 4(c) presents our numerical results of the transfer rate for a
$\pi$-pulse sequence. Once more, we observe a rather bad performance due to
the detuning caused by the ion beam energy spread. Only about
$0.5\text{\,}\mathrm{\char 37\relax}$ are transferred to the isomeric state.
This is related to the very short pulse durations of the lasers and the
resulting broad spectral width. Larger transfer rates can be obtained either
by increasing the XFEL intensity, or by decreasing the ion energy spread. Once
the intensity is scaled up by a factor of approx. 25 we again reach transfer
rates approaching unity for large pulse overlaps. In turn, keeping the
intensity set to $I_{0,P/S}$, for $\Delta\gamma/\gamma=${10}^{-6}$$ transfer
rates of approx. $99\text{\,}\mathrm{\char 37\relax}$ can be reached with high
stability and the characteristic plateau for large delays. In the case of
$\Delta\gamma/\gamma=${10}^{-5}$$, transfer rates of unity are only achievable
with a up-scaling of the intensities and within a short time interval around
$15\text{\,}\mathrm{ps}$ with a large pulse overlap.

Figure 4: (a) STIRAP transfer rate as a function of delay (in the lab frame)
for different XFEL intensity values for $\Delta\gamma/\gamma=${10}^{-4}$$. (b)
Single STIRAP sequence (lab frame) for $0.8\cdot I_{0,P/S}$ and
$\Delta\tau=$7.5\text{\,}\mathrm{fs}$$. (c) $\pi$-pulse transfer rate as a
function of delay $|\Delta\tau|$ (lab frame) for different ion energy spreads.
(d) Single $\pi$-pulse sequence (lab frame) for
$\Delta\gamma/\gamma=${10}^{-6}$$ and $\Delta\tau=$-90\text{\,}\mathrm{fs}$$.

Our simulations show that scenario B is problematic from several points of
view. The large intensities required for NCPT imply either a very high and so
far unavailable pulse energy, or a strong focus. For instance, for intensity
values of $0.8\cdot I_{0,P/S}$ and pulse energy 1 mJ, the required beam waists
are $w_{p}=$0.56\text{\,}\mathrm{nm}$$ and $w_{s}=$1.77\text{\,}\mathrm{nm}$$,
exceeding the typical focusing limits of a few nm Mimura et al. (2009); Yumoto
et al. (2013, 2020). With stronger focus the number of addressed isomers is
also diminishing. In this example, the effective number of ions significantly
decreases to $\approx$7\text{\times}{10}^{-6}$$, which means hardly any
thorium ions are addressed. Even using an unrealistic pulse energy input for
the calculation, the isomer population does not approach the same levels as
scenario A due to the much lower pulse repetition rates. Finally, setup B is
logistically challenging because it requires both an XFEL and an ion
accelerator at the same facility. We conclude that scenario B is less likely
to be experimentally implemented.

###  IV.3 NCPT with 229Th nuclei at rest

The last scenario investigates the case of a fixed thorium sample irradiated
by resonant and fully coherent x-ray pulses. These pulses could be delivered
by a next generation cavity-based FEL lasing source, the so-called XFEL
oscillator (XFELO) Kim et al. (2008); Adams et al. (2019); Qin et al. (2018a);
Lindberg et al. (2011). This lasing source is expected to provide radiation
with large temporal coherence times and photon energies up to
$25\text{\,}\mathrm{keV}$ in basic operation. Theoretical studies have even
shown that an XFELO can generate photons with energies up to
$60\text{\,}\mathrm{keV}$ through high harmonic generation Qin et al. (2018b).
In the following we consider the XFELO parameters for basic operation despite
requiring a photon energy of $29.19\text{\,}\mathrm{keV}$.

X-ray pumping of the 29 keV level in 229Th using nuclear resonant scattering
synchrotron radiation and a fixed target has been reported in Ref. Masuda et
al. (2019). The target comprised of 229Th-doped thorium oxide sealed in Be
cover plates. One could imagine also using a VUV-transparent target such as
229Th-doped CaF2, which has an experimentally measured band gap of
$\approx$12\text{\,}\mathrm{eV}$$ Tsujibayashi et al. (2002); Dessovic et al.
(2014). This would have the advantage that IC is not allowed from the isomeric
state and in addition, due to transparency, VUV photons from the radiative
decay of the isomer can be detected. It is expected that both types of samples
would experience damage from the highly energetic and intense radiation, such
that a tape station for the target shifting the impact point after each pulse
sequence would be required.

Since the 229Th atoms or ions in the sample are not highly charged, the IC
decay channel of the 29 keV state is energetically permitted. Thus, all nuclei
which are pumped to the second excited state will decay via IC to both isomer
and ground states. The total half-life of the 29 keV state derived from theory
using the parameters $q_{s}=g_{R}=0.6$ is $T_{1/2}=$84\text{\,}\mathrm{ps}$$,
in good agreement with the experimental value reported in Ref. Masuda et al.
(2019). In the following we use the theoretical $B(M1)$ and branching ratio
values for the calculation. In addition, we neglect detuning and set
$\Delta_{P}=\Delta_{S}=0$. The transfer rate is given by
$\eta\approx\rho_{22}+\text{BR}_{32}\rho_{33}$, where we use
$\text{BR}_{32}=$92.5\text{\,}\mathrm{\char 37\relax}$$.

Set | $t_{\text{pul}}[$\mathrm{ps}$]$ | $E_{\text{pul}}[$\mathrm{\SIUnitSymbolMicro J}$]$ | Method | $I_{0,P}[$\mathrm{W}\text{\,}{\mathrm{m}}^{-2}$]$ | $I_{0,S}[$\mathrm{W}\text{\,}{\mathrm{m}}^{-2}$]$ | Scaling | $\Delta\tau[$\mathrm{ps}$]$ | $\eta[$\mathrm{\char 37\relax}$]$ | $w_{P}[$\mathrm{nm}$]$
---|---|---|---|---|---|---|---|---|---
Qin et al. (2018a) | 0.7 | 28 | STIRAP | $8.20\text{\times}{10}^{25}$ | $5.48\text{\times}{10}^{24}$ | 3-0.04 | 0.7-0.01 | 100-97.7 | 0.23-1.97
| 0.7 | 28 |  $\pi$-pulses | $2.58\text{\times}{10}^{24}$ | $1.72\text{\times}{10}^{23}$ | 1 | -1.15 | 99.8 | 2.2
Agapov (2021) | 12 | 5 | STIRAP | $2.79\text{\times}{10}^{23}$ | $1.87\text{\times}{10}^{22}$ | 3-0.1 | 12.0-4.0 | 100-98.7 | 0.4-2.2
| 12 | 5 |  $\pi$-pulses | $8.76\text{\times}{10}^{21}$ | $5.86\text{\times}{10}^{20}$ | 1 | -16 | 98.2 | 3.9
Table 7: XFELO pulse duration $t_{\text{pul}}$ and energy $E_{\text{pul}}$ for
two parameter sets discussed in the literature (left). The corresponding laser
intensities required for NCPT via STIRAP and $\pi$-pulses for theoretical
$B(M1)$ values are given in the middle column split. The calculated ranges of
NCPT transfer rates $\eta$ are presented in the right column split. In
vicinity of the adiabatic criterion intensity value,
$\Delta\tau=t_{\text{pul}}$ corresponds to the optimum delay, while for
shrinking intensity the maximum shifts towards smaller delays. The beam waist
of the Stokes field can be determined via $w_{S}=\sqrt{I_{0,P}/I_{0,S}}\cdot
w_{P}$. Figure 5: (a) STIRAP transfer rate as a function of pulse delay with
scaled intensities (XFELO parameters from Qin et al. (2018a)). (b) Related
transfer rate as a function of intensity for different pulse delays. The
vertical lines on the right and left-hand side correspond to $3\cdot
I_{0,P/S}$ and $0.01\cdot I_{0,P/S}$.(c) $\pi$-pulse transfer rate as a
function of pulse delay $|\Delta\tau|$ for XFELO parameters Qin et al. (2018a)
(solid line), and Agapov (2021) (dashed line). (d) Single $\pi$-pulse sequence
for set Agapov (2021) with delay $\Delta\tau=$-16\text{\,}\mathrm{ps}$$.

We proceed once more to calculate the x-ray pulse intensities required by the
adiabatic and $\pi$-pulse criteria. The results are listed in Table 7 for two
sets of XFELO parameters which mainly differ in their x-ray pulse duration and
pulse energy. We first address NCPT via STIRAP and investigate the population
transfer behaviour for different intensity scaling starting from the generic
values $I_{0,P/S}$ in Table 7. The transfer rate as a function of delay for
the parameter set of Ref. Qin et al. (2018a) is presented in Fig. 5(a). For a
large pulse intensity of $3\cdot I_{0,P/S}$, STIRAP appears to be robust and
100% transfer rates are reached for a plateau of time delay values on the
scale of the pulse duration. Once the intensity is decreased towards
$I_{0,P/S}$ and less, the plateau narrows visibly and the peak of 100% NCPT
shifts towards smaller pulse delays. Transfer rates of almost unity are still
reached for intensities of as low as $0.04\cdot I_{0,P/S}$
($\eta=$97.7\text{\,}\mathrm{\char 37\relax}$$).

The overall behaviour presented in Fig. 5(a) displays some interesting
features. For instance at $\Delta\tau=$0.01\text{\,}\mathrm{ps}$$ and an
intensity scaling of 1, the transfer rate is almost vanishing. Moreover, the
transfer rates for large time delays between pulses saturate at different
values for different intensities. To understand better this behaviour, the
transfer rate as a function of intensity $\eta\left(I\right)$ is calculated
for three different pulse delays $\Delta\tau$ marked by bullet points in Fig.
5(a). The numerical results are shown in Fig. 5(b). For small (i) and large
(iii) delay between pulses, $\eta\left(I\right)$ shows an oscillatory
behaviour. However, due to the large pulse overlap, transfer rates of unity
can be reached for small delays, while for largely delayed pulses transfer
rates of only $92.5\text{\,}\mathrm{\char 37\relax}$ are reached. That is
because the Stokes pulse does not affect the system in this case. Depending on
the pump pulse intensity, the population is either completely pumped to
$\left|3\right\rangle$ ($\pi$-pulse) or stays in the ground state
($2\pi$-pulse) or it is only partially pumped. In case of a single $\pi$-pulse
configuration the population is only pumped to the intermediate state
$\left|3\right\rangle$ from which it subsequently decays. Due to the branching
ratio of the in-band transition, only $92.5\text{\,}\mathrm{\char 37\relax}$
of the nuclei can be promoted to the isomeric state.

In case of an optimal robust STIRAP delay $(\Delta\tau\approx t_{\text{pul}})$
(ii), $\eta\left(I\right)$ shows the oscillatory behaviour only for small
intensities. Once $I_{0,P/S}$ is exceeded, the oscillation stops and the
transfer rate saturates at $\approx$100\text{\,}\mathrm{\char 37\relax}$$
which then corresponds to robust STIRAP. We note that the numerical results
for the XFELO parameters from Ref. Agapov (2021) show a similar behaviour. For
this case, the transfer rates drop below $99\text{\,}\mathrm{\char 37\relax}$
at approximately $0.1\cdot I_{0,P/S}$. We summarize the most important results
for both XFELO parameter sets in Tab. 7.

We now turn to the case of NCPT via two subsequent $\pi$-pules. Figure 5(c)
shows the transfer rate $\eta\left(\tau\right)$ for both XFELO parameter sets.
For both cases, the transfer rates can be large, and decrease for large delay
times. The reason for this is the short half-life of the intermediate state
due to the open IC channel of state $\left|3\right\rangle$. This decay channel
affects the population transfer already on ps-time scale. Therefore, the total
population in the isomeric state decreases due to losses from
$\left|3\right\rangle\rightarrow\left|1\right\rangle$ and re-pumping from
$\left|2\right\rangle\rightarrow\left|3\right\rangle$ and subsequent decay.
This effect is also the reason why the set Qin et al. (2018a) achieves
slightly higher transfer rates, since it operates on a shorter timescale
compared to set Agapov (2021). In Fig. 5(d) one can observe the exponential
decay due to IC during the population transfer. In this sequence the small
slope in the population of $\left|2\right\rangle$ and $\left|3\right\rangle$
corresponds to an exponential decay/gain since for small times the exponential
factor approaches to $e^{\pm\Gamma t}\approx 1\pm\Gamma t$.

We note here that also for this scenario, the required laser intensities are
very large and correspond to very small focal spots. For instance, the pump
pulse requires a beam waist of $2.2\text{\,}\mathrm{nm}$
($3.9\text{\,}\mathrm{nm}$) for the parameters in Ref. Qin et al. (2018a)
(Ref. Agapov (2021)) to fulfill the $\pi$-pulse criterion. To avoid different
laser beam waists we once more can choose a lower pulse energy for the Stokes
field. The very tight focusing also means that just a small volume in the
solid-state sample is addressed by the laser pulses. However, the large dopant
density of the sample leads to a number of irradiated nuclei comparable to the
ion beam case. Considering the sample used in Ref. Masuda et al. (2019)
(thickness $D=$0.2\text{\,}\mathrm{mm}$$, diameter
$L=$0.4\text{\,}\mathrm{mm}$$, dopant number $N=$6.3\text{\times}{10}^{14}$$)
the number of irradiated nuclei can be estimated under the assumption of a
disk-shaped sample and a homogeneous dopant distribution as
$N_{\text{irradiated}}=N\frac{w^{2}}{L^{2}}$. For a beam waist of
$2.2\text{\,}\mathrm{nm}$ ($3.9\text{\,}\mathrm{nm}$) one can expect to
irradiate up to $1.9\text{\times}{10}^{4}$ ($6\text{\times}{10}^{4}$) thorium
atoms.

##  V Conclusions

The second nuclear excited state of 229Th opens the possibility to indirectly
reach the isomeric state at 8 eV via x-ray pumping. We have investigated this
possibility following two directions of study. First, we have focused on the
nuclear transition rates which characterize the first three levels of the
229Th nucleus. These values have been deduced within a nuclear structure
model, and discussed in the context of the few available experimental data
values. Our analysis points out at inconsistencies between the central
experimental values deduced from experiments, allowing to identify the error
bar interval which can accommodate all measured data. Our findings summarized
in Table 3 will be useful for future experiments employing x-ray pumping.

Second, we have studied NCPT for isomer population in x-ray quantum optics
schemes involving STIRAP and $\pi$-pulses configurations. These schemes are
experimentally challenging and typically require large pulse intensities. From
the three possible setups addressed, our simulations have identified the GF
scenario with two UV laser beams interacting with relativistically accelerated
229Th ions as the most promising one. Our results show that NCPT with two
subsequent $\pi$-pulses could be implemented with less experimental effort
since the required intensities are smaller than in the STIRAP case. However,
the $\pi$-pulse excitation scheme necessitates almost full resonance which is
usually not provided in storage ring experiments due to the ion beam energy
spread. Thus, STIRAP yields a robust alternative for coherent excitation
although a detuning is present. Due to an advantageous in-band branching
ratio, direct x-ray pumping to the 29.19 keV state might prove to be
competitive, provided strong XFEL pulses can be used at this energy.

###### Acknowledgements.

We thank Brenden S. Nickerson for fruitful discussions. AP gratefully
acknowledges funding from the DFG in the framework of the Heisenberg Program.
This work is 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). NM
gratefully acknowledges the support by the Bulgarian National Science Fund
(BNSF) under contract No. KP-06-N48/1.

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