# Cancellation effects as a fingerprint of quantum collapse models at atomic
scale

Kristian Piscicchia  Centro Ricerche Enrico Fermi – Museo Storico della Fisica
e Centro Studi e Ricerche “Enrico Fermi”, Italy  Laboratori Nazionali di
Frascati, INFN, Italy;  Sandro Donadi  Centre for Quantum Materials and
Technologies, School of Mathematics and Physics, Queen’s University, Belfast
BT7 1NN, United Kingdom;  Simone Manti  Laboratori Nazionali di Frascati,
INFN, Italy;  Angelo Bassi  Department of Physics, University of Trieste,
Italy  Istituto Nazionale di Fisica Nucleare, Section of Trieste, Italy;
Maaneli Derakhshani  Department of Mathematics, Rutgers University – New
Brunswick, New Jersey, USA;  Lajos Diósi  Eötvös Loránd University, Budapest,
Hungary  Wigner Research Centre for Physics, Budapest, Hungary  Catalina
Curceanu  Laboratori Nazionali di Frascati, INFN, Italy;  IFIN-HH, Institutul
National pentru Fizica si Inginerie Nucleara Horia Hulubei, Romania.

###### Abstract

In this work the spontaneous electromagnetic radiation from atomic systems,
induced by dynamical wave-function collapse, is investigated in the X-rays
domain. Strong departures are evidenced with respect to the simple cases
considered until now in the literature, in which the emission is either
perfectly coherent (protons in the same nuclei) or incoherent (electrons). In
this low-energy regime the spontaneous radiation rate strongly depends on the
atomic species under investigation and, for the first time, is found to depend
on the specific collapse model.

Introduction. —

Quantum mechanics is our most successful physical theory, allowing us to
understand and predict a large number of phenomena with extreme precision Fan
_et al._ (2023). At the core of the theory lies the superposition principle,
according to which systems are allowed to be in superposition of different
states. The thoery does not set any boundary on the limit of validity of this
principle; yet, we don’t observe superpositions of macroscopic objects. This
well-known problem is exemplified by the Schördinger’s cat Gedankenexperiment
Schrödinger (1935).

It has been suggested that the linearity of quantum mechanics, from which the
superposition principle directly follows, may break down at a certain scale
Leggett (1980); Weinberg (1989); Bell (2004). This idea has been extensively
developed in the framework of collapse models Pearle (1976); Ghirardi  _et
al._ (1986); Bassi and Ghirardi (2003); Bassi  _et al._ (2013), which are
phenomenological models that modify the Schrödinger equation by adding non-
linear and stochastic terms that naturally collapse the wave function in
space. According to these models, microscopic systems are very weakly affected
by the non-linearities which, however, become dominant when atoms glue
together to form larger and larger systems, this way solving the measurement
problem. Among the several collapse models proposed, two are of particular
relevance: the Continuous Spontaneous Localization (CSL) model Ghirardi  _et
al._ (1990) and the so-called Diósi-Penrose (DP) model Diósi (1989); Penrose
(1996).

Spontaneous collapses must be random, in order to avoid the possibility of
faster than light signaling Gisin (1989); this randomness manifests as a
diffusive motion of the system Donadi  _et al._ (2023), which corresponds to a
random acceleration of the atoms and, hence, to the emission of radiation from
their charged constituents. The experimental search of this spontaneous
radiation was performed, for both CSL Donadi  _et al._ (2021a) and DP Donadi
_et al._ (2021b), in the energy domain of the $\gamma$-rays, by comparing the
measured radiation spectrum by high purity germanium crystals with the
spontaneous emission rate predicted by the models for the atomic systems which
constitute the experimental setup. The obtained strong bounds, combined with
constraints provided by other experimental tests and theoretical
considerations, are leading to a progressive falsification of the models in
their Markovian formulation.

Theoretical efforts have being devoted to the development of non-Markovian
collapse models Adler and Ramazanoğlu (2007); Bassi and Dürr (2009); Adler
_et al._ (2013); Bassi and Donadi (2014); Donadi  _et al._ (2014), in order to
counteract the runaway energy increase. These new models require the
introduction of a cutoff frequency in the stochastic noise spectrum; for this
reason, a systematic scan of the spontaneous radiation phenomenon, as a
function of the decreasing energy, is mandatory.

The search for spontaneous radiation emitted by Germanium crystals was
performed in the X-rays domain in Piscicchia  _et al._ (2017) (for
$E\in(15-50)$ keV), and more recently in Arnquist  _et al._ (2022) (for
$E\in(19-100)$ keV). In Piscicchia  _et al._ (2017) a formula for the expected
spontaneous radiation rate from quasi-free electrons was applied: the expected
radiation depends on the energy as $1/E$ and is proportional to the number of
quasi-free electrons Fu (1997). This formula is not suitable to describe the
more complex phenomenology of the spontaneous radiation emitted by the whole
atomic system; more refined calculations are presented in Donadi  _et al._
(2021a, b), where the CSL and the DP rates are calculated for an atomic
system, in the limit in which the spontaneous photon wave-length
$\lambda_{\gamma}$ is between the nuclear dimension and the mean radius of the
lower laying atomic orbit. Consistently, in Donadi  _et al._ (2021a, b) the
data analyses are performed for photons energies in the range (1-3.8) MeV. The
rate results to be proportional, for both CSL and DP, to
$(N_{p}^{2}+N_{e})/E$, where $N_{p}$ and $N_{e}$ are, respectively, the number
of protons and electrons of the atom under study. In this regime, different
collapse models share the same expected shape for the energy distribution of
the spontaneous emission rate, the scaling factor being proportional to
combinations of constants of nature with the characteristic parameters of the
models, which are $\lambda$ and $r_{C}$ (strength and correlation length of
the collapse noise) for CSL and the correlation length $R_{0}$ for DP (the
role of the strength being played by the gravitational constant $G$ in this
case). The latter theoretical rates were also applied in Arnquist  _et al._
(2022), where the X-rays spectrum measured by Germanium crystals is analyzed
in the energy range (19-100) keV, in which $\lambda_{\gamma}$ is comparable
with the mean radii of the atomic orbits of Germanium.

Given the importance of integrating the search of the spontaneous radiation
signal in the high-energy domain of the $\gamma$-rays to the one in the
X-Rays, we derive in this work the general expression of the radiation
emission rates, for both Markovian and non-Markovian formulations of the CSL
and DP models. We adopt a semiclassical approach which is valid above 1 keV
(see e.g. Adler and Ramazanoğlu (2007)), appropriate to the current
experimental surveys; for lower energies a fully quantum mechanical analysis
is required, which is under development. The general rate is found to exhibit
a non-trivial energy dependence, which is strongly influenced by the interplay
between the photons wavelengths, the radii of the electronic orbits and the
correlation length of the model under study.

Interestingly, the spontaneous radiation energy spectrum is found, at the
atomic $\lambda_{\gamma}$ scale, to depend on the specific model of wave
function collapse under scrutiny. This finding opens new scenarios in the
experimental investigation of the spontaneous radiation: a measurement
sensitive to this signature of the collapse, would be able to recognize the
most probable pattern of dynamical wave function reduction.

CSL spontaneous emission rate, general expression. — The rate of the
spontaneous radiation emitted by an atomic system, in the context of the
Markovian CSL model, was derived in Donadi  _et al._ (2021a):

$\left.\frac{d\Gamma}{dE}\right|_{t}^{CSL}=\frac{\hbar\,\lambda}{6\,\pi^{2}\,\epsilon_{0}\,c^{3}\,m_{0}^{2}\,E}\sum_{i,j}\frac{q_{i}\,q_{j}}{m_{i}\,m_{j}}\cdot
f_{ij}\cdot\frac{\sin(b_{ij})}{b_{ij}},$ (1)

where $b_{ij}=2\pi|\mathbf{r}_{i}-\mathbf{r}_{j}|/\lambda_{\gamma}$, $q_{j}$
and $m_{j}$ represent, respectively, the charge and the mass of the $j$-th
particle, at position $\mathbf{r}_{j}$. $m_{0}$ denotes the nucleon mass,
$\epsilon_{0}$ the vacuum permittivity, $\hbar$ and $c$ are, as usual, the
reduced Planck constant and the speed of light, $E$ is the energy of the
spontaneously emitted photon.

The term $f_{ij}$ encodes the balance between the emitters’ distances and the
correlation length $r_{C}$. A generalized expression for $f_{ij}$ is provided
in Appendix A:

$f_{ij}=\frac{m_{i}\,m_{j}}{2r_{C}^{2}}\,e^{-\frac{(\mathbf{r}_{i}-\mathbf{r}_{j})^{2}}{4r_{C}^{2}}}\left(3-\frac{(\mathbf{r}_{i}-\mathbf{r}_{j})^{2}}{2r_{C}^{2}}\right).$
(2)

Using Eq. (2), and analyzing the contributions to the spontaneous radiation by
the protons in the nucleus, the orbital electrons and due the combined
electrons-protons emission - see Appendix B \- Eq. (1) turns to:

$\displaystyle\left.\frac{d\Gamma}{dE}\right|_{t}^{CSL}\\!\\!=\frac{\hbar\,e^{2}\,\lambda}{12\,\pi^{2}\,\epsilon_{0}\,c^{3}\,m_{0}^{2}\,r_{C}^{2}\,E}\left\\{3\,N_{p}^{2}+3\,N_{e}+2\sum_{o\,o^{\prime}\,\textrm{pairs}}N_{o}\,N_{o^{\prime}}\,\frac{\textrm{sin}\left[\frac{\beta|\rho_{o}-\rho_{o^{\prime}}|\,E}{\hbar\,c}\right]}{\left[\frac{\beta|\rho_{o}-\rho_{o^{\prime}}|\,E}{\hbar\,c}\right]}e^{-\frac{\beta^{2}(\rho_{o}-\rho_{o^{\prime}})^{2}}{4r_{C}^{2}}}\,\left(3-\frac{\beta^{2}(\rho_{o}-\rho_{o^{\prime}})^{2}}{2r_{C}^{2}}\right)-\right.$
$\displaystyle\left.-2N_{p}\sum_{o}N_{o}\,\frac{\textrm{sin}\left(\frac{\rho_{o}\,E}{\hbar\,c}\right)}{\left(\frac{\rho_{o}\,E}{\hbar\,c}\right)}\cdot
e^{-\frac{\rho_{o}^{2}}{4r_{C}^{2}}}\,\left(3-\frac{\rho_{o}^{2}}{2r_{C}^{2}}\right)+\sum_{o}N_{o}\,(N_{o}-1)\,e^{-\frac{(\alpha\,\rho_{o})^{2}}{4r_{C}^{2}}}\cdot\frac{\textrm{sin}\left(\frac{\alpha\,\rho_{o}\,E}{\hbar\,c}\right)}{\left(\frac{\alpha\,\rho_{o}\,E}{\hbar\,c}\right)}\cdot\left(3-\frac{(\alpha\,\rho_{o})^{2}}{2r_{C}^{2}}\right)\right\\},$
(3)

where the electron pairs distances are parametrized in terms of the mean radii
of the atomic orbits $\rho_{o}$ ($\rho_{o^{\prime}}$) by means of the
constants $\alpha$ and $\beta$ which are given below. $N_{o}$ represents the
number of electrons in the $o$-th orbit of the atom. Eq. (Cancellation effects
as a fingerprint of quantum collapse models at atomic scale) represents a
generalization of the spontaneous emission rate which was derived in Ref.
Donadi  _et al._ (2021a):

$\left.\frac{d\Gamma}{dE}\right|_{t}^{CSL}=\frac{\hbar\,e^{2}\,\lambda}{4\,\pi^{2}\,\epsilon_{0}\,c^{3}\,r_{C}^{2}\,m_{0}^{2}\,E}\left(N_{p}^{2}+N_{e}\right),$
(4)

under the condition that $\lambda_{\gamma}$ is intermediate between the
nuclear and atomic dimensions, i.e. the energy range under scrutiny belongs to
the $\gamma$-rays domain. In that case protons emit coherently (proportionally
to the square of their number) and electrons emit independently (linear
dependence). More complex is the situations described by Eq. (Cancellation
effects as a fingerprint of quantum collapse models at atomic scale).

If $r_{C}$ exceeds the distance between the emitters (as confirmed by
radiation experiments for a Markovian CSL Donadi  _et al._ (2021a); Piscicchia
_et al._ (2017)) then the stochastic field “shakes” them coherently. If
$\lambda_{\gamma}$ becomes also of the order of the mean orbit radii of the
atom, then the electrons of the corresponding orbits start to emit coherently,
i.e. quadratically. Nonetheless, the corresponding increase in the expected
spontaneous emission rate is counteracted by the _cancellation_ , among
oppositely charged particles whose distance is smaller than
$\lambda_{\gamma}$. In the limit in which $\lambda_{\gamma}$ is also much
bigger than the atomic size, Eq. (Cancellation effects as a fingerprint of
quantum collapse models at atomic scale) reduces to:

$\left.\frac{d\Gamma}{dE}\right|_{t}^{CSL}=\frac{\hbar\,e^{2}\,\lambda}{4\,\pi^{2}\,\epsilon_{0}\,c^{3}\,m_{0}^{2}\,r_{C}^{2}\,E}\left[N_{p}^{2}-2\cdot
N_{p}\,N_{e}+N_{e}^{2},\right]$ (5)

which vanishes for neutral atoms.

Intermediate regimes for $\lambda_{\gamma}$ and $r_{C}$, in comparison with
$|\mathbf{r}_{i}-\mathbf{r}_{j}|$, give rise to a new, interesting pattern, in
which the shape of the expected spontaneous radiation spectrum exhibits a non-
trivial energy dependence, which is influenced by the atomic structure. This
is exemplified in Fig. 1. The top panel of Fig. 1 shows in red the general
spontaneous emission rate predicted by Eq. (Cancellation effects as a
fingerprint of quantum collapse models at atomic scale) (red), compared to the
simple (blue) case described by Eq. (4), for Germanium (Ge targets were indeed
used in various experiments, e.g. Donadi  _et al._ (2021a); Arnquist  _et al._
(2022)). The red shaded area corresponds to the values $\beta=1.04$ and
$\alpha$ spanning in the range (1 - 1.5), according to the literature Gill
_et al._ (2003); Coulson and Neilson (1961), see Appendix B. The same rates
for a Xenon target are shown in the bottom panel of Fig. 1. In Eq
(Cancellation effects as a fingerprint of quantum collapse models at atomic
scale) the value of $r_{C}$ is set to $1.15\cdot 10^{-8}$ m, consistently with
the results of Ref. Donadi  _et al._ (2021a), which are obtained by applying
Eq. (4) in the $\gamma$-rays regime ($(1-3.8)$ MeV), where Eq. (4) is an
excellent approximation. $r_{C}=1.15\cdot 10^{-8}$ m corresponds to the
intersection among the experimental bound and the theoretical constraint
(corresponding respectively to the orange and gray lines in Fig. 4 of Ref.
Donadi  _et al._ (2021a)). The mean radii of the orbits are obtained based on
a density functional theory (DFT) Kohn and Sham (1965) all-electron
calculation, for an isolated atom; the DFT code GPAW Enkovaara  _et al._
(2010) is adopted. The distributions are normalized to the common constant
pre-factors to evidence differences in shape. As expected, the simple (blue)
and the general (red) rates converge for high energies (above 200 keV), where
$\lambda_{\gamma}$ becomes sizably smaller than the lower atomic orbit radii.
Since $r_{C}$ is much greater than the size of the Germanium atom, the X-Rays
regime is, instead, characterized by a balance among electrons and protons
coherent emission and the cancellation of their contributions. The analysis
performed in Ref. Arnquist  _et al._ (2022) should be re-considered based on
this low-energy complex pattern.

The general expression of the spontaneous emission rate in Eq. (Cancellation
effects as a fingerprint of quantum collapse models at atomic scale) encodes
the phenomenology for future investigations of the spontaneous radiation at
low energies (X-rays). Comparison of the theoretical expectation with the
measured spectra requires a recursive analysis: in the first step, a suitable
prior has to be assumed for $r_{C}$, an updated value for $r_{C}$ will be
obtained, which will serve as input for the new prior. The analysis should
then be iterated till convergence of the $r_{C}$ values, within the
experimental sensitivity, is reached.

The dependence of the expected rate on the atomic structure becomes evident
comparing the top panel of Fig. 1 with the bottom one, which shows in red
colour the general spontaneous emission rate given by Eq. (Cancellation
effects as a fingerprint of quantum collapse models at atomic scale) for Xenon
(high sensitivity bounds on the spontaneous collapse could be set by the XENON
experiment Aprile  _et al._ (2019) by exploiting a Xenon target); the blue
line describes again the simple rate of Eq. (4). Eq. (Cancellation effects as
a fingerprint of quantum collapse models at atomic scale) predicts a strong
dependence of the spontaneous radiation yield on the atomic number $Z$; as
such a survey of the spontaneous collapse induced emission over $Z$ would
greatly improve the experimental sensitivity on this, new physics, phenomenon.

Figure 1: The top panel of the Figure shows in blue the $1/E$ dependence Eq.
(4), for the spontaneous radiation rate of a CSL model, which is only valid in
the high-energy domain. This is compared to the general rate in Eq.
(Cancellation effects as a fingerprint of quantum collapse models at atomic
scale) (red curve) for a prior value of the correlation length
$r_{C}=1.15\cdot 10^{-8}$ m. The distributions are calculated for a Germanium
atom and normalized to the common constant pre-factors. The bottom panel of
the Figure shows the shapes of the same rates, calculated for a Xenon atom.

The generalization of Eq. (Cancellation effects as a fingerprint of quantum
collapse models at atomic scale) to the non-Markovian case requires to
multiply the right-hand side by the Fourier transform of the noise correlation
function (see e.g. Adler and Ramazanoğlu (2007); Adler  _et al._ (2013); Bassi
and Donadi (2014); Donadi  _et al._ (2014); Carlesso  _et al._ (2018) and the
derivation of the colored DP model emission rate below). Assuming e.g. an
exponentially decaying noise correlation function
($f(t-s)=\frac{\Omega}{2}\,e^{-\Omega|t-s|}$), characterized by a correlation
time $\Omega^{-1}$, the rate becomes:

$\left.\frac{d\Gamma}{dE}\right|_{t}^{cCSL}=\left.\frac{d\Gamma}{dE}\right|_{t}^{CSL}\times\frac{E_{c}^{2}}{E_{c}^{2}+E^{2}},$
(6)

where $E_{c}=\hbar\Omega$ and cCSL denote results for a colored (non-
Markovian) CSL model.

DP spontaneous emission rate, general expression. — The rate of the
spontaneous radiation emitted by an atomic system, according to a Markovian DP
model, was derived in Ref. Donadi  _et al._ (2021b):

$\left.\frac{d\Gamma}{dE}\right|_{t}^{DP}=\frac{Ge^{2}}{12\pi^{5/2}\epsilon_{0}c^{3}R_{0}^{3}E}\left(N_{p}^{2}+N_{e}\right).$
(7)

assuming a spontaneous photon wavelength which is much bigger than the nuclear
size and much smaller than the lower lying atomic orbit mean radius. Note that
Eq. (7) differs from the result presented in Donadi  _et al._ (2021b) by a
factor $8\pi$. This is because in Donadi  _et al._ (2021b) we adopted the
convention introduced in Penrose (1996) while we refer here to the original
model introduced by Diósi Diosi (1987), in which the factor $8\pi$ was not
present.

The general structure of the rate is derived in Appendix C, where the non-
Markovianity of the noise time correlation is also considered. For an
exponential time correlation we have:

$\displaystyle\left.\frac{d\Gamma}{dE}\right|_{t}^{cDP}=\frac{G}{6\pi^{2}\epsilon_{0}c^{3}E}\sum_{i,j}q_{i}\,q_{j}f_{ij}\frac{\textrm{sin}(b_{ij})}{b_{ij}}\frac{E_{c}^{2}}{E_{c}^{2}+E^{2}}=$
$\displaystyle=\left.\frac{d\Gamma}{dE}\right|_{t}^{DP}\frac{E_{c}^{2}}{E_{c}^{2}+E^{2}},$
(8)

with $G$ the Newton constant. The rate for the Markovian model is recovered in
the limit $E_{c}\rightarrow\infty$. In analogy with the expected rate for the
CSL model (Eq. (1)), the interplay between the particles mean distance and the
wavelength of the spontaneously emitted photon is contained in the terms
$\textrm{sin}(b_{ij})/b_{ij}$. The dependence on the particles distances in
relation to the correlation length of the model $R_{0}$ is instead specified
by the terms $f_{ij}$. As it is shown in Appendix D, $f_{ij}$ is a measure of
the overlap between the mass densities of the particles $i$-th and $j$-th
($g_{i,j}$), whose spatial resolution is measured by $R_{0}$. In formulae:

$\displaystyle f_{ij}=4\pi\int
d\boldsymbol{r}g_{i}(\boldsymbol{r}-\boldsymbol{r}_{i},R_{0})g_{j}(\boldsymbol{r}-\boldsymbol{r}_{j},R_{0}).$
(9)

Assuming Gaussian mass density profiles (following e.g. Ghirardi  _et al._
(1990); Donadi  _et al._ (2021b)
$g_{i}(\boldsymbol{r}-\boldsymbol{r}_{i},R_{0})=(2\pi
R_{0}^{2})^{-3/2}e^{-\frac{(\boldsymbol{r}-\boldsymbol{r_{i}})^{2}}{2R_{0}^{2}}}$)
and specifying the rate for the mean radii of the atomic orbits we obtain (see
Appendix D):

$\displaystyle\left.\frac{d\Gamma}{dE}\right|_{t}^{DP}$
$\displaystyle=\frac{Ge^{2}}{12\pi^{5/2}\epsilon_{0}c^{3}R_{0}^{3}E}\left\\{N_{p}^{2}+N_{e}+2\sum_{o\,o^{\prime}\,\textrm{pairs}}N_{o}\,N_{o^{\prime}}\,\frac{\textrm{sin}\left[\frac{\beta|\rho_{o}-\rho_{o^{\prime}}|\,E}{\hbar\,c}\right]}{\left[\frac{\beta|\rho_{o}-\rho_{o^{\prime}}|\,E}{\hbar\,c}\right]}e^{-\frac{\beta^{2}(\rho_{o}-\rho_{o^{\prime}})^{2}}{4R_{0}^{2}}}+\right.$
$\displaystyle\left.+\sum_{o}N_{o}\,(N_{o}-1)\,e^{-\frac{(\alpha\,\rho_{o})^{2}}{4R_{0}^{2}}}\cdot\frac{\textrm{sin}\left(\frac{\alpha\,\rho_{o}\,E}{\hbar\,c}\right)}{\left(\frac{\alpha\,\rho_{o}\,E}{\hbar\,c}\right)}-2N_{p}\sum_{o}N_{o}\,\frac{\textrm{sin}\left(\frac{\rho_{o}\,E}{\hbar\,c}\right)}{\left(\frac{\rho_{o}\,E}{\hbar\,c}\right)}\cdot
e^{-\frac{\rho_{o}^{2}}{4R_{0}^{2}}}\right\\}$ (10)

Again, the simple rate described by equation (7) is a good approximation of
Eq. (Cancellation effects as a fingerprint of quantum collapse models at
atomic scale) for energies belonging to the $\gamma$-rays domain.

Figure 2: Top panel of the Figure shows in blue the $1/E$ dependence Eq. (7),
for the spontaneous radiation rate of the DP model, which is only valid in the
high-energy domain. This is compared to the general rate Eq. (Cancellation
effects as a fingerprint of quantum collapse models at atomic scale) (red
curve) for a prior value of the correlation length $R_{0}=0.54$ Å. The
distributions are calculated for a Germanium atom and normalized to the common
constant pre-factors. The bottom panel of the Figure shows the shapes of the
same rates, calculated for a Xenon atom.

Figure 2 compares the general expression of the DP spontaneous emission rate
(Eq. (Cancellation effects as a fingerprint of quantum collapse models at
atomic scale)) given by the red curve, with the simple expression in Eq. (7)
given by the blue curve. The prior value $R_{0}=0.54\;\AA$ is chosen,
consistently with the result Donadi  _et al._ (2021b), which is obtained by
applying Eq. (7) in the $\gamma$-rays regime ($(1-3.8)$ MeV), where Eq. (4) is
an excellent approximation. A strong departure from the approximate expression
is evident, at low-energy, for both Germanium (top panel) and Xenon (bottom
panel) atoms. A significant dependence of the generalized rate on the atomic
number is evident for the DP model as well.

Discussion. — Figures 1 and 2 summarize our findings and unveil the most
interesting consequence of the cancellation phenomenon, for which the
contribution to the spontaneous radiation emission from oppositely charged
particles, whose distance is exceeded by the model’s correlation length and by
the observed photon wavelength, cancels. In the low-energy regime, for
correlation lengths of the models of the order, or bigger, than the atomic
orbits radii, the shapes predicted for the spontaneous emission rates
distributions of the CSL and DP models strongly differ. This is both due to
the different mathematical structure of the terms $f_{ij}$ and the different
values of the correlation lengths, $r_{C}$ and $R_{0}$, of the two models. In
the simple scenario in which $\lambda_{\gamma}$ is much smaller than the
atomic size, any difference is washed-out and the shapes of the spontaneous
radiation rates of the two models just differ by a scaling factor.

An experimental investigation of the spontaneous radiation emitted by the
atoms, in the energy range going from few to tens of keV, exploiting the
outlined phenomenological analysis scheme, and based on the predicted rates
Eqs. (Cancellation effects as a fingerprint of quantum collapse models at
atomic scale) and (Cancellation effects as a fingerprint of quantum collapse
models at atomic scale), may be able to disentangle which model describes the
data with higher probability. If the measurement is sensitive to the signal of
collapse, it should then be also able to recognize the physical mechanism
which is responsible for it. Moreover, the experimental sensitivity may be
improved, and the systematic effects minimized, by performing a survey on
targets of different atomic numbers, exploiting the peculiar impact of the
atomic structure on the cancellation phenomenon.

In this paper we derived the fundamental theoretical formulas for these future
investigations, both for the white noise models and their non-Markovian
generalizations.

###### Acknowledgements.

This publication was made possible through the support of Grant 62099 from the
John Templeton Foundation. The opinions expressed in this publication are
those of the authors and do not necessarily reflect the views of the John
Templeton Foundation. We acknowledge support from the Foundational Questions
Institute and Fetzer Franklin Fund, a donor advised fund of Silicon Valley
Community Foundation (Grants No. FQXi-RFP-CPW-2008 and FQXi-MGA-2102), and
from the H2020 FET TEQ (Grant No. 766900). We thank: the INFN Institute, for
supporting the research presented in this article and, in particular, the Gran
Sasso underground laboratory of INFN, INFN-LNGS, and its Director, Ezio
Previtali, the LNGS staff, and the Low Radioactivity laboratory for the
experimental activities dedicated to the search for spontaneous radiation. We
thank the Austrian Science Foundation (FWF) which supports the VIP2 project
with the grants P25529-N20, project P 30635-N36 and W1252-N27 (doctoral
college particles and interactions). K.P. acknowledges support from the Centro
Ricerche Enrico Fermi - Museo Storico della Fisica e Centro Studi e Ricerche
“Enrico Fermi” (Open Problems in Quantum Mechanics project). S.D. acknowledges
support from the Marie Sklodowska Curie Action through the UK Horizon Europe
guarantee administered by UKRI. A.B. acknowledges financial support from the
University of Trieste, INFN, the PNRR MUR project PE0000023-NQSTI, and the EIC
Pathfinder project QuCoM (GA No. 101046973).

## Appendix A

We reconsider here the general expression for $f_{ij}$ Donadi  _et al._
(2021a):

$f_{ij}=\sum_{k=x,y,z}\int d^{3}s\int
d^{3}s^{\prime}\,e^{-\frac{(\mathbf{r}_{i}-\mathbf{r}_{j}+\mathbf{s}^{\prime}-\mathbf{s})^{2}}{4r_{C}^{2}}}\,\frac{\partial\mu_{i}(\mathbf{s})}{\partial
s^{k}}\frac{\partial\mu_{j}(\mathbf{s}^{\prime})}{\partial s^{\prime k}},$
(11)

$\mu_{i}$ denoting the mass density of the $i$-th particle, but we don’t
assume $r_{C}\gg|\mathbf{r}_{i}-\mathbf{r}_{j}|$, condition which was
experimentally tested only in the case of a Markovian CSL. Hence

$f_{ij}=\int d^{3}s\int
d^{3}s^{\prime}\,\mu_{i}(\mathbf{s})\mu_{j}(\mathbf{s}^{\prime})\left\\{e^{-\frac{(\mathbf{r}_{i}-\mathbf{r}_{j}+\mathbf{s}^{\prime}-\mathbf{s})^{2}}{4r_{C}^{2}}}\frac{1}{2r_{C}^{2}}\left[3-\frac{(\mathbf{r}_{i}-\mathbf{r}_{j}+\mathbf{s}^{\prime}-\mathbf{s})^{2}}{2r_{C}^{2}}\right]\right\\}.$
(12)

which reduces to Eq. (2) for point-like mass densities
$\mu_{i}(\mathbf{r})=m_{i}\,\delta(\mathbf{r})$.

## Appendix B

By substituting the general expression for $f_{ij}$ given by Eq. (2) in Eq.
(1), and considering the separate contributions of the protons
($i_{p},j_{p}$), of the electrons ($i_{e},j_{e}$) and of the combined
electrons-protons emission ($i_{p},j_{e}$) and ($i_{e},j_{p}$) we have:

$\left.\frac{d\Gamma}{dE}\right|_{t}^{CSL}=\frac{\hbar\,\lambda}{12\,\pi^{2}\,\epsilon_{0}\,c^{3}\,m_{0}^{2}\,r_{C}^{2}\,E}\cdot$

$\left[\sum_{ip,jp}q_{ip}\,q_{jp}\cdot\frac{\sin(2\pi|\mathbf{r}_{ip}-\mathbf{r}_{jp}|/\lambda_{\gamma})}{2\pi|\mathbf{r}_{i}-\mathbf{r}_{j}|/\lambda_{\gamma}}\,e^{-\frac{(\mathbf{r}_{ip}-\mathbf{r}_{jp})^{2}}{4r_{C}^{2}}}\left(3-\frac{(\mathbf{r}_{ip}-\mathbf{r}_{jp})^{2}}{2r_{C}^{2}}\right)\,+\right.$

$+\sum_{ip,je}q_{ip}\,q_{je}\cdot\frac{\sin(2\pi|\mathbf{r}_{ip}-\mathbf{r}_{je}|/\lambda_{\gamma})}{2\pi|\mathbf{r}_{ip}-\mathbf{r}_{je}|/\lambda_{\gamma}}\,e^{-\frac{(\mathbf{r}_{ip}-\mathbf{r}_{je})^{2}}{4r_{C}^{2}}}\left(3-\frac{(\mathbf{r}_{ip}-\mathbf{r}_{je})^{2}}{2r_{C}^{2}}\right)\,+$

$+\,\sum_{ie,jp}q_{ie}\,q_{jp}\cdot\frac{\sin(2\pi|\mathbf{r}_{ie}-\mathbf{r}_{jp}|/\lambda_{\gamma})}{2\pi|\mathbf{r}_{ie}-\mathbf{r}_{jp}|/\lambda_{\gamma}}\,e^{-\frac{(\mathbf{r}_{ie}-\mathbf{r}_{jp})^{2}}{4r_{C}^{2}}}\left(3-\frac{(\mathbf{r}_{ie}-\mathbf{r}_{jp})^{2}}{2r_{C}^{2}}\right)\,+$

$+\,\left.\sum_{ie,je}q_{ie}\,q_{je}\cdot\frac{\sin(2\pi|\mathbf{r}_{ie}-\mathbf{r}_{je}|/\lambda_{\gamma})}{2\pi|\mathbf{r}_{ie}-\mathbf{r}_{je}|/\lambda_{\gamma}}\,e^{-\frac{(\mathbf{r}_{ie}-\mathbf{r}_{je})^{2}}{4r_{C}^{2}}}\left(3-\frac{(\mathbf{r}_{ie}-\mathbf{r}_{je})^{2}}{2r_{C}^{2}}\right)\right]$
(13)

Let us analyze the three contributions separately:

  1. 1.

The first sum in Eq. (13) is performed on the protons in the nucleus. In the
regime that we are analyzing we have
$|\mathbf{r}_{ip}-\mathbf{r}_{jp}|/\lambda_{\gamma}\ll 1$ and
$(\mathbf{r}_{ip}-\mathbf{r}_{jp})^{2}/r_{C}^{2}\ll 1$ hence the sum reduces
to:

$3\sum_{ip,jp}q_{ip}\,q_{jp}=3\,\left(\sum_{ip}q_{ip}\right)^{2}=3\,e^{2}\,N_{p}^{2}.$
(14)

  2. 2.

The arguments of the second and third sums are symmetric under the exchange
$i\rightarrow j$. For any electron $j_{e}$ belonging to the $o$-th orbit of
the atom we approximate:

$|\mathbf{r}_{ip}-\mathbf{r}_{je}|\sim\rho_{o}\,\,,\,\,\forall\,ip\,,$ (15)

$\rho_{o}$ representing the mean radius of the orbit hosting the $je$-th
electron. Hence, in terms of the energy of the spontaneously emitted photon,
and indicating with $N_{o}$ the number of electrons in the $o$-th orbit, the
second and third sum yield:

$-2\,e^{2}\,N_{p}\,\sum_{o}N_{o}\,e^{-\frac{\rho_{o}^{2}}{4r_{C}^{2}}}\,\left(3-\frac{\rho_{o}^{2}}{2r_{C}^{2}}\right)\,\frac{\textrm{sin}\left(\frac{\rho_{o}\,E}{\hbar\,c}\right)}{\left(\frac{\rho_{o}\,E}{\hbar\,c}\right)}.$
(16)

  3. 3.

Concerning the last sum, if $ie=je$ then

$3\sum_{ie,je\,,\,ie=je}q_{ie}q_{je}=3\,\sum_{ie}q_{ie}^{2}=3\,e^{2}\,N_{e}.$
(17)

For $i_{e},j_{e}$ belonging to the same orbit $o$, with $ie\neq je$ we
parametrize their distance as
$|\mathbf{r}_{ie}-\mathbf{r}_{je}|=\alpha\rho_{o}$. Considered that the total
number of pairs in the $o$-th orbit is $N_{o}\,(N_{o}-1)$ we find:

$e^{2}\,\sum_{o}N_{o}\,(N_{o}-1)\,e^{-\frac{(\alpha\,\rho_{o})^{2}}{4r_{C}^{2}}}\cdot\frac{\textrm{sin}\left(\frac{\alpha\,\rho_{o}\,E}{\hbar\,c}\right)}{\left(\frac{\alpha\,\rho_{o}\,E}{\hbar\,c}\right)}\cdot\left(3-\frac{(\alpha\,\rho_{o})^{2}}{2r_{C}^{2}}\right).$
(18)

For $i_{e},j_{e}$ belonging to different orbits $o$ and $o^{\prime}$ we adopt
the parametrization
$|\mathbf{r}_{ie}-\mathbf{r}_{je}|=\beta|\rho_{o}-\rho_{o^{\prime}}|$. The
corresponding contribution to the sum is:

$2e^{2}\,\sum_{o\,o^{\prime}\,\textrm{pairs}}N_{o}\,N_{o^{\prime}}\,\frac{\textrm{sin}\left[\frac{\beta|\rho_{o}-\rho_{o^{\prime}}|\,E}{\hbar\,c}\right]}{\left[\frac{\beta|\rho_{o}-\rho_{o^{\prime}}|\,E}{\hbar\,c}\right]}e^{-\frac{\beta^{2}(\rho_{o}-\rho_{o^{\prime}})^{2}}{4r_{C}^{2}}}\,\left(3-\frac{\beta^{2}(\rho_{o}-\rho_{o^{\prime}})^{2}}{2r_{C}^{2}}\right).$
(19)

By substituting Eqs. (14)-(19) in Eq. (13) Eq. (Cancellation effects as a
fingerprint of quantum collapse models at atomic scale) is obtained.
Concerning $\alpha$ and $\beta$ their values can be, in principle, estimated
from the average distance between two electrons in an atom, extracted from the
position intracule function Gill  _et al._ (2003). This function provides the
probability distribution of finding two electrons at a given distance.
Intracule function calculations are available in the literature for helium,
lithium, and beryllium. In helium, the average distance between two electrons
in the 1s shell is known at different levels of approximation for the
correlation among electrons Coulson and Neilson (1961). Using the correlated
value for the inter-electronic distance, $\alpha=1.47$ for the 1s shell. From
calculations in lithium and beryllium Gill  _et al._ (2003) both $\alpha$ and
$\beta$ can be obtained. For lithium it is found $\alpha=1.03$ for the 1s
shell and $\beta=1.04$ for the 1s and 2s shells distance. In the case of
beryllium $\alpha=1.20$ for the 1s shell, $\alpha=1.11$ for the 2s shell, and
$\beta=1.04$ for the 1s and 2s shells distance. According to the calculated
values, the spontaneous emission rate is shown as a red shaded area in Figure
1 for $\beta=1.04$ and $\alpha$ spanning in the range (1 - 1.5), the red line
corresponds to $\alpha=1.25$. The implementation of specific intracule
functions for Germanium and Xenon, integrated into a full quantum calculation
of the spontaneous radiation rate, is currently under development.

## Appendix C

By analogy with the proof outlined in Donadi  _et al._ (2021a) the starting
point of the semi-classical calculation of the rate emitted by the charged
particles of the atomic system, for a coloured DP model, is the classical
expression of the total emitted power:

$P(t)=R_{sp}^{2}\int d\Omega S(R_{sp}^{2}\hat{n},t)$ (20)

with $S$ being the Poynting vector at time $t$, and the integration being
performed on a spherical surface of radius $R_{sp}$. This turns out to be

$P(t)=\frac{1}{64\,\pi^{4}\,\epsilon_{0}\,c^{3}}\int_{-\infty}^{+\infty}d\omega\int_{-\infty}^{+\infty}d\nu\,e^{i(\omega+\nu)(t-R_{sp}/c)}\,\sum_{i,j}q_{i}q_{j}\,\mathbb{E}[J_{ij}(\omega,\nu)]$
(21)

with

$J_{ij}(\omega,\nu)=4\pi\,\ddot{\mathbf{r}}_{i}(\omega)\cdot\ddot{\mathbf{r}}_{j}(\nu)\frac{(b_{ij}^{2}-1)\,\mathrm{sin}(b_{ij})+b_{ij}\,\mathrm{cos}(b_{ij})}{b_{ij}^{3}}-4\pi\,\ddot{{r}}_{i}^{z}(\omega)\ddot{{r}}_{j}^{z}(\nu)\frac{(b_{ij}^{2}-3)\,\mathrm{sin}(b_{ij})+3b\,\mathrm{cos}(b_{ij})}{b_{ij}^{3}},$
(22)

where $b_{ij}=|\omega\mathbf{r}_{i}+\nu\mathbf{r}_{j}|/c$.

In the case of the DP model, the acceleration induced by the collapse is:

$\ddot{\boldsymbol{r}}_{i}(t)=-\frac{1}{m_{i}}\int
d\boldsymbol{r}\,\left[\nabla_{\boldsymbol{r}_{i}(t)}\mu_{i}(\boldsymbol{r}-\boldsymbol{r}_{i}(t))\right]\phi(\boldsymbol{r},t)$
(23)

where

$\mu_{i}(\boldsymbol{r})=m_{i}g_{i}(\boldsymbol{r},R_{0}),$ (24)

with $g_{i}$ a function describing the mass density of size $R_{0}$ of the
$i$-th particle (which will be specified below, see Eq. (34)) and
$\phi(\boldsymbol{r},t)$ a Gaussian noise field with zero average and
correlation:

$\mathbb{E}\left[\phi(\boldsymbol{r},t)\phi(\boldsymbol{r}^{\prime},t^{\prime})\right]=G\hbar\frac{\delta(t-t^{\prime})}{|\boldsymbol{r}-\boldsymbol{r}^{\prime}|}.$
(25)

A delta correlation in time implies a Markovian dynamics. Similarly to the
analysis done for the CSL model, non-Markovianity can be introduced by
replacing this correlation function with one describing e.g. an exponentially
decaying correlation in time, characterized by a cutoff frequency $\Omega$,
i.e.:

$\mathbb{E}\left[\phi(\boldsymbol{r},t)\phi(\boldsymbol{r}^{\prime},t^{\prime})\right]=\frac{G\hbar}{|\boldsymbol{r}-\boldsymbol{r}^{\prime}|}\frac{\Omega}{2}e^{-\Omega|t-t^{\prime}|}.$
(26)

The Fourier transform of the $k$-th component of the acceleration takes the
form:

$\displaystyle\ddot{r}_{i}^{k}(\omega)$ $\displaystyle=-\int dte^{-i\omega
t}\int d\boldsymbol{r}\,\left[\frac{\partial}{\partial
r_{i}^{k}}g_{i}(\boldsymbol{r}-\boldsymbol{r}_{i},R_{0})\right]\phi(\boldsymbol{r},t)$
$\displaystyle=\int d\boldsymbol{s}\,\left[\frac{\partial}{\partial
s^{k}}g_{i}(\boldsymbol{s},R_{0})\right]\tilde{\phi}(\boldsymbol{s}+\boldsymbol{r}_{i},\omega)$
(27)

where in the second line we performed the change of variables
$\mathbf{s}=\mathbf{r}-\mathbf{r}_{i}$ and introduced
$\tilde{\phi}(\boldsymbol{r},\omega):=\int dte^{-i\omega
t}\phi(\boldsymbol{r},t)$, with zero average and correlation

$\mathbb{E}\left[\tilde{\phi}(\boldsymbol{r},\omega)\tilde{\phi}(\boldsymbol{r}^{\prime},\nu)\right]=2\pi
G\hbar\left(\frac{\Omega^{2}}{\omega^{2}+\Omega^{2}}\right)\frac{\delta(\nu+\omega)}{|\boldsymbol{r}-\boldsymbol{r}^{\prime}|}.$

We can now focus on the average of the scalar product of two accelerations,
necessary for computing $J_{ij}(\omega,\nu)$ in Eq. (22):

$\mathbb{E}\left[\ddot{\mathbf{r}}_{i}(\omega)\cdot\ddot{\mathbf{r}}_{j}(\nu)\right]=2\pi\hbar
G\delta(\omega+\nu)\frac{\Omega^{2}}{\Omega^{2}+\omega^{2}}f_{ij}$ (28)

where $f_{ij}=\sum_{k=x,y,z}f_{ij}^{k}$ with

$f_{ij}^{k}=\int d^{3}s\int
d^{3}s^{\prime}\frac{1}{|\mathbf{r}_{i}-\mathbf{r}_{j}+\mathbf{s}-\mathbf{s}^{\prime}|}\frac{\partial
g_{i}(\mathbf{s},R_{0})}{\partial s^{k}}\frac{\partial
g_{j}(\mathbf{s}^{\prime},R_{0})}{\partial s^{\prime k}}.$ (29)

By assuming $f_{ij}^{z}=f_{ij}/3$, which holds for spherical symmetric mass
distributions, we have:

$\mathbb{E}\left[J_{ij}(\omega,\nu)\right]=8\pi^{2}\hbar
G\delta(\omega+\nu)\frac{\Omega^{2}}{\Omega^{2}+\omega^{2}}f_{ij}\frac{2}{3}\frac{\textrm{sin}(b_{ij})}{b_{ij}}$
(30)

and

$P(t)=\frac{G\hbar}{12\pi^{2}\epsilon_{0}c^{3}}\int_{-\infty}^{+\infty}d\omega\,\,\sum_{i,j}q_{i}\,q_{j}f_{ij}\frac{\textrm{sin}(b_{ij})}{b_{ij}}\frac{\Omega^{2}}{\Omega^{2}+\omega^{2}}.$
(31)

By equating the last result for the power to the following expression

$P(t)=\int_{0}^{+\infty}d\omega\,\hbar\,\omega\frac{d\Gamma_{t}}{d\omega},$
(32)

the spontaneous emission rate in Eq. (Cancellation effects as a fingerprint of
quantum collapse models at atomic scale) is obtained.

## Appendix D

By substituting $\mathbf{s}=\mathbf{r}-\mathbf{r}_{i}$ and
$\mathbf{s}^{\prime}=\mathbf{r}^{\prime}-\mathbf{r}_{j}$ in Eq. (29) and
summing over $k=x,y,z$ we have

$\displaystyle f_{ij}$ $\displaystyle=\sum_{k=x,y,z}\int d\boldsymbol{r}\int
d\boldsymbol{r}^{\prime}\left[\frac{\partial}{\partial
r^{k}}g_{i}(\boldsymbol{r}-\boldsymbol{r}_{i},R_{0})\right]\left[\frac{\partial}{\partial
r^{\prime
k}}g_{j}(\boldsymbol{r}^{\prime}-\boldsymbol{r}_{j},R_{0})\right]\frac{1}{|\boldsymbol{r}-\boldsymbol{r}^{\prime}|}=$
$\displaystyle=\int d\boldsymbol{r}\int
d\boldsymbol{r}^{\prime}g_{i}(\boldsymbol{r}-\boldsymbol{r}_{i},R_{0})g_{j}(\boldsymbol{r}^{\prime}-\boldsymbol{r}_{j},R_{0})\sum_{k=x,y,z}\left[\frac{\partial}{\partial
r^{\prime k}}\frac{\partial}{\partial
r^{k}}\frac{1}{|\boldsymbol{r}-\boldsymbol{r}^{\prime}|}\right]=$
$\displaystyle=\int d\boldsymbol{r}\int
d\boldsymbol{r}^{\prime}g_{i}(\boldsymbol{r}-\boldsymbol{r}_{i},R_{0})g_{j}(\boldsymbol{r}^{\prime}-\boldsymbol{r}_{j},R_{0})\left[-\nabla_{r}^{2}\frac{1}{|\boldsymbol{r}-\boldsymbol{r}^{\prime}|}\right]=$
$\displaystyle=\int d\boldsymbol{r}\int
d\boldsymbol{r}^{\prime}g_{i}(\boldsymbol{r}-\boldsymbol{r}_{i},R_{0})g_{j}(\boldsymbol{r}^{\prime}-\boldsymbol{r}_{j},R_{0})\left[4\pi\delta(\boldsymbol{r}-\boldsymbol{r}^{\prime})\right]=$
$\displaystyle=4\pi\int
d\boldsymbol{r}g_{i}(\boldsymbol{r}-\boldsymbol{r}_{i},R_{0})g_{j}(\boldsymbol{r}-\boldsymbol{r}_{j},R_{0}).$
(33)

For Gaussian mass density profiles:

$g_{i}(\boldsymbol{r}-\boldsymbol{r}_{i},R_{0})=\frac{1}{(2\pi
R_{0}^{2})^{3/2}}e^{-\frac{(\boldsymbol{r}-\boldsymbol{r_{i}})^{2}}{2R_{0}^{2}}},$
(34)

simple steps lead to

$f_{ij}=\frac{1}{2\pi^{1/2}R_{0}^{3}}e^{-\frac{(\mathbf{r}_{i}-\mathbf{r}_{j})^{2}}{4R_{0}^{2}}}.$
(35)

By substituting Eq. (35) in Eq. (Cancellation effects as a fingerprint of
quantum collapse models at atomic scale) we get:

$\left.\frac{d\Gamma}{dE}\right|_{t}^{DP}=\frac{G}{12\pi^{5/2}\epsilon_{0}c^{3}R_{0}^{3}E}\sum_{i,j}q_{i}\,q_{j}\,e^{-\frac{(\mathbf{r}_{i}-\mathbf{r}_{j})^{2}}{4R_{0}^{2}}}\,\frac{\textrm{sin}\left(\frac{|\mathbf{r}_{i}-\mathbf{r}_{j}|\,E}{\hbar\,c}\right)}{\left(\frac{|\mathbf{r}_{i}-\mathbf{r}_{j}|\,E}{\hbar\,c}\right)}.$
(36)

By repeating the steps (1)-(3) which lead from Eq. (13) to (Cancellation
effects as a fingerprint of quantum collapse models at atomic scale), Eq.
(Cancellation effects as a fingerprint of quantum collapse models at atomic
scale) is obtained.

## References

  * Fan  _et al._ (2023) X. Fan, T. Myers, B. Sukra, and G. Gabrielse, Physical review letters 130, 071801 (2023).
  * Schrödinger (1935) E. Schrödinger, Naturwissenschaften 23, 844 (1935).
  * Leggett (1980) A. J. Leggett, Progress of Theoretical Physics Supplement 69, 80 (1980).
  * Weinberg (1989) S. Weinberg, Physical Review Letters 62, 485 (1989).
  * Bell (2004) J. S. Bell,  _Speakable and unspeakable in quantum mechanics: Collected papers on quantum philosophy_ (Cambridge university press, 2004).
  * Pearle (1976) P. Pearle, Physical Review D 13, 857 (1976).
  * Ghirardi  _et al._ (1986) G. C. Ghirardi, A. Rimini, and T. Weber, Physical review D 34, 470 (1986).
  * Bassi and Ghirardi (2003) A. Bassi and G. Ghirardi, Physics Reports 379, 257 (2003).
  * Bassi  _et al._ (2013) A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, Reviews of Modern Physics 85, 471 (2013).
  * Ghirardi  _et al._ (1990) G. Ghirardi, R. Grassi, and A. Rimini, Physical Review A 42, 1057 (1990).
  * Diósi (1989) L. Diósi, Physical Review A 40, 1165 (1989).
  * Penrose (1996) R. Penrose, Gen. Rel. Grav. 28: 581-600 (1996).
  * Gisin (1989) N. Gisin, Helv. Phys. Acta 62, 363 (1989).
  * Donadi  _et al._ (2023) S. Donadi, L. Ferialdi, and A. Bassi, Physical Review Letters 130, 230202 (2023).
  * Donadi  _et al._ (2021a) S. Donadi, K. Piscicchia, R. Del Grande, C. Curceanu, M. Laubenstein, and A. Bassi, The European Physical Journal C 81, 1 (2021a).
  * Donadi  _et al._ (2021b) S. Donadi, K. Piscicchia, C. Curceanu, L. Diósi, M. Laubenstein, and A. Bassi, Nature Physics 17, 74 (2021b).
  * Adler and Ramazanoğlu (2007) S. L. Adler and F. M. Ramazanoğlu, Journal of Physics A: Mathematical and Theoretical 40, 13395 (2007).
  * Bassi and Dürr (2009) A. Bassi and D. Dürr, Journal of Physics A: Mathematical and Theoretical 42, 485302 (2009).
  * Adler  _et al._ (2013) S. L. Adler, A. Bassi, and S. Donadi, Journal of Physics A: Mathematical and Theoretical 46, 245304 (2013).
  * Bassi and Donadi (2014) A. Bassi and S. Donadi, Physics Letters A 378, 761 (2014).
  * Donadi  _et al._ (2014) S. Donadi, D.-A. Deckert, and A. Bassi, Annals of Physics 340, 70 (2014).
  * Piscicchia  _et al._ (2017) K. Piscicchia, A. Bassi, C. Curceanu, R. D. Grande, S. Donadi, B. C. Hiesmayr, and A. Pichler, Entropy 19, 319 (2017).
  * Arnquist  _et al._ (2022) I. J. Arnquist, F. T. Avignone, A. S. Barabash, C. J. Barton, K. H. Bhimani, E. Blalock, B. Bos, M. Busch, M. Buuck, T. S. Caldwell, Y.-D. Chan, C. D. Christofferson, P.-H. Chu, M. L. Clark, C. Cuesta, J. A. Detwiler, Y. Efremenko, H. Ejiri, S. R. Elliott, G. K. Giovanetti, M. P. Green, J. Gruszko, I. S. Guinn, V. E. Guiseppe, C. R. Haufe, R. Henning, D. Hervas Aguilar, E. W. Hoppe, A. Hostiuc, I. Kim, R. T. Kouzes, T. E. Lannen V., A. Li, A. M. Lopez, J. M. López-Castaño, E. L. Martin, R. D. Martin, R. Massarczyk, S. J. Meijer, T. K. Oli, G. Othman, L. S. Paudel, W. Pettus, A. W. P. Poon, D. C. Radford, A. L. Reine, K. Rielage, N. W. Ruof, D. Tedeschi, R. L. Varner, S. Vasilyev, J. F. Wilkerson, C. Wiseman, W. Xu, C.-H. Yu, and B. X. Zhu (Majorana Collaboration), Phys. Rev. Lett. 129, 080401 (2022).
  * Fu (1997) Q. Fu, Physical Review A 56, 1806 (1997).
  * Gill  _et al._ (2003) P. M. Gill, D. P. O’Neill, and N. A. Besley, Theoretical Chemistry Accounts 109, 241 (2003).
  * Coulson and Neilson (1961) C. Coulson and A. Neilson, Proceedings of the Physical Society (1958-1967) 78, 831 (1961).
  * Kohn and Sham (1965) W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
  * Enkovaara  _et al._ (2010) J. Enkovaara, C. Rostgaard, J. J. Mortensen, J. Chen, M. Dułak, L. Ferrighi, J. Gavnholt, C. Glinsvad, V. Haikola, H. A. Hansen, H. H. Kristoffersen, M. Kuisma, A. H. Larsen, L. Lehtovaara, M. Ljungberg, O. Lopez-Acevedo, P. G. Moses, J. Ojanen, T. Olsen, V. Petzold, N. A. Romero, J. Stausholm-Møller, M. Strange, G. A. Tritsaris, M. Vanin, M. Walter, B. Hammer, H. Häkkinen, G. K. H. Madsen, R. M. Nieminen, J. K. Nørskov, M. Puska, T. T. Rantala, J. Schiøtz, K. S. Thygesen, and K. W. Jacobsen, Journal of Physics: Condensed Matter 22, 253202 (2010).
  * Aprile  _et al._ (2019) E. Aprile, J. Aalbers, F. Agostini, M. Alfonsi, L. Althueser, F. Amaro, V. C. Antochi, F. Arneodo, D. Barge, L. Baudis, _et al._ , Journal of Instrumentation 14 (07), P07016.
  * Carlesso  _et al._ (2018) M. Carlesso, L. Ferialdi, and A. Bassi, The European Physical Journal D 72, 1 (2018).
  * Diosi (1987) L. Diosi, Physics letters A 120, 377 (1987).