# Fluorescence calorimetry of an ion crystal

Marvin Gajewski<EMAIL_ADDRESS>Theoretische Physik,
Universität des Saarlandes, 66123 Saarbrücken, Germany  Wenbing Li
<EMAIL_ADDRESS>QUANTUM, Institut für Physik, Johannes Gutenberg
Universität Mainz, 55099 Mainz, Germany  Helmholtz-Institut Mainz, 55099
Mainz, Germany  Sebastian Wolf  QUANTUM, Institut für Physik, Johannes
Gutenberg Universität Mainz, 55099 Mainz, Germany  Walter Hahn  Helmholtz-
Institut Mainz, 55099 Mainz, Germany  Institute for Quantum Optics and Quantum
Information of the Austrian Academy of Sciences, Innsbruck, Austria  Christoph
E. Düllmann  Helmholtz-Institut Mainz, 55099 Mainz, Germany  PRISMA Cluster of
Excellence, Johannes Gutenberg University Mainz, 55099 Mainz, Germany
Department Chemie - Standort TRIGA, Johannes Gutenberg University Mainz, 55099
Mainz, Germany  GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291
Darmstadt, Germany  Dmitry Budker  QUANTUM, Institut für Physik, Johannes
Gutenberg Universität Mainz, 55099 Mainz, Germany  Helmholtz-Institut Mainz,
55099 Mainz, Germany  PRISMA Cluster of Excellence, Johannes Gutenberg
University Mainz, 55099 Mainz, Germany  Department of Physics, University of
California, Berkeley, California 94720-7300, USA  Giovanna Morigi
Theoretische Physik, Universität des Saarlandes, 66123 Saarbrücken, Germany
Ferdinand Schmidt-Kaler  QUANTUM, Institut für Physik, Johannes Gutenberg
Universität Mainz, 55099 Mainz, Germany  Helmholtz-Institut Mainz, 55099
Mainz, Germany  PRISMA Cluster of Excellence, Johannes Gutenberg University
Mainz, 55099 Mainz, Germany

###### Abstract

Motivated by the challenge of identifying intruder ions in a cold ion crystal,
we investigate calorimetry from emitted fluorescence light. Under continuous
Doppler cooling, the ion crystal reaches a temperature equilibrium with a
fixed level of fluorescence intensity and any change in the motional energy of
the crystal results in a modification of this intensity. We theoretically
determine the fluorescence rate of an ion crystal as a function of the
temperature, assuming that laser light is scattered along a two-level
electronic transition, which couples to the crystal’s vibrations via the
mechanical effects of light. We analyse how the heat dissipated by collisions
of an incoming intruder ion alters the scattering rate. We argue that an
energy change by an incoming 229Th10+ can be unambiguously detected within 50
$\mu$s via illuminating 50 ions out of a 103 ion crystal. This method enables
applications including capture and spectroscopy of charged states of thorium
isotopes and investigation of highly charged ions.

##  I Introduction

Cold single trapped ions and ion crystals are well established workhorses in
quantum optics, precision spectroscopy and quantum information. Ordered
structures emerge from the combination of trapping forces and Coulomb
repulsion at laser-cooling temperatures and realize Wigner crystals in the
laboratory Dubin and O’Neil (1999). Due to the Coulomb interactions and the
rarefied charge distribution, the thermodynamic properties are expected to
exhibit peculiar features Morigi and Fishman (2004); Baughman  _et al._
(2000). Due to the isolation from the environment, moreover, measurements of
temperatures, and in general, of the thermodynamic functions cannot be
implemented as in normal solids. This question acquires particular relevance
for calorimetric studies.

Calorimetry, namely, the determination of a transient change in temperature,
might feature various application cases including the detection of sudden
heating events. The latter may be caused by collisions or by the injection of
an ion or an ion bunch from an external source, coined here intruder ions.
Apart from such applications, calorimetry of ion crystals may allow
investigating the distribution and thermalization of impact energy over
several motional degrees of freedom.

Similar studies have been performed for a single trapped ion. In the so-called
Doppler-recooling technique, time-dependent fluorescence emitted by a single
trapped ion is recorded for characterizing the heating in the trap during a
time interval when cooling light is turned off Wesenberg  _et al._ (2007). In
a similar fashion, one may sense large coherent motional excitation after
rapid ion shuttles via measuring the Doppler-recooling time Huber  _et al._
(2008). Recently, the fluorescence intensity emitted by a laser cooled crystal
was numerically simulated for revealing information about intruding ions which
heat up the crystal Poindron  _et al._ (2021).

In this work, we develop a theoretical model that allows us to systematically
connect the mean vibrational energy of the crystal with the intensity of the
scattered light. By means of our theoretical model we identify the optimal
working conditions where the resonance fluorescence allows for a calorimetric
measurement. Our starting point is the master equation for the internal and
external degrees of freedom of $N$ ions forming a three dimensional ion
crystal (section II). In the regime in which the motion can be described by
semiclassical equations Stenholm (1986); Morigi and Eschner (2001), we
calculate the fluorescence emitted as a function of the temperature (section
II.4). We study the dependence on laser detuning and laser power and identify
optimal parameters for thermometry. We then apply this knowledge to reveal the
injected energy from the change in the fluorescence (section III.2). As a
specific application of fluorescence calorimetry of an ion crystal, we
consider nondestructive detection of a single thorium ion stochastically
entering (intruding) into an ion crystal of $\sim 10^{3}$ 40Ca+ ions. This is
relevant for the "Trapped And sympathetically Cooled Thorium Ions in Calcium
crystals” (TACTICa) experiment (Fig. 1) Groot-Berning  _et al._ (2019); Stopp
_et al._ (2019); Haas  _et al._ (2020a), aiming at high-precision laser
spectroscopic studies of thorium isotopes including the exotic low-energy
nuclear isomer in 229Th von der Wense  _et al._ (2016); Thielking  _et al._
(2018). This unique system is of interest for the study of effects unexplained
by the standard model of particle physics Flambaum (2006); Peik  _et al._
(2021) and for nuclear physics Beeks  _et al._ (2021).

While 229Th is a long-lived isotope and is available in macroscopic
quantities, the 229mTh is not Seiferle  _et al._ (2017); it is most readily
available as the $\alpha$-decay daughter of 233U Haas  _et al._ (2020b). This
isotope’s $\alpha$ decay populates the isomeric state in about 2$\%$ of all
cases. The decay produces ions with 84 keV kinetic energy Haas  _et al._
(2020a), which needs to be removed for the successful trapping of the ion. In
the TACTICa setup, this is achieved by putting the 233U recoil ion source at a
potential of about -84/$q$ keV, against which the daughter ions emitted in
charge state $q$\+ run up to arrive at the trap at near rest. Successful
trapping relies on the ion entering the trap and being reflected by the exit
end cap, with the entrance side being closed before the ion round trip time
has elapsed. Taking a realistic case example of thorium ions with charge
states $q=1,\ldots,10$, we determine the dissipated energy of the intruding
ion in an ion crystal of calcium (section III.3). We show that ion crystal
fluorescence calorimetry will provide a non-destructive signal indicating
intrusion of a 229Th ion, which can be used for triggering electrostatic
closure of the trap.

Figure 1: Schematic of the setting considered. A thorium ion of charge state
$q$ and velocity $v_{\text{i}}$ enters a Paul trap containing a laser-cooled
Ca+-ion crystal and decelerates. The crystal heats up, which leads to a change
of intensity of the laser-induced fluorescence. The black solid circle in the
left part of the fluorescence image of the ion crystal indicates a
successfully trapped thorium ion.

##  II Measuring the temperature of large crystals with light

In this section we derive the dependence of the fluorescence intensity on the
thermal energy of an ion crystal. This could in principle allow one to measure
the temperature of an ion crystal. Moreover, it can be also applied to
measure, e.g., the energy released by interaction with an intruder ion. For
the purpose of developing a theory for performing thermometry using light, in
this section, we discuss a simplified model, which can be systematically
derived from the full quantum mechanical master equation of the ion crystal
interacting with light. We argue that, despite the large number of degrees of
freedom, a model based on only a few degrees of freedom allows one to describe
most experimental scenarios of Doppler-cooled ion crystals.

###  II.1 Model and basic assumptions

We first start from the full quantum mechanical model. We consider the motion
of the ion crystal composed of $N$ ions, which here can range from one to
thousands Dubin and O’Neil (1999). Let ${\bf p_{j}},{\bf r_{j}}$ be the
canonically-conjugated momentum and position of ion $j$ and $M$ be the ion
mass (without loss of generality we assume the ions to have equal mass, see
Sec. II.5), the mechanical energy is described by the Hamiltonian

$H_{\rm crystal}=\sum\frac{{\bf p_{j}}^{2}}{2M}+V_{\rm pot}({\bf
r_{1}},\ldots,{\bf r_{N}})\,,$ (1)

where the potential energy $V_{\rm pot}$ includes the repulsive Coulomb
interaction and the trap potential. For sufficiently cold ion crystals, the
Hamiltonian (1) can be reduced to describe the energy of a harmonic crystal,
whose eigenmodes are the harmonic collective vibrations of the ions about the
equilibrium positions of potential $V_{\rm pot}$.

The motion couples with light via the mechanical effects of atom-photon
interactions. Laser light drives an optical transition of the ions, with
ground state $|g\rangle$ and excited state $|e\rangle$. Let ${\bf d}$ be the
dipole moment and $\gamma$ the spectral linewidth of the transition. We denote
by $\Delta=\omega_{0}-\omega_{L}$ the detuning of the laser frequency
$\omega_{L}$ (for Doppler cooling $\omega_{L}<\omega_{0}$ Wineland and Itano
(1979); Stenholm (1986); Eschner  _et al._ (2003)). The total Hamiltonian
reads $H=H_{\rm crystal}+H_{\rm ions}+H_{\rm AL}$, where $H_{\rm
ions}=\sum_{j}\hbar\Delta|e\rangle_{j}\langle e|$ and the laser-dipole
interactions –in the electric-dipole approximation– take the form:

$H_{\rm AL}=\sum_{j}\frac{\hbar\Omega({\bf
r_{j})}}{2}[\sigma_{j}^{\dagger}\exp(-i\mathbf{k\cdot
r_{j}})+\sigma_{j}\exp(i\mathbf{k\cdot r_{j}})]\,.$ (2)

In this description the laser is a classical travelling wave with wave vector
$\mathbf{k}$ and position-dependent Rabi frequency $\Omega({\bf r_{j}})=-{\bf
d}\cdot{\bf E_{L}({\bf r_{j}})}/\hbar$, where ${\bf E_{L}}({\bf r_{j}})$ is
the slowly-varying amplitude of the laser electric field. Here,
$\sigma_{j}=|g\rangle_{j}\langle e|$ is the lowering operator for the internal
electronic transition of ion $j$ and $\sigma_{j}^{\dagger}$ is its adjoint.

The dynamics of internal and external degrees of freedom of the crystal
results from the interplay between the coherent dynamics, governed by the
Hamiltonian $H$, and the incoherent one due to the spontaneous decay. It is
described by the master equation for the density matrix $\varrho$:

$\frac{\partial\varrho}{\partial
t}=-\frac{i}{\hbar}\left[H,\varrho\right]+L(\varrho)\,,$ (3)

where
$L\varrho=\gamma/2\sum_{j}[2\sigma_{j}\tilde{\varrho}\sigma_{j}^{\dagger}-\sigma_{j}^{\dagger}\sigma_{j}\varrho-\varrho\sigma_{j}^{\dagger}\sigma_{j}]$
is the superoperator for the spontaneous decay, which includes the diffusion
due to the stochastic mechanical forces associated with the emission of a
photon with wave vector ${\bf k^{\prime}}$, namely,
$\tilde{\varrho}\propto\sum_{j}\int{\rm d}{\bf k^{\prime}}f({\bf
k^{\prime}})e^{i\mathbf{k^{\prime}\cdot r_{j}}}\rho e^{-i\mathbf{k\cdot
r_{j}}}$. The probability density $f({\bf k^{\prime}})$ weights the process
and depends on the dipole’s polarization. We here discard collective effects
in the spontaneous emission as the interparticle distances between ions are of
the order of several optical wavelengths.

In this work, we determine the rate of photon emission,

$S=\gamma\sum_{j}{\rm Tr}\\{|e\rangle_{j}\langle e|\varrho(t)\\}\,,$ (4)

and its dependence on the temperature of the crystal when the crystal is in a
thermal-equilibrium state.

###  II.2 Experimental parameters

Modelling the dynamics of resonance fluorescence requires taking into account
the coupled equations of internal and external degrees of freedom of the ions
forming the crystal. For a large crystal this is a formidable task. This
complexity can be significantly reduced in the case of Doppler cooling, and
more generally, when the characteristic frequency scales are orders of
magnitude larger than the recoil frequency $\omega_{R}=\hbar k^{2}/(2M)$.
These assumptions are valid for the parameters of typical experimental ion-
trap settings. For example, in the TACTICa experiment Groot-Berning  _et al._
(2019); Stopp  _et al._ (2019); Haas  _et al._ (2020a) the crystals are
composed of 40Ca+ trapped ions. The secular trap frequencies
$\omega_{x,y,z}/(2\pi)$ are lower than 0.5 MHz and determine the smallest
characteristic frequency of the motion (provided the system is sufficiently
far away from structural instabilities Birkl  _et al._ (1994); Dubin and
O’Neil (1999); Fishman  _et al._ (2008)). The natural linewidth of the S1/2 to
P1/2 transition in 40Ca+ near 397 nm is $\gamma/(2\pi)$ = 21.6 MHz which
corresponds to a lifetime of 6.904(26) ns Hettrich  _et al._ (2015). The
recoil frequency is $\omega_{R}=2\pi\times 31.5$ kHz, thus it is one order of
magnitude smaller than the characteristic frequencies determining the crystal
dynamics. The time scale $\tau_{cool}$ of the mechanical effects of light is
of the order of 100 $\mu$s for a single ion driven below saturation, see
Appendix C. When $N$ ions of the crystals are driven below saturation, then
the cooling time scale is of the order of $\tau_{cool}/N$ but still
significantly longer than the lifetime of the excited state Wunderlich  _et
al._ (2005). This time scale separation is the regime at the basis of our
treatment.

###  II.3 Time-scale separation

This parameter regime allows us to make some simplifications based on the
following observations. In the first place, the internal degrees of freedom
evolve over a faster time scale than the time scale $\tau_{cool}$ of the
external degrees of freedom. In this limit internal and external degrees of
freedom are not entangled to good approximation, and the density matrix of the
ions is separable $\varrho\approx\rho\otimes\mu$, with $\rho$ and $\mu$ the
density matrices for the internal and the external degrees of freedom,
respectively. We can hence solve the equations of motion for $\rho$ over a
time scale such the internal degrees of freedom relax to the steady state,
while the external degrees of freedom do not have time to evolve. Let
$\\{|{\cal E},\ell\rangle\\}$ be a basis of eigenstates of $H_{\rm crystal}$
with mechanical energy $\cal E$, where $\ell$ is a set of quantum numbers
which uniquely determines the quantum state. We now write the optical Bloch
equations for $\rho$ assuming that only one ion (say, ion $j$) of the crystal
is coupled to the laser light and that the spatial displacement of this ion
couples to all normal modes. We use the notation

$\rho_{pq}(\mathcal{E},\mathcal{E}^{\prime})\equiv\sum_{\ell}\langle
p_{j},\mathcal{E},\ell|\varrho|q_{j},\mathcal{E}^{\prime},\ell\rangle\,,$

with $p,q=e,g$, and obtain the corresponding equations of motion:

$\displaystyle\frac{d}{dt}\rho_{ee}(\mathcal{E},\mathcal{E}^{\prime})$
$\displaystyle=-\gamma\rho_{ee}(\mathcal{E},\mathcal{E}^{\prime})$ (5)
$\displaystyle+\frac{i\Omega}{2}\sum_{\mathcal{E}_{1}}\left(T(\mathcal{E},\mathcal{E}_{1})\rho_{ge}(\mathcal{E}_{1},\mathcal{E}^{\prime})-c.c.\right)$
$\displaystyle\frac{d}{dt}\rho_{eg}(\mathcal{E},\mathcal{E}^{\prime})$
$\displaystyle=-\left(\frac{\gamma}{2}-i\Delta-i\frac{\mathcal{E}-\mathcal{E}^{\prime}}{\hbar}\right)\rho_{eg}(\mathcal{E},\mathcal{E}^{\prime})$
(6)
$\displaystyle+\frac{i\Omega}{2}\sum_{\mathcal{E}_{1}}T(\mathcal{E},\mathcal{E}_{1})\left[\rho_{gg}(\mathcal{E}_{1},\mathcal{E}^{\prime})-\rho_{ee}(\mathcal{E},\mathcal{E}^{\prime})\right]\,,$

where
$\rho_{ge}(\mathcal{E},\mathcal{E}^{\prime})=\rho_{eg}(\mathcal{E}^{\prime},\mathcal{E})$,
and the coefficient

$T(\mathcal{E},\mathcal{E}^{\prime})=\langle\mathcal{E}|\exp(i{\bf k}\cdot{\bf
r_{j}})|\mathcal{E}^{\prime}\rangle\ $ (7)

is the probability amplitude of a transition from the mechanical state
$|\mathcal{E}^{\prime}\rangle$ to the mechanical state $|\mathcal{E}\rangle$
by absorption of a photon. Moreover, under the time-scale-separation ansatz,
we neglect the effect of the recoil of spontaneous emission on the motion,
which gives rise to diffusion Stenholm (1986); Morigi and Eschner (2001).
Therefore, the number of coupled equations to solve is finite. Since
$\gamma\tau_{\rm cool}\gg 1$, we can find a time scale $\Delta t$ such that
$\Delta t\gg 1/\gamma$ and $\Delta t\ll\tau_{\rm cool}$. Hence, we can neglect
a change of the motional state over the time scale $\Delta t$ and determine
the internal state by setting
$\dot{\rho}_{ee}^{(SS)}(\mathcal{E},\mathcal{E}^{\prime})=\dot{\rho}_{eg}^{(SS)}(\mathcal{E},\mathcal{E}^{\prime})=0$.
The resonance fluorescence signal over the time scale $\Delta t$ is

$S=\gamma\sum_{\mathcal{E}}\rho_{ee}^{(SS)}(\mathcal{E},\mathcal{E})p(\mathcal{E})\,,$
(8)

where $p(\mathcal{E})$ is the probability that the crystal has mechanical
energy $\mathcal{E}$.

###  II.4 Semiclassical limit

We now perform a further simplification that reduces the number of equations
to solve to two. This simplification relies on (i) a coarse-graining in
energy, which can be made in the limit where the ion-crystal temperature $T$
is $k_{B}T\gg\hbar\omega_{R}$ and (ii) the weak binding regime, when the
spectral linewidth $\gamma\gg{\rm max}_{n}\omega_{n}$, where $\omega_{n}$ are
the vibrational frequencies of the crystal and $n$ labels the mode Stenholm
(1986); Morigi and Eschner (2001), see Appendix B. The equations are
determined for an average energy $\mathcal{E}$ over the coarse grained energy
scale $\delta E$ such that $k_{B}T\gg\delta E\gg\hbar\omega_{R}$ Morigi and
Eschner (2001). At the leading order in the expansion in the small parameter
$\hbar\omega_{R}/k_{B}T$, the solution of the corresponding optical Bloch
equations reads

$\displaystyle\rho_{\text{ee}}^{(SS)}(E)=\frac{\Omega^{2}/2}{\Delta_{\text{eff}}(E)^{2}+\gamma^{2}/4+\Omega^{2}}\,,$
(9)

where

$\displaystyle\Delta_{\text{eff}}(E)=\frac{E}{\hbar}+\Delta\,,$ (10)

and $E=E_{e}-E_{g}$ is the energy gap between the ground and excited states
coupled by the absorption of a laser photon. In the weak-binding limit these
energies can be written as:

$\displaystyle E_{g}=$ $\displaystyle\frac{\textbf{p}^{2}}{2m}+V(\textbf{r}),$
$\displaystyle E_{e}=$
$\displaystyle\frac{(\textbf{p}-\hbar\textbf{k})^{2}}{2m}+V(\textbf{r})+\hbar\Delta\,.$

We emphasize that these expressions for the energy are valid when one can
neglect the spatial displacement of the ions during a photon-scattering event.
Let us summarize when this is a valid assumption. For this purpose, we
consider a single ion in one dimension, and remark that the argument can be
straightforwardly extended to the case when the motion is in three dimensions
and the scattering ion is embedded in a crystalline structure. For a single
ion in a trap at frequency $\omega$, the spatial displacement after a photon-
scattering event is given by $\Delta x\sim p/(M\gamma)$, where
$p\sim\sqrt{k_{B}TM}$. This is much smaller than the oscillation amplitude
$\mathcal{A}_{x}\sim\sqrt{k_{B}T/(M\omega^{2})}$, when $\omega\ll\gamma$
Stenholm (1986). As a result, in this regime the population of the excited
state solely depends on the ion-momentum distribution at leading order in an
expansion in $\omega/\gamma$ (for an ion crystal, the small parameter is
$\omega_{n}/\gamma$).

Figure 2: Fluorescence $S=\gamma\rho_{ee}$ of an ion crystal when a single ion
of the crystal is illuminated. The plot displays $\rho_{ee}$, the average
probability be in the excited state of the illuminated ion as a function of
the temperature. The left and central panels are color plots of $\rho_{ee}(T)$
as a function of the temperature $T$ and of the detuning $\Delta$ (in units of
$\gamma$) for a fixed saturation parameter (a) $s=0.25$ and (b) $s=1$ [Note
that, as we vary $\Delta$, we accordingly vary $\Omega$ in order to keep $s$
constant, see Eq. (12)]. Panel (c) displays $\rho_{ee}(T)$ as a function of
$T$ for $\Delta=\gamma/2$ and for saturation parameters $s=0.25$, $s=0.5$,
$s=1$, $s=2$ and $s=5$ (lowest to highest curve). The behavior for $s\ll 1$
applies also to the case in which $N$ ions of the crystal are illuminated
below saturation, see Sec. II.5.

In the following we assume that the ion phase-space function is a Boltzmann
distribution, $\exp(-\beta H_{\rm crystal})$, with $\beta^{-1}=k_{B}T$ and $T$
the temperature. Then, the total excited state population is given by the
expression Wineland and Itano (1979)

$\displaystyle\rho_{\text{ee}}(T)$
$\displaystyle=\int\rho_{\text{ee}}^{(SS)}(\bar{E})\frac{e^{-\beta H_{\rm
crystal}}}{\mathcal{Z}}d^{3N}\textbf{p}d^{3N}\textbf{r}\,,$ (11)

where $\mathcal{Z}$ is the partition function. We insert Eq. (9) into Eq.
(11), discard higher orders in $\omega_{R}$, consistent with the validity of
the semiclassical expansion, and rescale the variables using the saturation
parameter $s$, which is defined as

$s=\frac{\Omega^{2}/2}{\gamma^{2}/4+\Delta^{2}}\,.$ (12)

Since the population of the excited state of ion $j$ solely depends on the
momentum of the same ion, all other degrees of freedom can be integrated out
and Eq. (11) is then brought to the form

$\displaystyle\rho_{\text{ee}}(T)$
$\displaystyle=\sqrt{\frac{1}{\pi}}\int_{-\infty}^{\infty}\frac{s^{2}e^{-p_{z}^{2}}}{1+2s^{2}+\frac{(\zeta
p_{z})^{2}-2\Delta\zeta p_{z}}{\gamma^{2}/4+\Delta^{2}}}dp_{z}\,,$ (13)

where $\zeta=\sqrt{\omega_{R}/(\hbar\beta)}$. Without loss of generality, in
the above equation we assumed that laser light propagates along the $z$
direction. From expression (13) we can directly find the resonance
fluorescence: $S(T)=\gamma\rho_{\text{ee}}(T)$. Figures 2 (a) and (b) display
Eq. (13) for two values of the saturation parameter $s$ and as a function of
the temperature $T$ and of the detuning $\Delta$. At low saturation (left
panel) the resonance fluorescence signal reaches a maximum as a function of
$T$. This maximum depends on the detuning and moves towards higher values of
$T$ with increasing $\Delta$. On the contrary, at saturation (central panel)
the resonance fluorescence signal is a monotonously decreasing function of
$T$. The temperature dependence of the resonance fluorescence at fixed
$\Delta$ and different saturation parameters $s$ is illustrated in the right
panel.

We now use a simplified model in order to estimate the temperature, at which
the fluorescence of the ion crystal is maximal for a given set of parameters.
We first observe that, at this temperature, the integral in Eq. (13) is
maximal. The integral can be interpreted as the overlap between the Gaussian
distribution and the ion’s absorption spectrum. In what follows, we
approximate the Gaussian, $\exp(-p_{z}^{2})$, with a step function for $p_{z}$
in $(-1,1)$. Then, the maximum temperature parameter $\zeta_{\text{max}}$
satisfies

$\displaystyle\arctan$
$\displaystyle\left(\frac{\Delta+\zeta_{\text{max}}}{\bar{s}}\right)-\arctan\left(\frac{\Delta-\zeta_{\text{max}}}{\bar{s}}\right)$
$\displaystyle=$
$\displaystyle\frac{2\zeta_{\text{max}}(\sqrt{\bar{s}}^{3}+\sqrt{\bar{s}}(\Delta^{2}+\zeta_{\text{max}}^{2}))}{\Delta^{4}+2\Delta^{2}({\bar{s}}-\zeta_{\text{max}}^{2})^{2}+({\bar{s}}+\zeta_{\text{max}}^{2})^{2}}$
(14)

with

${\bar{s}}=(1+2s^{2})(\gamma^{2}/4+\Delta^{2}).$ (15)

This approximation generally gives a good estimate of $T_{\text{max}}$. The
furthest deviation from exact numerical results is in the regime $s\approx 0$,
where, e.g., for $\Delta=\gamma/2$ we obtain
$T_{\text{max}}^{\text{num}}=10.58\,$mK and
$T_{\text{max}}^{\text{approx}}=9.64\,$mK.

###  II.5 Discussion

We now comment on the validity of our model and on the limits of its
applicability. The model has been derived under the assumption that the
relevant electronic degrees of freedom are two. In this limit we could find a
closed expression for the fluorescence as a function of the temperature. This
strictly holds when a single ion of the crystal is illuminated. Nevertheless,
from this result one can extract the resonance fluorescence for the case when
the ion crystal is illuminated, as long as the laser drives the ions below
saturation. In that case, given the saturation parameter $s$ for a single ion,
our model requires that $Ns\ll 1$. 111When the laser drives the ions well
below saturation, in fact, one can reduce the problem to a two-level systems,
consisting of the state $|G\rangle$, where all the ions are in the ground
state, and the Dicke state $|E\rangle$, that is coupled to $|G\rangle$ by
absorption of a laser photon and that contains one (delocalized) electronic
excitation. Then, the resonance fluorescence over the full space is determined
by the population of $|E\rangle$ as given by Eq. (9) after replacing
$\Omega\to\sqrt{N}\Omega$. We remark that this is strictly correct provided
that the laser parameters are homogeneous across the crystal and that $Ns\ll
1$, with $s$ given in Eq. (12), so that, on average, only one ion can be
excited at a time Morigi and Eschner (2001, 2003). The ions can be considered
as indipendent scatterers also at moderately larger intensities, as long as
the temperature is sufficiently large. The extension to the case in which the
laser parameters are not homogeneous can be performed along the lines of the
study in Ref. Schmit  _et al._ (2021): In this case $\rho_{ee}$, Eq. (9),
becomes a function of the position inside the crystal and the resonance
fluorescence is found by averaging $\rho_{ee}$ also over the spatial
distribution of the laser parameter across the crystal. We remark that our
model does not capture the general case, when the internal state of the
emitters can be multiply excited at the same time and multiple absorptions of
laser photons create correlations between internal and external degrees of
freedom. In this regime, in fact, the set of optical Bloch equations that
shall be solved is of the order of $2^{N}$ and we expect that the dependence
on the mean energy and on the saturation parameter takes a different
functional form.

Our model neglects the ions’ displacement during the scattering process and
thus requires that the lifetime of the excited state is much smaller than the
oscillation frequency, more specifically, $\omega_{n}\ll\gamma$. This is the
so-called weak binding limit Wineland and Itano (1979), which is consistent
with a semiclassical description provided that the recoil frequency is smaller
than the linewidth Stenholm (1986); Morigi and Eschner (2001). In this limit,
the result does not depend on the mode spectra, and thus not on the ion masses
within the crystal, as long as the modes are at the same temperature and the
scattering ions are of the same species.

We finally comment on whether the approximations adopted here describe the
setup of TACTICa. The semiclassical limit requires temperatures
$T_{\text{min}}=\hbar\omega_{R}/k_{B}$. For the parameters of the TACTICa
experiment, $T_{\text{min}}\sim 1\,\mu$K, which is three orders of magnitude
smaller than the Doppler cooling temperature. The weak-binding regime requires
that $\omega_{n}\ll\gamma$. According to our estimates, this is approximately
fulfilled since $\max\omega_{n}\sim\gamma/3$.

##  III Detection scenario of an intruder ion

We consider the scenario of identifying a sudden addition of motional energy
to the ion crystal experiencing Coulomb collisions of an intruder ion. While
the ion crystal is kept under continuous Doppler cooling, this impact will
change the fluorescence rate of the emitted light which is recorded, see Fig.
1. We discuss the example of a 40Ca+ ion crystal under Doppler cooling on the
S1/2 to P1/2 dipole-allowed transition near 397 nm and a single incoming
229Thq+ ion as the intruder. In this regime the motion can be described by the
semiclassical model, see Sec. II.5. Let the thorium charge state $q$ range
from +1 to about +10 elementary charges using the uranium recoil source Haas
_et al._ (2020a). The analysis of the detection scenario is based on three
main steps: First, we estimate the energy increase of the crystal from the
impact by the intruder, then we use the results of the previous Sec. II.4 to
calculate the corresponding change of fluorescence emission and finally, we
discuss the feasibility of the proposed detection scheme taking into account
the experimental parameters. Our results, as outlined below, confirm a
recently published numerical simulation Poindron  _et al._ (2021). The
analytical solution, moreover, may pave the way for an understanding of heat
capacity in such model systems and intruder-induced heating effects.

###  III.1 Energy increase of the crystal due to an intruder ion

We model the deceleration of a Thq+ ion in a Ca+-ion crystal using Poth
(1990); Zwicknagel  _et al._ (1999)

$\frac{dE}{dx}\approx-\frac{q^{2}e^{2}}{4\pi\epsilon_{0}}\frac{\nu_{\text{p}}^{2}}{v^{2}(x)}\Lambda_{\text{C}},$
(16)

where $E$ is the kinetic energy of the Thq+ ion at velocity $v(x)$, $x$ is the
distance traveled in the ion crystal, $\nu_{\text{p}}$ is the plasma frequency
of the ion crystal and $\Lambda_{\text{C}}$ is a constant usually referred to
as Coulomb logarithm (see Appendix A). The plasma frequency $\nu_{\text{p}}$
describes the characteristic time scale for the response of the ion crystal to
external perturbations and can be expressed as
$\nu_{\text{p}}=\sqrt{e^{2}n_{\text{d}}/(\epsilon_{0}M)}$, where
$n_{\text{d}}$ is the density of the ions of mass $M$ in the ion crystal
(40$u$ for Ca+ ions). The expression in Eq. (16) is based on linear response
Poth (1990); Zwicknagel  _et al._ (1999) and was shown to provide reliable
estimates 222Expression (16) remains valid until the velocity of the thorium
ion becomes comparable to the mean thermal velocity of individual calcium
ions, but this limitation is of minor importance here. of the order of
magnitude by means of numerical simulations Bussmann  _et al._ (2006).

The voltage on the entrance-endcap electrode, through which the intruder ion
enters, is initially adjusted to just allow it entering and the potential on
the opposite electrode is set for reflection. The goal is to detect the
intruder ion and pulse the potential on the entrance-endcap electrode to close
the trap before the intruder ion escapes again after its round trip. Since the
energy transfer becomes more efficient for intruders at lower energy
[$dE/dx\propto 1/v^{2}(x)$, cf. Eq. (16)], the maximal energy deposited by the
thorium ion after a round trip in the crystal is when it would completely stop
after the round trip, an optimal case on which we focus now.

To estimate the characteristic energy of the above process, we obtain from Eq.
(16) $v(x)\approx(v_{\text{i}}^{4}-c_{0}x)^{1/4}$, where $v_{\text{i}}$ is the
initial velocity of the ${}^{229}\text{Th}^{q+}$ ion, $c_{0}\equiv
q^{2}e^{2}\nu_{\text{p}}^{2}\Lambda_{\text{C}}/(\pi\epsilon_{0}M_{\text{Th}})$,
and $M_{\text{Th}}=229$ u is the mass of ${}^{229}\text{Th}$. Assuming an ion
trajectory on the trap axis, the thorium ion is stopped when its initial
velocity is $v_{\text{i}2}\approx(2Lc_{0})^{1/4}$, where $L$ is the length of
the crystal in axial direction. Substituting $\nu_{\text{p}}\approx 3$ MHz and
$L\approx 1$ cm for the TACTICa experiment, we obtain $v_{\text{i}2}\approx
2.5\ 10^{2}$ m/s for $q=1$ corresponding to an energy of $E_{\text{i}2}\approx
75$ meV. For $q=10$, the energy increases to $E_{\text{i}2}\approx 0.75\,$eV
because $E_{\text{i}2}\propto q$. In case of a uranium $\alpha$-recoil ion
source, thorium ions are delivered at kinetic energies of 70 to 100 keV. For
efficient Coulomb coupling with the Ca crystal, we plan to set the source at
high attractive electrostatic potential for decelerating emitted Th ions Haas
_et al._ (2020a).

In the Coulomb crystal, ion motion is described by collective modes, such that
the absorbed energy is quickly distributed, faster than the Doppler cooling
dynamics. For the particular case of a single Th+ ion stopped after traversing
the ion crystal twice and the above parameters, we estimate a temperature
increase $\delta T\sim 10\,$mK, while for a Th10+ ion, we obtain $\delta T\sim
100\,$mK.

###  III.2 Change of fluorescence

We have investigated the dependence of fluorescence intensity on crystal
temperature in Sec. II.4. A convenient observable in order to measure the
change in fluorescence is the relative decrease, quantified by

$R(T_{i},\delta T)\equiv\frac{S(T_{i}+\delta T)-S(T_{i})}{S(T_{i})},$ (17)

where $S(T)=\gamma\rho_{ee}(T)$ is the measured intensity of fluorescence at
temperature $T$, and $T_{i}$ and $\delta T$ are the initial temperature of the
ion crystal and the temperature shift. We illuminate a section of 50 ions of a
$10^{3}$ ion crystal with a saturation per ion of $s=0.01$, such that $Ns<1$.
Taking into account the case example of a single Th10+ ion intruding a Coulomb
crystal at temperature $T_{i}=50\,$mK, with detuning $\Delta=15\,$MHz, then
the relative decrease of fluorescence is $R(T_{i},100\text{mK})=0.25$, see
Fig. 3. The change takes place over a time of about $100\,\mu$s, until the
deposited energy is removed by Doppler cooling over a timescale of $5$ ms. The
relative decrease of fluorescence $R(T_{i},10\text{mK})=0.04$ is less
significant for a singly charged Th intruder.

###  III.3 Efficiency of detection

The Ca crystal fluorescence is continuously observed by a photon counting unit
to discriminate the intruder event of a single Th10+ from detecting the
relative change of fluorescence of $R(T_{i},\delta T)$. Assuming a scattering
rate of $\rho_{ee}\tau$, taking into account an overall photon detection
efficiency of $\sim 5\%$, illuminating $50$ ions of 103-ion crystal, with a
relative fluorescence change of R=0.25, and employing a Poissonian counting
statistics, the tagging signal would exceed noise by $\sim$ 1.3 standard
deviations already within a counting time of 50$\,\mu$s, which is clearly
shorter as compared to the Doppler recooling time. In the case of a singly
charged intruder ion, however, the realtive change in fluorescence R is 0.04,
and the event detection with $\sim$ 1.2 standard deviation would require a
2$\,m$s counting time. Thus, for singly charged intruder ion, the necessary
detection time would be too long, as this time may possibly exceed the Doppler
recooling time when 50 ions are illuminated. An increase in illuminated ions
$N$ would go along with a better optical signal, but also with an even shorter
maximum observation time, because of the scaling of the recooling time by
$N^{-1}$. Therefore, for an efficient detection of a singly charged intruder
ion, we propose improving the detection optics e.g. using high NA objectives,
and the quantum efficiency of the detector, furthermore reducing the initial
cooling limit of the crystal to enlarge R.

Note, that tagging of small bunches of Ar+ ions at 4 keV has been
demonstrated, based on their fly-by thought a narrow electrode structure and
the detection of an induced voltage. For bunches of about 300 ions, the
detection signal exceeds the amplifier noise level Raecke  _et al._ (2018).
Thus, our method features an alternative for loading Paul traps
deterministically at the single particle level with intruder ions, that might
be stemming either from EBIT sources Micke  _et al._ (2018); Andelkovic  _et
al._ (2015) or from radioactive sources Haas  _et al._ (2020a) with a
perspective of precision quantum logic spectroscopy using exotic matter Micke
_et al._ (2020).

Figure 3: Relative decrease of fluorescence intensity $R(T_{i},\delta T)$ [cf.
Eq. (17)] over an increase of temperature $\delta T$. The left and central
panels show $R(T_{i},\delta T)$ as a function of initial temperature $T_{i}$
(in mK) and $\Delta$ (in units of $\gamma$) and for (a) $s=0.25$ and $\delta
T=10$ mK and (b) $s=1$ and $\delta T=100$ mK. The scenario of a 229Th10+
intruder ion discussed in Sec. III is marked with a black cross in panel (b).
Panel (c) shows $R(T_{i},\delta T)$ as a function of $\delta T$ (in mK) for a
initial temperature of $T_{i}=50\,$mK, detuning $\Delta=3/4\gamma$ and
$s=0.25$ (bolt) and $s=1$ (dashed). The curves have been evaluated for the
case when a single ion of the crystal is illuminated. For $s\ll 1$ they also
describe the regime in which all ions of the crystal are uniformly driven by
the laser below saturation, see Sec. II.5.

##  IV Outlook

We have developed an analytical model that allows us to determine the
dependence of the resonance fluorescence of an ion crystal on its temperature,
and thus to access the heat capacity of the crystal. We apply this model to
investigate intruder-induced heating, as it is detected from the modification
of laser-induced fluorescence. We envision direct applications for the capture
of singly charged thorium ions and eventually also of more highly charged
ions.

Our model is based on the assumption that the signal is collected over a time
that is smaller than the time scale of the mechanical effects of light.
Analysis of the time scales show that during the recooling period a state-of-
the-art trapped-ion setup will be able to collect a sufficiently large number
of photons.

Our analysis is performed in the regime where the motion can be treated
semiclassically and is valid in the Doppler cooling regime. In the future we
will extend our theoretical studies to the sub-Doppler regime, where the
calorimetry signal may become even more sensitive to temperature changes,
e.g., when the ion crystal is undergoing polarization-gradient cooling
Wineland  _et al._ (1992); Joshi  _et al._ (2020); Li  _et al._ (2022). Under
these conditions, we conjecture that the quantum fluctuations can no longer be
neglected. Moreover, we expect that one shall include the modification of the
photon emission due to collective effects in the ordered array of ion emitters
Wolf  _et al._ (2020); Richter  _et al._ (2021).

We finally remark that the intensity of the scattered light allows one to
solely determine the mean energy, but gives no information on whether the
crystal is in thermal equilibrium neither whether the crystal eigenmodes are
thermalized at the same temperature. In future work we will investigate how
the latter information could be extracted by measuring the light emitted from
different ions within the crystal, employing detectors which selectively
collect the light from individual emitters within the crystal. Alternatively,
this information could be extracted from impurity ions embedded in the crystal
Morigi and Walther (2001); Sosnova  _et al._ (2021).

## Acknowledgements

We thank the TACTICa collaboration members for useful discussions. This work
was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation), Project-ID 429529648 TRR 306 QuCoLiMa ("Quantum Cooperativity of
Light and Matter"). This work was also partly supported by the Cluster of
Excellence "Precision Physics, Fundamental Interactions, and Structure of
Matter" (PRISMA+ EXC 2118/1) funded by the German Research Foundation (DFG)
within the German Excellence Strategy (Project ID 39083149), by the DFG
Reinhart Koselleck project and through the DIP program (Grant Nos. SCHM
1049/7-1 and FO 703/2-1). W.H. acknowledges support by a research fellowship
from the German Research Foundation (DFG, HA 8894/1-1). W.L. acknowledges
financial support by the “China-Germany Postdoctoral Exchange Program”.

## Appendix

##  Appendix A Estimate for the Coulomb logarithm

The factor $\Lambda_{\text{C}}$ in Eq. (16) can be approximated Poth (1990);
Zwicknagel  _et al._ (1999) by
$\Lambda_{\text{C}}\approx\int_{b_{\text{min}}}^{b_{\text{max}}}\frac{db}{b}$,
where $b$ is the scattering impact parameter, $b_{\text{max}}$ is usually
estimated by the screening length of the Coulomb potential
$b_{\text{max}}\approx v_{\text{i}}/\nu_{\text{p}}$ and $b_{\text{min}}$ by
the maximal momentum transfer
$b_{\text{min}}=e^{2}/(4\pi\epsilon_{0}M_{\text{Th}}v_{\text{i}}^{2})$.
Substituting the parameters of the TACTICa experiment and
$v_{\text{i}}=v_{\text{i}2}$, we obtain
$\Lambda_{\text{C}}\approx\ln\left(\frac{b_{\text{max}}}{b_{\text{min}}}\right)\approx
9$.

##  Appendix B Normal modes of the ion crystal

The range of the normal-mode frequencies $\omega_{m}$ can be estimated as
follows (see also Refs. James (1998); Rohde  _et al._ (2001)). The highest-
frequency modes usually correspond to the ions oscillating anti-phase with
respect to their nearest neighbors. This implies that, for the highest-
frequency modes, $\omega_{m}$ can be estimated by $\sqrt{2}\omega_{h}$, where
$\omega_{h}$ is the frequency of the quasi-harmonic potential that each ion
feels in the ion crystal. This potential is mainly governed by the Coulomb
repulsion from the nearest neighbors, hence the frequency is
$\omega_{h}=\sqrt{e^{2}/(\pi\epsilon_{0}Ml^{3})}$, where $l$ is the average
distance between the ions and $M$ is the mass of a crystal ion. For the
lowest-frequency modes, $\omega_{m}$ can be approximated by the external
harmonic trapping potential with frequency $\omega_{\text{a}}$ along the axial
direction and $\omega_{\text{r}}$ along the radial direction.

The difference of the trapping frequencies along the axial and the radial
directions usually lead to cigar-shaped ion crystals. We assume that due to
this asymmetry, the normal modes of the crystal split into two groups having a
preferred direction of oscillation either along the axial or the radial
direction Morigi and Fishman (2004). Along each direction, we assume the
dispersion relation to be linear similar to the Debye model Ashcroft and
Mermin (1976) such that the mode frequencies can acquire the values
$\omega_{\min}+n(\omega_{\max}-\omega_{\min})/(N_{\text{Ca}}-1)$, where $n$ is
an integer $n\in[0,N_{\text{Ca}}-1]$ and $\omega_{\min}$ ($\omega_{\max}$) is
the minimal (maximal) mode frequency in the given direction.

Using $l\approx 4\mu$m, $M=M_{\text{Ca}}$, $\omega_{\text{a}}\sim 2\pi\times
100$ kHz and $\omega_{\text{r}}\sim 2\pi\times 1$ MHz, we obtain
$\omega_{h}\sim 2\pi\times 15$ MHz and, hence, for the approximate range of
normal-mode frequencies $\omega_{m}/(2\pi)\\!\in$[100 kHz,20 MHz] for the
axial direction and $\omega_{m}/(2\pi)\\!\in$[1 MHz, 20 MHz] for the radial
direction, respectively.

Let us also remark that micromotion Berkeland  _et al._ (1998), which is
expected to be strong in three-dimensional ion crystals, where the equilibrium
positions of ions can be far from the trap axis, is included in our estimates
of temperatures achievable with Doppler cooling Poindron  _et al._ (2021).

##  Appendix C Timescale for Doppler cooling

In the semiclassical model the cooling of the ion crystal is described with
the radiation pressure force F exerting on the individual ions, such that
their momentum decreases as

$\frac{d\textbf{p}}{dt}=\textbf{F}=\hbar\textbf{k}\gamma\rho_{ee}.$ (18)

In the process, one photon momentum $\hbar\textbf{k}$ is taken away from each
ion at a rate of $\gamma\rho_{ee}$, whereby $\rho_{ee}$ is the average
probability to find an ion in the excited state Stenholm (1986). This yields a
rate of change of the motional energy $E$ of the ion crystal as

$\tau_{\text{cool}}=\frac{dE}{dt}/E=2\sqrt{\frac{\hbar\omega_{R}}{E}}\gamma\rho_{ee}.$
(19)

For the parameters of the TACTICa experiment and temperatures in the order of
$100\,$mK, this rate of change is on the order of $100\,\mu$s for a saturation
parameter of $s=0.25$, see Fig. 4.

Figure 4: Rate of change of the motional energy per ion (cf. Eq. (19)) as a
function of the initial temperature, and for the parameters of the TACTICa
experiment and $\Delta=\gamma/2$. The saturation is $s=0.25$ (blue) and
$s=1.5$ (orange).

## References

  * Dubin and O’Neil (1999) D. H. E. Dubin and T. M. O’Neil, Rev. Mod. Phys. 71, 87 (1999).
  * Morigi and Fishman (2004) G. Morigi and S. Fishman, Phys. Rev. Lett. 93, 170602 (2004).
  * Baughman  _et al._ (2000) R. H. Baughman, S. O. Dantas, S. Stafstrom, A. A. Zakhidov, T. B. Mitchell, and D. H. Dubin, Science 288, 2018 (2000).
  * Wesenberg  _et al._ (2007) J. H. Wesenberg, R. J. Epstein, D. Leibfried, R. B. Blakestad, J. Britton, J. P. Home, W. M. Itano, J. D. Jost, E. Knill, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, Phys. Rev. A 76, 053416 (2007).
  * Huber  _et al._ (2008) G. Huber, F. Schmidt-Kaler, S. Deffner, and E. Lutz, Phys. Rev. Lett. 101, 070403 (2008).
  * Poindron  _et al._ (2021) A. Poindron, J. Pedregosa-Gutierrez, C. Jouvet, M. Knoop, and C. Champenois, J. Chem. Phys. 154, 184203 (2021).
  * Stenholm (1986) S. Stenholm, Rev. Mod. Phys. 58, 699 (1986).
  * Morigi and Eschner (2001) G. Morigi and J. Eschner, Phys. Rev. A 64, 063407 (2001).
  * Groot-Berning  _et al._ (2019) K. Groot-Berning, F. Stopp, G. Jacob, D. Budker, R. Haas, D. Renisch, J. Runke, P. Thörle-Pospiech, C. E. Düllmann, and F. Schmidt-Kaler, Phys. Rev. A 99, 023420 (2019).
  * Stopp  _et al._ (2019) F. Stopp, K. Groot-Berning, G. Jacob, D. Budker, R. Haas, D. Renisch, J. Runke, P. Thörle-Pospiech, C. E. Düllmann, and F. Schmidt-Kaler, Hyperfine interact. 240, 33 (2019).
  * Haas  _et al._ (2020a) R. Haas, T. Kieck, D. Budker, C. E. Düllmann, K. Groot-Berning, W. Li, D. Renisch, F. Schmidt-Kaler, F. Stopp, and A. Viatkina, Hyperfine interact. 241, 25 (2020a).
  * 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, N. G. Trautmann, and P. G. Thirolf, Nature 533, 47 (2016).
  * Thielking  _et al._ (2018) J. Thielking, M. V. Okhapkin, P. Glowacki, D. M. Meier, L. von der Wense, B. Seiferle, C. E. Düllmann, P. G. Thirolf, and E. Peik, Nature 556, 321 (2018).
  * Flambaum (2006) V. V. Flambaum, Phys. Rev. Lett. 97, 092502 (2006).
  * Peik  _et al._ (2021) E. Peik, T. Schumm, M. Safronova, A. Pálffy, J. Weitenberg, and P. G. Thirolf, Quantum Science and Technology (2021).
  * Beeks  _et al._ (2021) K. Beeks, T. Sikorsky, T. Schumm, J. Thielking, M. V. Okhapkin, and E. Peik, Nat. Rev. Phys. 3, 238 (2021).
  * Seiferle  _et al._ (2017) B. Seiferle, L. von der Wense, and P. G. Thirolf, Phys. Rev. Lett. 118, 042501 (2017).
  * Haas  _et al._ (2020b) R. Haas, M. Hufnagel, R. Abrosimov, C. E. Düllmann, D. Krupp, C. Mokry, D. Renisch, J. Runke, and U. W. Scherer, Radiochim. Acta 108, 923 (2020b).
  * Wineland and Itano (1979) D. J. Wineland and W. M. Itano, Phys. Rev. A 20, 1521 (1979).
  * Eschner  _et al._ (2003) J. Eschner, G. Morigi, F. Schmidt-Kaler, and R. Blatt, J. Opt. Soc. Am. B 20, 1003 (2003).
  * Birkl  _et al._ (1994) G. Birkl, J. A. Yeazell, R. Rückerl, and H. Walther, Europhys. Lett. 27, 197 (1994).
  * Fishman  _et al._ (2008) S. Fishman, G. De Chiara, T. Calarco, and G. Morigi, Phys. Rev. B 77, 064111 (2008).
  * Hettrich  _et al._ (2015) M. Hettrich, T. Ruster, H. Kaufmann, C. Roos, C. Schmiegelow, F. Schmidt-Kaler, and U. Poschinger, Phys. Rev. Lett. 115, 143003 (2015).
  * Wunderlich  _et al._ (2005) C. Wunderlich, G. Morigi, and D. Reiß, Phys. Rev. A 72, 023421 (2005).
  * Note (1) When the laser drives the ions well below saturation, in fact, one can reduce the problem to a two-level systems, consisting of the state $|G\rangle$, where all the ions are in the ground state, and the Dicke state $|E\rangle$, that is coupled to $|G\rangle$ by absorption of a laser photon and that contains one (delocalized) electronic excitation. Then, the resonance fluorescence over the full space is determined by the population of $|E\rangle$ as given by Eq. (9) after replacing $\Omega\to\sqrt{N}\Omega$. We remark that this is strictly correct provided that the laser parameters are homogeneous across the crystal and that $Ns\ll 1$, with $s$ given in Eq. (12), so that, on average, only one ion can be excited at a time Morigi and Eschner (2001, 2003). The ions can be considered as indipendent scatterers also at moderately larger intensities, as long as the temperature is sufficiently large.
  * Schmit  _et al._ (2021) T. Schmit, L. Giannelli, A. S. Sørensen, and G. Morigi, Phys. Rev. A 104, 062607 (2021).
  * Poth (1990) H. Poth, Physics Reports 196, 135 (1990).
  * Zwicknagel  _et al._ (1999) G. Zwicknagel, C. Toepffer, and P.-G. Reinhard, Physics Reports 309, 117 (1999).
  * Note (2) Expression (16) remains valid until the velocity of the thorium ion becomes comparable to the mean thermal velocity of individual calcium ions, but this limitation is of minor importance here.
  * Bussmann  _et al._ (2006) M. Bussmann, U. Schramm, D. Habs, V. Kolhinen, and J. Szerypo, International Journal of Mass Spectrometry 251, 179 (2006).
  * Raecke  _et al._ (2018) P. Raecke, D. Spemann, J. Gerlach, B. Rauschenbach, and J. Meijer, Sci. Rep. 8, 9781 (2018).
  * Micke  _et al._ (2018) P. Micke, S. Kuehn, L. Buchauer, J. Harries, and M. e. a. Buecking, Rev. Sci. Instr. 89, 063109 (2018).
  * Andelkovic  _et al._ (2015) Z. Andelkovic, F. Herfurth, N. Kotovskiy, K. König, B. Maaß, T. Murböck, D. Neidherr, S. Schmidt, J. Steinmann, M. Vogel, and G. Vorobjev, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 795, 109 (2015).
  * Micke  _et al._ (2020) P. Micke, T. Leopold, S. King, E. Benkler, L. Spiess, L. Schmoeger, M. Schwarz, J. Crespo, and P. Schmidt, Nature 578, 60 (2020).
  * Wineland  _et al._ (1992) D. J. Wineland, J. Dalibard, and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 9, 32 (1992).
  * Joshi  _et al._ (2020) M. K. Joshi, A. Fabre, C. Maier, T. Brydges, D. Kiesenhofer, H. Hainzer, R. Blatt, and C. F. Roos, New J. Phys. 22, 103013 (2020).
  * Li  _et al._ (2022) W. Li, S. Wolf, L. Klein, D. Budker, C. E. Düllmann, and F. Schmidt-Kaler, New J. Phys. 24, 10 (2022).
  * Wolf  _et al._ (2020) S. Wolf, J. v. Z. S. Richter, and F. Schmidt-Kaler, Phys. Rev. Lett. 124, 063603 (2020).
  * Richter  _et al._ (2021) S. Richter, S. Wolf, J. von Zanthier, and F. Schmidt-Kaler, Phys. Rev. Lett. 126, 173602 (2021).
  * Morigi and Walther (2001) G. Morigi and H. Walther, The European Physical Journal D-Atomic, Molecular, Optical and Plasma Physics 13, 261 (2001).
  * Sosnova  _et al._ (2021) K. Sosnova, A. Carter, and C. Monroe, Phys. Rev. A 103, 012610 (2021).
  * James (1998) D. James, Appl. Phys. B: Lasers and Optics 66, 181 (1998).
  * Rohde  _et al._ (2001) H. Rohde, S. T. Gulde, C. F. Roos, P. A. Barton, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and R. Blatt, Journal of Optics B: Quantum and Semiclassical Optics 3, S34 (2001).
  * Ashcroft and Mermin (1976) N. W. Ashcroft and D. N. Mermin,  _Solid State Physics_ (Saunders College Publishing, 1976).
  * Berkeland  _et al._ (1998) D. J. Berkeland, J. D. Miller, J. C. Bergquist, W. M. Itano, and D. J. Wineland, J. Appl. Phys. 83, 5025 (1998).
  * Morigi and Eschner (2003) G. Morigi and J. Eschner, J. Phys. B: At. Mol. Opt. Phys. 36, 1041 (2003).