# Decoherence of a matter-wave interferometer due to dipole-dipole
interactions

Paolo Fragolino  Van Swinderen Institute for Particle Physics and Gravity,
University of Groningen, 9747 AG Groningen, the Netherlands  Dipartimento di
Fisica e Astronomia, Università di Bologna, via Irnerio 46, I-40126 Bologna,
Italy  Martine Schut  Van Swinderen Institute for Particle Physics and
Gravity, University of Groningen, 9747 AG Groningen, the Netherlands
Bernoulli Institute for Mathematics, Computer Science and Artificial
Intelligence, University of Groningen, 9747 AG Groningen, the Netherlands
Marko Toroš  School of Physics and Astronomy, University of Glasgow, Glasgow,
G12 8QQ, UK  Sougato Bose  Department of Physics and Astronomy, University
College London, Gower Street, WC1E 6BT London, UK  Anupam Mazumdar  Van
Swinderen Institute for Particle Physics and Gravity, University of Groningen,
9747 AG Groningen, the Netherlands

###### Abstract

Matter-wave interferometry with nanoparticles will enable the development of
quantum sensors capable of probing ultraweak fields with unprecedented
applications for fundamental physics. The high sensitivity of such devices
however makes them susceptible to a number of noise and decoherence sources
and as such can only operate when sufficient isolation from the environment is
achieved. It is thus imperative to model and characterize the interaction of
nanoparticles with the environment and to estimate its deleterious effects.
The aim of this paper will be to study the decoherence of the matter-wave
interferometer due to dipole-dipole interactions which is one of the
unavoidable channels for decoherence even for a neutral micro-crystal. We will
start the analysis from QED and show that it reduces to the scattering model
characterized by the differential cross-section. We will then obtain simple
expressions for the decoherence rate in the short and long wavelength limits
that can be readily applied to estimate the available coherence time. We will
conclude by applying the obtained formulae to estimate the dipole-dipole
decoherence rate for the Quantum Gravity-induced Entanglement of Masses (QGEM)
protocol and discuss if the effects should be mitigated.

##  I Introduction

The idea that matter can behave as a wave is a key conceptual leap of modern
physics de1923waves , with matterwave interferometry one of the central
experimental techniques of quantum mechanics thomson1927diffraction ;
davisson1927diffraction ; davisson1928reflection . It is the basis for the
notion of quantum superposition Schrodinger:1935zz  and it is the building
block of quantum entanglement einstein1935can ; bell1964einstein , two
features that cannot be mimicked by a classical theory Horodecki:2009zz .
Matter-wave interferometry has been also used in a series of fundamental
experiments to demonstrate gravitationally-induced interference with neutrons
and atoms colella1975observation ; nesvizhevsky2002quantum ; fixler2007atom ;
asenbaum2017phase ; overstreet2022observation . Furthermore, matter-wave
interferometers can be excellent quantum sensors Kilian:2022kgm ; Wu:2022rdv
and can act as probes of physics beyond the standard model Barker:2022mdz .

It was further suggested that the next generation of matter-wave
interferometers with nanoparticles will be sensitive enough to probe
gravitationally-induced entanglement. Known as a quantum gravity-induced
entanglement of masses (QGEM) Bose:2017nin  111The first report of the results
of the QGEM protocol Bose:2017nin  was in a conference in 2016, Bangalore
workshop ICTS . See also Ref. Marletto:2017kzi ., the scheme shows that if
gravity is inherently a quantum entity then the masses of two nearby
interferometers will entangle when placed sufficiently close. The key
observation is that as long as we follow the standard relativistic quantum
mechanics, locality/causality, and general relativity in an effective field
theory of quantum gravity the two quantum superposed masses will inevitably
entangle each other via the quantum gravitational interaction Bose:2017nin ;
Marshman:2019sne ; Bose:2022uxe ; Gunnink:2022ner ; Elahi:2023ozf ;
Vinckers:2023grv ; Carney_2019 ; Belenchia:2018szb ; Carney:2021vvt ;
Danielson:2021egj ; Christodoulou:2022vte , while classical gravity cannot
entangle the two quantum systems as formalized by the local operation and
classical communication (LOCC) theorem Bennett:1996gf ; Bose:2017nin ;
Marshman:2019sne ; Bose:2022uxe . Recently, the QGEM protocol was also
extended to test the quantum nature of gravity in an optomechanical setup
where we can test the quantum gravitational entanglement between matter and
photon Biswas:2022qto . However, there are many experimental challenges to be
resolved before interferometry with nanoparticles can be implemented. To name
a few, creating spatial quantum superpositions Bose:2017nin ;
vandeKamp:2020rqh ; Bose:2017nin ; PhysRevLett.111.180403 ;
PhysRevLett.117.143003 ; Margalit:2020qcy ; Marshman:2021wyk ; Folman2013 ;
folman2019 ; PhysRevLett.125.023602 ; Folman2018 ; Zhou:2022frl ; Zhou:2022jug
; Zhou:2022epb ; Marshman:2023nkh , ensuring sufficiently long coherence times
Bose:2017nin ; Schut:2021svd ; Tilly:2021qef ; PhysRevLett.125.023602 ;
Rijavec:2020qxd ; RomeroIsart2011LargeQS ; Chang_2009 ; Schlosshauer:2014pgr ;
bassi2013models ; Toros:2020krn , and protecting the experiment from external
jitters, gravity gradient noise and seismic noise Toros:2020dbf .

The aim of this paper will be to investigate dipole-dipole decoherence in
matter-wave interferometry with nanoparticles. Such channel of decoherence is
unavoidable and must be taken into account even with neutral micrcrystals
Casimir:1947kzi ; Casimir:1948dh . In section II, we set up the formalism for
describing the evolution of the density matrix of a matter-wave interferometer
and the environment which are coupled via a generic EM interaction. We do this
by first constructing the interaction Hamiltonian for QED (subsection II.1)
and then we use this Hamiltonian to find the Born-Markov master equation
(subsection II.2). In section III, we will discuss the dipole-dipole
interaction and study the decoherence of the matter-wave interferometer in a
short and long wavelength limit (subsections III.1, III.2 respectively).
Section IV discusses the application of the decoherence model from dipole
interactions to the specific case of the QGEM experiment. There we discuss all
possible dipole-dipole interactions that play a role in the QGEM experiment,
such as: the interaction of an induced dipole in the QGEM test mass with an
environmental dipole (subsection IV.1), the interaction between a permanent
dipole in the test mass with an environmental dipole (subsection IV.2), and
the interaction between the permanent dipole of the test mass with the induced
dipole in the environmental particles that it induces (subsection IV.3). From
the three cases we put constraints on the allowed dipoles in the QGEM
experiment, which will be discussed in section V.

##  II Modelling decoherence from electromagnetic interactions

We start by modelling the decoherence caused by a generic electromagnetic
interaction between an object in superposition (the matter-wave
interferometer) and an environmental particle. We will consider a charged
neutral crystal which is responsible for creating the spatial quantum
superposition for the matter-wave interferometer as a fermion. We will also
treat the environmental particle as a fermion field. Hence the two fermions
are interacting within the Standard Model by virtue of the photon field. This
is the model we will use to study the effect of decoherence in the Born-Markov
approximation Schlosshauer:2014pgr ; joos2003decoherence  222In this section,
we will use the natural units $\hbar=c=1$. We will switch to SI units in
section IV when we discuss the application to the QGEM experiment..

Let us first consider a generic QED interaction Lagrangian between the two
fermions. The fermion field $\psi$ that represents the environmental particle
will interact electromagnetically via a photon field $A_{\mu}$. The fermion-
photon interaction term (the three-vertex) is given by:

$\mathcal{L}_{\text{int}}=-e\bar{\psi}\gamma^{\mu}\psi A_{\mu},$ (1)

where $\gamma^{\mu}$ are the Dirac’s matrices and $e$ is the electric charge
unit. Here $\mu=0,1,2,3$, and we take the signature $(-,+,+,+)$ of the metric
$\eta_{\mu\nu}$. The corresponding Hamiltonian interaction is given by:

$H_{\text{int}}=-\int
d^{3}\textbf{x}\;\mathcal{L}_{\text{int}}(\textbf{x})=\int
d^{3}\textbf{x}\;e\bar{\psi}\gamma^{\mu}\psi A_{\mu}.$ (2)

We first evaluate the fermion current $\bar{\psi}\gamma^{\mu}\psi$ in terms of
the fermion field’s creation and annihilation operators, $\hat{a}$ and
$\hat{a}^{\dagger}$, respectively:

$\begin{split}\bar{\psi}\gamma^{\mu}\psi&=\sum_{s,s^{\prime}}\int\frac{d^{3}\textbf{p}_{i}d^{3}\textbf{p}_{f}}{(2\pi)^{6}\sqrt{4E_{p_{i}}E_{p_{f}}}}\\\
&\times\;\hat{a}_{p_{f}}^{s^{\prime}\;{\dagger}}\hat{a}_{p_{i}}^{s}e^{-i(\textbf{p}_{i}-\textbf{p}_{f})\cdot\textbf{x}}\bar{u}(\boldsymbol{p}_{f})^{s^{\prime}}\gamma^{\mu}u(\boldsymbol{p}_{i})^{s},\end{split}$
(3)

where $u$ is the spinor that the creation/annihilation operators are
associated to, p are the momenta of the particle and $s=1,2$ labels the two
component spinors. Here, we will assume that $u({\boldsymbol{p}})^{s}$
corresponds to the fermionic field of the environment. Moreover, we also
assume that there is no negative-frequency spinors (anti-particle with
associated $b$-creation and annihilation operators) in the on-shell state.

The other quantum field that appears in Eq. (1) is a generic electromagnetic
field $A_{\mu}(\textbf{x})$. In our case this term can be related to the
crystal in spatial superposition by considering its fermion current
$J_{\mu}(\textbf{x})$ associated with the object. In particular, from the
Maxwell’s equations, we have:

$A_{\mu}(\textbf{x})=\Box^{-1}J_{\mu}(\textbf{x}),$ (4)

where $\Box=\eta_{\mu\nu}\partial^{\mu}\partial^{\nu}$ is the D’Alembertian
operator and the inverse of the virtual photon propagator. Substituting
$A_{\mu}$ into Eq. (1), we can study the interaction between two fermions by
the exchange of a virtual photon, e.g., between the environmental and crystal
fermions.

We will consider the crystal to be a heavy fermion, and denote the spinor as
$U({\boldsymbol{k}})^{r}$, where $r=1,2$ labels the two component spinors. In
the momentum space Eq. (4) becomes:

$\tilde{A}_{\mu}(\textbf{q})=\frac{-i\;g_{\mu\nu}}{q^{2}}\tilde{J}^{\nu}(\textbf{q}),$
(5)

where q represents the transferred momentum.

The fermion current of Eq. (4) can be expressed by the crystal’s fermion
$\Psi$:

$\begin{split}J^{\mu}(\textbf{x})&=\bar{\Psi}\gamma^{\mu}\Psi=\int\frac{d^{3}\textbf{k}_{i}d^{3}\textbf{k}_{f}}{(2\pi)^{6}\sqrt{4E_{k_{i}}E_{k_{f}}}}\hat{c}_{k_{f}}^{r^{\prime}\;{\dagger}}\hat{c}_{k_{i}}^{r}\\\
&\times\sum_{r,r^{\prime}}e^{-i(\textbf{k}_{i}-\textbf{k}_{f})\cdot\textbf{x}}\overline{U}(\boldsymbol{k}_{f})^{r^{\prime}}\gamma^{\mu}U(\boldsymbol{k}_{i})^{r}\end{split}$
(6)

where $\hat{c}$ and $\hat{c}^{\dagger}$ are respectively the creation and
annihilation operators associated to the spinor $U(\boldsymbol{k})^{r}$ of the
fermion field that describes the crystal. This leads to:

$\begin{split}\tilde{J}^{\nu}(\textbf{q})=\int
d^{3}\textbf{x}\;e^{i\textbf{q}\cdot\textbf{x}}J^{\nu}(\textbf{x})=\int\frac{d^{3}\textbf{k}_{i}d^{3}\textbf{k}_{f}}{(2\pi)^{6}\sqrt{4E_{k_{i}}E_{k_{f}}}}\\\
\times\sum_{r,r^{\prime}}\hat{c}_{k_{f}}^{r^{\prime}\;{\dagger}}\hat{c}_{k_{i}}^{r}\overline{U}(\boldsymbol{k}_{f})^{r^{\prime}}\gamma^{\mu}U(\boldsymbol{k}_{i})^{r}\;\delta^{3}(\textbf{q}+\textbf{k}_{i}-\textbf{k}_{f}),\end{split}$
(7)

where the Dirac’s delta $\delta^{3}(\textbf{q}+\textbf{k}_{i}-\textbf{k}_{f})$
encodes the conservation of momentum during the scattering process.

Plugging Eq. (7) into Eq. (2) through:

$\begin{split}A_{\mu}(\textbf{x})&=\int
d^{3}\textbf{q}\;e^{-i\textbf{q}\cdot\textbf{x}}\tilde{A}_{\mu}(\textbf{q})\\\
&=\int
d^{3}\textbf{q}\;e^{-i\textbf{q}\cdot\textbf{x}}\frac{-i\;g_{\mu\nu}}{q^{2}}\tilde{J}^{\nu}(\textbf{q}),\end{split}$
(8)

one obtains:

$\displaystyle\hat{H}_{\text{int}}$
$\displaystyle=(2\pi)^{3}e\int\frac{d^{3}\textbf{p}_{i}d^{3}\textbf{p}_{f}d^{3}\textbf{k}_{i}d^{3}\textbf{k}_{f}}{(2\pi)^{12}\sqrt{16\;E_{p_{i}}E_{p_{f}}E_{k_{i}}E_{k_{f}}}}$
$\displaystyle\times\sum_{\begin{subarray}{c}s,s^{\prime}\\\
r,r^{\prime}\end{subarray}}\hat{a}_{p_{f}}^{s^{\prime}\;{\dagger}}\hat{a}_{p_{i}}^{s}\;\hat{c}_{k_{f}}^{r^{\prime}\;{\dagger}}\hat{c}_{k_{i}}^{r}\;\delta^{3}(\textbf{k}_{i}+\textbf{p}_{i}-\textbf{k}_{f}-\textbf{p}_{f})$
$\displaystyle\times\;\bar{u}(\boldsymbol{p}_{f})^{s^{\prime}}\gamma^{\mu}u(\boldsymbol{p}_{i})^{s}\frac{-i\;g_{\mu\nu}}{q^{2}}\overline{U}(\boldsymbol{k}_{f})^{r^{\prime}}\gamma^{\nu}U(\boldsymbol{k}_{i})^{r},$
(9)

where now the transferred momentum q appearing in the propagator is explicitly
given by $\textbf{q}=\textbf{k}_{f}-\textbf{k}_{i}$ because of the Dirac’s
delta appearing in Eq. (7).

Notice that the last term in the RHS is exactly the quantum electrodynamics
(QED) matrix element for the fermion-fermion interaction
$\mathcal{M}_{p_{i},p_{f},k_{i},k_{f}}^{s,s^{\prime},r,r^{\prime}}$
Schwartz:2014sze :

$\begin{split}&\hat{H}_{\text{int}}=(2\pi)^{3}e\int\frac{d^{3}\textbf{p}_{i}d^{3}\textbf{p}_{f}d^{3}\textbf{k}_{i}d^{3}\textbf{k}_{f}}{(2\pi)^{12}\sqrt{16\;E_{p_{i}}E_{p_{f}}E_{k_{i}}E_{k_{f}}}}\sum_{\begin{subarray}{c}s,s^{\prime},r,r^{\prime}\end{subarray}}\\\
&\hat{a}_{p_{f}}^{s^{\prime}\;{\dagger}}\hat{a}_{p_{i}}^{s}\;\hat{c}_{k_{f}}^{r^{\prime}\;{\dagger}}\hat{c}_{k_{i}}^{r}\delta^{3}(\textbf{k}_{i}+\textbf{p}_{i}-\textbf{k}_{f}-\textbf{p}_{f})\;\mathcal{M}_{p_{i},p_{f},k_{i},k_{f}}^{s,s^{\prime},r,r^{\prime}}.\end{split}$
(10)

with 333Further note that the Dirac delta
$\delta^{3}(\textbf{k}_{i}+\textbf{p}_{i}-\textbf{k}_{f}-\textbf{p}_{f})$
appearing in Eq. (10) encodes the conservation of momentum between the initial
and the final states of the scattering diagram between the environmental
fermion and the crystal’s fermion, i.e.
$\textbf{k}_{i}+\textbf{p}_{i}=\textbf{k}_{f}+\textbf{p}_{f}$.

$\begin{split}\mathcal{M}_{p_{i},p_{f},k_{i},k_{f}}^{s,s^{\prime},r,r^{\prime}}=&\bar{u}(\boldsymbol{p}_{f})^{s^{\prime}}\gamma^{\mu}u((\boldsymbol{p}_{i}))^{s}\\\
&\times\frac{-i\;g_{\mu\nu}}{q^{2}}\overline{U}(\boldsymbol{k}_{f})^{r^{\prime}}\gamma^{\nu}U(\boldsymbol{k}_{i})^{r}.\end{split}$
(11)

###  II.1 Hamiltonian construction

At this point, one can express the creation and annihilation operators
($\hat{c}$’s and $\hat{a}$’s) in terms of one particle states
($\ket{\textbf{k},r}$ and $\ket{\textbf{p},s}$ respectively 444Here, the
notation $\ket{\textbf{k},r}$ means formally
$\ket{\textbf{k}}\otimes\ket{r}\equiv\ket{\textbf{k},r}$, i.e. the state can
be separated in the momentum space and the spin space.) through the relations:

$\ket{\textbf{k},r}=\sqrt{2E_{k}}\hat{c}_{k}^{r\;{\dagger}}\ket{0}_{S}\,;\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\ket{\textbf{p},s}=\sqrt{2E_{p}}\hat{a}_{p}^{s\;{\dagger}}\ket{0}_{\mathcal{E}}$
(12)

where $\ket{0}_{\mathcal{E}}$ and $\ket{0}_{S}$ represent respectively the
vacuum of the environmental particle and the vacuum of the crystal. In
particular, the operator
$\hat{A}\equiv\hat{a}_{p_{f}}^{s^{\prime}\;{\dagger}}\hat{a}_{p_{i}}^{s}$
destroys an environmental particle with initial momentum $\boldsymbol{p}_{i}$
and initial spin component $s$ and creates an environmental particle with
final momentum $\boldsymbol{p}_{f}$ and final spin component $s^{\prime}$.
This means that the action of this operator can be written as:

$\hat{A}:\ket{\textbf{p}_{i},s}\;\longrightarrow\ket{\textbf{p}_{f},s^{\prime}}.$
(13)

Now, we can write the creation operator $\hat{a}_{p}^{s\;\dagger}$ (which acts
on the vacuum as
$\ket{\textbf{p},s}=\sqrt{2E_{p}}\hat{a}_{p}^{s\;{\dagger}}\ket{0}_{\mathcal{E}}$)
in the momentum basis representation $\left\\{\ket{\textbf{p},s}\right\\}$ as:

$\hat{a}_{p}^{s\;\dagger}=\frac{1}{\sqrt{2E_{p}}}\ket{\textbf{p},s}\bra{0}_{\mathcal{E}}.$
(14)

Because of this property, the operator $\hat{A}$ becomes:

$\begin{split}\hat{A}\equiv\hat{a}_{p_{f}}^{s^{\prime}\;{\dagger}}\hat{a}_{p_{i}}^{s}=\frac{1}{\sqrt{4E_{p_{i}}E_{p_{f}}}}\ket{\textbf{p}_{f},s^{\prime}}\bra{\textbf{p}_{i},s},\end{split}$
(15)

which satisfies Eq.(13). The same relations hold for the crystal field, with
$\ket{\textbf{p},s}\rightarrow\ket{\textbf{k},r}$, $\hat{a}\rightarrow\hat{c}$
and $\ket{0}_{\mathcal{E}}\rightarrow\ket{0}_{S}$. This means that Eq.(10)
gives the following form for the Hamiltonian interaction
$\hat{H}_{\text{int}}$:

$\begin{split}\hat{H}_{\text{int}}&=\int\frac{d^{3}\textbf{p}_{i}d^{3}\textbf{p}_{f}d^{3}\textbf{k}_{i}}{(2\pi)^{9}16\;E_{p_{i}}E_{p_{f}}E_{k_{i}}E_{k_{i}+p_{i}-p_{f}}}\\\
&\times\sum_{\begin{subarray}{c}s,s^{\prime},r,r^{\prime}\end{subarray}}\mathcal{M}_{p_{i},p_{f},k_{i},k_{i}+p_{i}-p_{f}}^{s,s^{\prime},r,r^{\prime}}\;\ket{\textbf{p}_{f},s^{\prime}}\bra{\textbf{p}_{i},s}\\\
&\otimes
e^{i(\textbf{p}_{i}-\textbf{p}_{f})\cdot\hat{\textbf{x}}}\ket{\textbf{k}_{i},r^{\prime}}\bra{\textbf{k}_{i},r},\end{split}$
(16)

where
$\delta^{3}(\textbf{k}_{i}+\textbf{p}_{i}-\textbf{k}_{f}-\textbf{p}_{f})$ has
been absorbed by performing the integral over $d^{3}\textbf{k}_{f}$. The
vector state
$\ket{\textbf{k}_{f}}\rightarrow\ket{\textbf{k}_{i}+\textbf{p}_{i}-\textbf{p}_{f}}$
has been expressed in terms of the translation operator, which generates the
translations in the momenta space through the position operator
$\hat{\textbf{x}}$, i.e.
$\ket{\textbf{k}_{i}+\textbf{p}_{i}-\textbf{p}_{f}}=e^{i(\textbf{p}_{i}-\textbf{p}_{f})\cdot\hat{\textbf{x}}}\ket{\textbf{k}_{i}}$.

To simplify the expression of the interaction Hamiltonian in Eq. (16) we make
some assumptions:

  * •

The environmental particle’s momenta are assumed to be very small, due to a
low ambient temperature in the original experiment, we will quantify that
later.

  * •

The crystal mass is assumed to be much heavier than the environmental
particle’s mass, so that its momentum does not change during the scattering
process with the environmental particles.

This means that
$\absolutevalue{\mathbf{p}_{i}},\absolutevalue{\mathbf{p}_{f}}\ll M$, where
$M$ is the crystal’s mass. As a result
$\absolutevalue{\mathbf{k}_{i}},\absolutevalue{\mathbf{k}_{f}}\sim 0$. These
assumptions are valid for most types of experiments, so the application of
their Hamiltonian is kept general while the approximations simplify the
expression. These assumptions allow us to approximate the matrix element in
Eq.(16) as

$\mathcal{M}_{p_{i},p_{f},k_{i},k_{i}+p_{i}-p_{f}}^{s,s^{\prime},r,r^{\prime}}\sim\mathcal{M}_{p_{i},p_{f},0,0}^{s,s^{\prime},r,r^{\prime}}\,.$
(17)

Since the crystal is heavy with respect to the ambient energy, we can further
approximate $2E_{k_{i}}2E_{k_{i}+p_{i}-p_{f}}\sim 4ME_{k_{i}}$. Finally,
integrating over $\boldsymbol{k}_{i}$ and using the identity
$\int\frac{d^{3}\textbf{k}_{i}}{(2\pi)^{3}2E_{k_{i}}}\ket{\textbf{k}_{i}}\bra{\textbf{k}_{i}}=\mathbb{I}$,
we can obtain the final expression for Eq.(2):

$\displaystyle\hat{H}_{int}$
$\displaystyle=\int\frac{d^{3}\textbf{p}_{i}d^{3}\textbf{p}_{f}}{(2\pi)^{6}2E_{p_{i}}2E_{p_{f}}}\sum_{\begin{subarray}{c}s,s^{\prime},r,r^{\prime}\end{subarray}}\frac{\mathcal{M}_{p_{i},p_{f}}^{s,s^{\prime},r,r^{\prime}}}{2M}$
(18)
$\displaystyle\times\ket{\textbf{p}_{f},s^{\prime}}\bra{\textbf{p}_{i},s}\otimes
e^{i(\textbf{p}_{i}-\textbf{p}_{f})\cdot\hat{\textbf{x}}}\,\ket{r^{\prime}}\bra{r}.$

###  II.2 Born-Markov Master Equation

To find a Master Equation that described the decoherence of the crystal due to
the environment, we start by supposing that the interaction Hamiltonian has
the general form Schlosshauer:2014pgr ; joos2003decoherence :

$\hat{H}_{\text{int}}=\sum_{\alpha}\hat{S}_{\alpha}\otimes\hat{E}_{\alpha},$
(19)

where $\hat{S}_{\alpha}$ and $\hat{E}_{\alpha}$ represent all the degrees of
freedom of the matter-wave interferometer and the environment respectively and
$\alpha$ represents schematically the sums over the momenta p and the two
component spinors $s$ 666 More explicitly, from Eq.(18) we have
$\sum_{\alpha}\equiv\int\frac{d^{3}\textbf{p}_{i}d^{3}\textbf{p}_{f}}{(2\pi)^{6}2E_{p_{i}}2E_{p_{f}}}\sum_{\begin{subarray}{c}s,s^{\prime},r,r^{\prime}\end{subarray}}$.
From Eq.(18), one can find the explicit expression for $\hat{S}_{\alpha}$ and
$\hat{E}_{\alpha}$:

$\displaystyle\hat{S}_{\alpha}$
$\displaystyle=\frac{e^{i(\textbf{p}_{i}-\textbf{p}_{f})\cdot\hat{\textbf{x}}}}{2M}\ket{r^{\prime}}\bra{r}$
$\displaystyle\hat{E}_{\alpha}$
$\displaystyle=\mathcal{M}_{p_{i},p_{f}}^{s,s^{\prime},r,r^{\prime}}\ket{\textbf{p}_{f},s^{\prime}}\bra{\textbf{p}_{i},s}$
(20)

which matches the result in Ref. Kilian:2022kgm  which described the
scattering between a neutrino and a heavy nucleus, mediated by the weak
interaction. The difference is that in Ref. Kilian:2022kgm  the matrix element
$\mathcal{M}_{p_{i},p_{f}}^{s,s^{\prime},r,r^{\prime}}$ is defined by the
scattering via the weak interaction, while we have defined it in terms of the
electromagnetic interaction as in Eq. (11).

The master equation in the Born-Markov approximation is given by
Schlosshauer:2019ewh :

$\begin{aligned}
\frac{d\rho_{S}}{dt}&=-\frac{i}{\hbar}\left[H_{S},\rho_{S}\right]-\bigg{\\{}\int_{0}^{\infty}d\tau\sum_{\alpha\beta}C_{\alpha\beta}(-\tau)\\\
&\times\left[S_{\alpha}S_{\beta}(-\tau)\rho_{S}-S_{\beta}(-\tau)\rho_{S}S_{\alpha}\right]+\text{
H.c. }\bigg{\\}}\end{aligned},$ (21)

where

$C_{\alpha\beta}=\frac{1}{\hbar^{2}}\operatorname{Tr}\left[\rho_{E}E_{\alpha}E_{\beta}(-\tau)\right]$
(22)

and

$\displaystyle S_{\beta}(-\tau)$
$\displaystyle=e^{-\frac{iH_{S}\tau}{\hbar}}S_{\beta}e^{\frac{iH_{S}\tau}{h}},$
(23) $\displaystyle E_{\beta}(-\tau)$
$\displaystyle=e^{-\frac{iH_{E}\tau}{\hbar}}E_{\beta}e^{\frac{iH_{E}\tau}{h}}.$

In Eq. (21) we have made the Born and Markov approximations. The Born
approximation considers the environment to be much larger than the system, and
the coupling between $S$ and $\mathcal{E}$ to be weak enough that it is
possible at all times to write the composite $S+\mathcal{E}$ system as a
tensor product:

$\hat{\rho}_{S+\mathcal{E}}(t)\approx\hat{\rho}_{S}(t)\otimes\hat{\rho}_{\mathcal{E}},$
(24)

where $\hat{\rho}_{\mathcal{E}}$ is approximately constant at all times.

The Markov approximation supposes that the memory effects in the environment
are negligible, i.e. any effect that the system has on the environment decays
rapidly compared to the evolution of the environment $\mathcal{E}$ itself.

Additionally we assume that the time evolution of the operator $S_{\beta}$ can
be neglected with respect to the correlation time scale of the environment,
which is applicable for example when the centre-of-mass of the crystal is
trapped in a very low-frequency trap, as we will see in th examples of the
QGEM experiment, see Marshman:2023nkh .

We also assume that the unitary time evolution of the crystal (given by
$-{i}\hbar^{-1}\left[H_{S},\rho_{S}\right]$ of Eq. (21)) is much slower than
the non-unitary time evolution (given by the second term of the right-hand-
side of Eq. (21)), which corresponds to changes of the system entirely due to
the decoherence.

This means that the decoherence due to the presence of the environment
modifies the state of the system faster than any free evolution of the system
itself. For this reason, in the derivation of the decoherence rate below, we
will neglect the unitary term $-{i}{\hbar^{-1}}\left[H_{S},\rho_{S}\right]$.

We consider the environmental particles to have a (normalized) wave function
with a localized momentum $\mathbf{p}_{0}$. We represent the environmental
particle using a Gaussian wavepacket centered in $\mathbf{p}_{0}$ in the
momentum space and with a width $\tilde{\sigma}$ and with spin component
$s_{0}$:

$\displaystyle|\psi_{E}\rangle$
$\displaystyle=\int\frac{d^{3}\mathbf{p}}{(2\pi)^{3}\sqrt{2E_{p}}}\tilde{\psi}(p)|\mathbf{p},s_{0}\rangle$
(25)
$\displaystyle=\int\frac{d^{3}p}{(2\pi)^{3}\sqrt{2E_{p}}}\frac{(2\pi)^{3/2}}{\left(\pi\tilde{\sigma}^{2}\right)^{3/4}}e^{-\frac{|\mathbf{p}-\mathbf{p}_{0}|^{2}}{2\tilde{\sigma}^{2}}}|\mathbf{p},s_{0}\rangle.$

From this wavefunction for the environmental particle we compute it’s density
matrix $\hat{\rho}_{E}=\ket{\Psi_{E}}\bra{\Psi_{E}}$.

Now, we can substitute this density matrix in Eq. (22) in order to compute the
coefficients $C_{\alpha,\beta}$. Notice that Eq. (22) depends on two indices,
$\alpha$ and $\beta$, so it is important to find a suitable notation for both
of them. In particular, we will define
$\left\\{\alpha\right\\}\equiv\left\\{\textbf{p}_{i},\textbf{p}_{f},s,s^{\prime},r,r^{\prime}\right\\}$
and
$\left\\{\beta\right\\}\equiv\left\\{\textbf{p}_{i}^{\prime},\textbf{p}_{f}^{\prime},n,n^{\prime},m,m^{\prime}\right\\}$.
At this point, we are ready to express the coefficients $C_{\alpha,\beta}$
explicitly:

$\displaystyle C_{\alpha,\beta}$
$\displaystyle=\Tr[\rho_{E}E_{\alpha}E_{\beta}(-\tau)]=\left\langle
E_{\alpha}E_{\beta}(-\tau)\right\rangle_{\psi_{E}}$ (26)
$\displaystyle=(2\pi)^{3}\mathcal{M}_{p_{i},p_{f}}^{s,s^{\prime},r,r^{\prime}}\left(\mathcal{M}_{p_{f}^{\prime},p_{i}^{\prime}}^{n,n^{\prime},m,m^{\prime}}\right)^{\ast}2E_{p_{i}}\delta^{3}\left(\textbf{p}_{i}-\textbf{p}_{f}^{\prime}\right)\delta_{sn^{\prime}}\bra{\psi_{E}}e^{-i\left(E_{p_{i}^{\prime}}-E_{p_{f}^{\prime}}^{\prime}\right)\tau}\ket{\textbf{p}_{f}}\left\langle\textbf{p}_{i}^{\prime}\mid\psi_{E}\right\rangle$
(27)
$\displaystyle=(2\pi)^{9}\pi^{-3/2}\tilde{\sigma}^{3}\mathcal{M}_{p_{i},p_{f}}^{s,s^{\prime},r,r^{\prime}}\left(\mathcal{M}_{p_{f}^{\prime},p_{i}^{\prime}}^{n,n^{\prime},m,m^{\prime}}\right)^{\ast}2E_{p_{i}}\delta^{3}\big{(}\textbf{p}_{i}-\textbf{p}_{f}^{\prime}\big{)}\delta_{sn^{\prime}}e^{-i\left(E_{p_{i}^{\prime}}-E_{p_{f}^{\prime}}\right)\tau}2E_{p_{i}^{\prime}}$
$\displaystyle\mbox{\quad\quad}\times\delta_{s_{0}s^{\prime}}\delta_{ns_{0}}\delta^{3}\left(\textbf{p}_{i}^{\prime}-\textbf{p}_{0}\right)\delta^{3}\left(\textbf{p}_{0}-\textbf{p}_{f}\right),$
(28)

where factors like $\delta_{s_{0}s^{\prime}}$ indicate Kronecker’s deltas,
because the spin components have discrete values 777The two component spinors
obey the Kronecker delta, e.g.
$\xi^{s_{0}\dagger}\xi^{s^{\prime}}=\delta^{s_{0}s^{\prime}}\equiv\delta_{s_{0}s^{\prime}}$,
where $\xi$ is the two component spinor.. Moreover, in Eq. (28) we have used
the following two properties:

$\begin{cases}\bra{\textbf{p},s}\ket{\textbf{k},r}=(2\pi)^{3}(2E_{p})\delta^{3}(\textbf{p}-\textbf{k})\delta_{sr}\\\
\delta^{3}(\textbf{p})=\lim_{\tilde{\sigma}\to
0}\frac{1}{\tilde{\sigma}^{3}\pi^{3/2}}e^{-|\textbf{p}|^{2}/\tilde{\sigma}^{2}}\end{cases}.$
(29)

The second property in Eq. (29) represents the assumption that the Gaussian
wavepacket of the environmental particle is sharp enough that can be
considered as a Dirac’s delta, i.e. the particle has a well-defined momentum
$\textbf{p}_{0}$.

We are now ready to find an explicit expression for the temporal evolution of
the density matrix $\hat{\rho}_{S}$ as given in Eq. (21). In particular,
plugging Eq. (20) into the second term of the right-hand-side of Eq. (22) (and
neglecting the unitary term $-\frac{i}{\hbar}\left[H_{S},\rho_{S}\right]$ as
discussed above), one obtains:

$\begin{split}\frac{d\hat{\rho}_{S}}{dt}=&-\frac{\pi^{-3/2}\tilde{\sigma}^{3}}{4M^{2}}\int
d\tau\frac{d^{3}\textbf{p}_{i}d^{3}\textbf{p}_{f}d^{3}\textbf{p}^{\prime}_{i}d^{3}\textbf{p}^{\prime}_{f}}{(2\pi)^{3}16E_{p_{i}}E_{p_{f}}E_{p^{\prime}_{i}}E_{p^{\prime}_{f}}}4E_{p_{i}}E_{p^{\prime}_{i}}\sum_{\begin{subarray}{c}s,s^{\prime},r,r^{\prime}\\\
n,n^{\prime},m,m^{\prime}\end{subarray}}\mathcal{M}_{p_{i},p_{f}}^{s,s^{\prime},r,r^{\prime}}\Big{(}\mathcal{M}_{p^{\prime}_{i},p^{\prime}_{f}}^{n,n^{\prime},m,m^{\prime}}\Big{)}^{*}\delta^{3}(\textbf{p}_{i}-\textbf{p}^{\prime}_{f})\\\
&\times\delta_{sn^{\prime}}e^{-i(E_{p^{\prime}_{i}}-E_{p^{\prime}_{f}})\tau}\delta^{3}(\textbf{p}_{0}-\textbf{p}_{f})\delta^{3}(\textbf{p}^{\prime}_{i}-\textbf{p}_{0})\delta_{s_{0}s^{\prime}}\delta_{ns_{0}}\big{\\{}-e^{i(\textbf{p}^{\prime}_{i}-\textbf{p}^{\prime}_{f})\cdot\hat{\textbf{x}}}\ket{m^{\prime}}\bra{m}\hat{\rho}_{S}e^{i(\textbf{p}_{i}-\textbf{p}_{f})\cdot\hat{\textbf{x}}}\ket{r^{\prime}}\bra{r}\\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+e^{i(\textbf{p}_{i}-\textbf{p}_{f})\cdot\hat{\textbf{x}}}\ket{r^{\prime}}\bra{r}e^{i(\textbf{p}^{\prime}_{i}-\textbf{p}^{\prime}_{f})\cdot\hat{\textbf{x}}}\ket{m^{\prime}}\bra{m}\hat{\rho}_{S}+\text{H.c.}\big{\\}},\end{split}$
(30)

which, after solving the Dirac and Kronecker deltas, leads to:

$\displaystyle\frac{d\hat{\rho}_{S}}{dt}=$
$\displaystyle-\frac{\pi^{-3/2}\tilde{\sigma}^{3}}{64\pi^{2}M^{2}E_{p_{0}}}\int\frac{d^{3}\textbf{p}^{\prime}_{f}}{E_{p^{\prime}_{f}}}\times\sum_{\begin{subarray}{c}s,r,r^{\prime}\\\
m,m^{\prime}\end{subarray}}\mathcal{M}_{p_{0},p^{\prime}_{f}}^{s,s_{0},r,r^{\prime}}\Big{(}\mathcal{M}_{p_{0},p^{\prime}_{f}}^{s_{0},s,m,m^{\prime}}\Big{)}^{*}\delta(E_{p_{0}}-E_{p^{\prime}_{f}})$
$\displaystyle\times\big{\\{}-\ket{m^{\prime}}\bra{m}e^{i(\textbf{p}_{0}-\textbf{p}^{\prime}_{f})\cdot\hat{\textbf{x}}}\hat{\rho}_{S}e^{i(\textbf{p}^{\prime}_{f}-\textbf{p}_{0})\cdot\hat{\textbf{x}}}\ket{r^{\prime}}\bra{r}+\delta_{rm^{\prime}}\ket{r^{\prime}}\bra{m}e^{i(\textbf{p}^{\prime}_{f}-\textbf{p}_{0})\cdot\hat{\textbf{x}}}e^{i(\textbf{p}^{\prime}_{i}-\textbf{p}^{\prime}_{f})\cdot\hat{\textbf{x}}}\hat{\rho}_{S}+\text{H.c.}\big{\\}},$
(31)

where the one-dimensional Dirac’s delta $\delta(E_{p_{0}}-E_{p^{\prime}_{f}})$
comes from the fact that we have integrated over $d\tau$ 888
$\delta(E_{p^{\prime}_{i}}-E_{p^{\prime}_{f}})=\int
d\tau\;e^{-i(E_{p^{\prime}_{i}}-E_{p^{\prime}_{f}})\tau}$, while
simultaneously solving the other Dirac delta
$\delta^{3}(\textbf{p}^{\prime}_{i}-\textbf{p}_{0})$ appearing in Eq.(30),
therefore giving
$\delta(E_{p^{\prime}_{i}}-E_{p^{\prime}_{f}})\rightarrow\delta(E_{p_{0}}-E_{p^{\prime}_{f}})$.
At this point, solving $\delta(E_{p_{0}}-E_{p^{\prime}_{f}})$ and using
$dp^{\prime}_{f}=({E_{p^{\prime}_{f}}}/{p^{\prime}_{f}})dE_{p^{\prime}_{f}}$
(we denote $\absolutevalue{\textbf{p}^{\prime}_{f}}$ as $p^{\prime}_{f}$),
gives:

$\begin{split}\frac{d\hat{\rho}_{S}}{dt}=&-\frac{\pi^{-3/2}\tilde{\sigma}^{3}}{64\pi^{2}M^{2}}\frac{p_{0}}{E_{p_{0}}}\int
d\Omega^{\prime}\sum_{\begin{subarray}{c}s,r,r^{\prime}\\\
m,m^{\prime}\end{subarray}}\mathcal{M}_{p_{0},p^{\prime}_{f}}^{s,s_{0},r,r^{\prime}}\Big{(}\mathcal{M}_{p_{0},p^{\prime}_{f}}^{s_{0},s,m,m^{\prime}}\Big{)}^{*}\\\
&\times\big{\\{}-\ket{m^{\prime}}\bra{m}e^{i(\textbf{p}_{0}-\textbf{p}^{\prime}_{f})\cdot\hat{\textbf{x}}}\hat{\rho}_{S}e^{i(\textbf{p}^{\prime}_{f}-\textbf{p}_{0})\cdot\hat{\textbf{x}}}\otimes\ket{r^{\prime}}\bra{r}+\delta_{rm^{\prime}}\ket{r^{\prime}}\bra{m}\hat{\rho}_{S}+H.c.\big{\\}},\end{split}$
(32)

where $d\Omega^{\prime}$ indicates the orientation angle of the final momentum
vector $d^{3}\textbf{p}_{f}=d\Omega^{\prime}dp_{f}p_{f}^{2}$. Note that
$E_{p_{0}}=E_{p^{\prime}_{f}}$ gives
$|\textbf{p}_{0}|=|\textbf{p}^{\prime}_{f}|$. This means also that the matrix
element $\mathcal{M}_{p_{0},p^{\prime}_{f}}$ will now depend only on
$|\textbf{p}_{0}|$ and the angle between $\textbf{p}_{0}$ and
$\textbf{p}^{\prime}_{f}$, i.e. on $\Omega^{\prime}$. We can also rewrite the
${p_{0}}/{E_{p_{0}}}$ term as ${p_{0}}/{E_{p_{0}}}=v_{0}$ 999This relation can
be easily derived from well known relativistic expressions, i.e.
$\frac{p_{0}}{E_{p_{0}}}=\frac{\frac{mv_{0}}{\sqrt{1-v_{0}^{2}}}}{\sqrt{p_{0}^{2}+m^{2}}}=\frac{\frac{mv_{0}}{\sqrt{1-v_{0}^{2}}}}{\sqrt{\frac{m^{2}v_{0}^{2}}{1-v_{0}^{2}}+m^{2}}}=v_{0}$..

The factor $\textbf{p}^{\prime}_{f}-\textbf{p}_{0}$ at the exponent of the
right-hand side of Eq. (32) can be rewritten as
$|\textbf{p}_{0}|(\hat{n}^{\prime}-\hat{n}_{0})$, where $\hat{n}^{\prime}$ and
$\hat{n}_{0}$ are the final and initial direction of the scattered
environmental particle (we will express their associated angles with,
respectively, $\Omega^{\prime}$ and $\Omega_{0}$). We can now use this
notation to rewrite the matrix
$\mathcal{M}_{p_{0},p^{\prime}_{f}}^{s,s_{0},r,r^{\prime}}$ in terms of the
scattering angles $\Omega^{\prime}$ and $\Omega_{0}$: in fact, it is always
possible to find a suitable parameterization of the momenta $\textbf{p}_{0}$
and $\textbf{p}^{\prime}_{f}$ in terms of the COM energy and their angles
$\Omega^{\prime}$ and $\Omega_{0}$. In particular, from now on we will write
$\mathcal{M}_{p_{0},p^{\prime}_{f}}^{s,s_{0},r,r^{\prime}}\rightarrow\mathcal{M}_{\Omega^{\prime},\Omega_{0},E_{0}}^{s,s_{0},r,r^{\prime}}$
Kilian:2022kgm . In this way, it is possible to have the matrix $\mathcal{M}$
with an explicit dependence on the integration variable appearing in Eq.(32),
i.e. $\Omega^{\prime}$.

We can thus rewrite the time evolution equation for $\hat{\rho}_{S}$ as:

$\begin{split}\frac{d\hat{\rho}_{S}}{dt}=&-\frac{\pi^{-3/2}\tilde{\sigma}^{3}}{64\pi^{2}M^{2}}v_{0}\\\
&\times\int d\Omega^{\prime}\sum_{\begin{subarray}{c}s,r,r^{\prime}\\\
m,m^{\prime}\end{subarray}}\mathcal{M}_{\Omega^{\prime},\Omega_{0},E_{0}}^{s,s_{0},r,r^{\prime}}\Big{(}\mathcal{M}_{\Omega^{\prime},\Omega_{0},E_{0}}^{s_{0},s,m,m^{\prime}}\Big{)}^{*}\\\
&\times\big{\\{}-\ket{m^{\prime}}\bra{m}e^{i(\textbf{p}_{0}-\textbf{p}^{\prime}_{f})\cdot\hat{\textbf{x}}}\hat{\rho}_{S}e^{i(\textbf{p}^{\prime}_{f}-\textbf{p}_{0})\cdot\hat{\textbf{x}}}\\\
&\otimes\ket{r^{\prime}}\bra{r}+\delta_{rm^{\prime}}\ket{r^{\prime}}\bra{m}\hat{\rho}_{S}+H.c.\big{\\}}.\end{split}$
(33)

The factor $\pi^{-3/2}\tilde{\sigma}^{3}$ is proportional to the inverse of
the volume ${1}/{V}$, see Appendix A.2. Using this proportionality in Eq.(33)
a term like ${v_{0}}/{V}=F_{1}$ will appear, which is the flux associated with
one environmental particle.

Now, to obtain the matrix elements of $S$, we need to perform the bracket of
the operator $\hat{\rho}_{S}$ with both the spatial degrees of freedom
$\big{\\{}\ket{\textbf{x}}\big{\\}}$ and the spin ones
$\big{\\{}\ket{r}\big{\\}}$. Therefore, contracting $\hat{\rho}_{S}$ with two
generic spin vectors and calling them $\ket{r_{i}}$ and $\ket{r_{f}}$, the RHS
of Eq. (33) becomes:

$\begin{split}\frac{d\hat{\rho}_{S}}{dt}=&-\frac{(2\pi)^{3/2}F_{1}}{64\pi^{2}M^{2}}\\\
&\times\int d\Omega^{\prime}\sum_{\begin{subarray}{c}s,r,r^{\prime}\\\
m,m^{\prime}\end{subarray}}\mathcal{M}_{\Omega^{\prime},\Omega_{0},E_{0}}^{s,s_{0},r,r^{\prime}}\Big{(}\mathcal{M}_{\Omega^{\prime},\Omega_{0},E_{0}}^{s_{0},s,m,m^{\prime}}\Big{)}^{*}\\\
&\times\big{\\{}-\delta_{r_{f}m^{\prime}}\delta_{rr_{i}}\bra{m}e^{i(\textbf{p}_{0}-\textbf{p}^{\prime}_{f})\cdot\hat{\textbf{x}}}\hat{\rho}_{S}e^{i(\textbf{p}^{\prime}_{f}-\textbf{p}_{0})\cdot\hat{\textbf{x}}}\ket{r^{\prime}}\\\
&+\delta_{rm^{\prime}}\delta_{r_{f}r^{\prime}}\bra{m}\hat{\rho}_{S}\ket{r_{i}}+H.c.\big{\\}}.\end{split}$
(34)

Note from Eq.(34), the terms inside the final curly brackets has four
Kronecker deltas, which will annihilate some indices in the summation 101010
In particular, the first two Kronecker deltas
$\delta_{r_{f}m^{\prime}}\delta_{rr_{i}}$ will give
$\sum_{\begin{subarray}{c}s,r,r^{\prime}\\\
m,m^{\prime}\end{subarray}}\rightarrow\sum_{\begin{subarray}{c}s,r^{\prime},m\end{subarray}}$,
while the second two kronecker deltas
$\delta_{rm^{\prime}}\delta_{r_{f}r^{\prime}}$ will give
$\sum_{\begin{subarray}{c}s,r,r^{\prime}\\\
m,m^{\prime}\end{subarray}}\rightarrow\sum_{\begin{subarray}{c}s,r,m\end{subarray}}$..
We can rename the $r^{\prime}$ index in the first summation
$\sum_{\begin{subarray}{c}s,r^{\prime},m\end{subarray}}\rightarrow\sum_{\begin{subarray}{c}s,r,m\end{subarray}}$
to obtain the same indices of the second one, giving us with the unique final
summation $\sum_{\begin{subarray}{c}s,r,m\end{subarray}}$. We can now make
some approximation to simplify Eq.(34). In particular, we will assume that the
interaction happens in the non-relativistic regime. This is a fair assumption,
if one considers the scenario in the QGEM proposal Bose:2017nin , where the
ambient temperature is very low to maintain coherence of the interferometer,
i.e. $T\sim 1\;\text{K}$, see the section below for the parameters of the QGEM
experiment. Therefore, we can use the fundamental properties of the QED at low
energy regime, where the spin components are preserved during the scattering
process 111111In the non-relativistic limit we have that
$p^{\mu}\rightarrow(m,\textbf{p})$ and thus a generic spinor
$u(\textbf{p})=\left(\begin{matrix}\sqrt{\textbf{p}\cdot\sigma}\xi\\\
\sqrt{\textbf{p}\cdot\bar{\sigma}}\xi\end{matrix}\right)\rightarrow\sqrt{m}\left(\begin{matrix}\xi\\\
\xi\end{matrix}\right)$, where $\sigma$ represents the Pauli matrices and
$\xi$ is a generic $2$ component spinors. This leads to the following spinor
contractions $\sum_{s,s^{\prime}}\bar{u}^{s^{\prime}}u^{s}\rightarrow
2m\delta_{ss^{\prime}}$ which is responsible to the existence of the spin
conservation term $\delta_{ss^{\prime}}$ inside the matrix
$\mathcal{M}^{ss^{\prime}}$. For further details, see tong .. This property
leads to the following form for the scattering matrix
$\mathcal{M}_{s,s_{0}}^{r,r^{\prime}}\rightarrow\mathcal{M}\;\delta^{rr^{\prime}}\delta^{ss_{0}}$.
Eq.(34) thus becomes:

$\begin{split}\frac{d\hat{\rho}_{S}}{dt}=&-\frac{(2\pi)^{3/2}F_{1}}{64\pi^{2}M^{2}}\int
d\Omega^{\prime}|\mathcal{M}(\Omega^{\prime},\Omega_{0},E_{0})|^{2}\\\
&\times\Big{\\{}-\bra{r_{f}}e^{i(\textbf{p}_{0}-\textbf{p}^{\prime}_{f})\cdot\hat{\textbf{x}}}\hat{\rho}_{S}e^{i(\textbf{p}^{\prime}_{f}-\textbf{p}_{0})\cdot\hat{\textbf{x}}}\ket{r_{i}}\\\
&+\bra{r_{f}}\hat{\rho}_{S}\ket{r_{i}}+H.c.\Big{\\}}\end{split}$ (35)

Now, contracting over the spatial components, $\ket{\textbf{x}}$, and defining
$\bra{\textbf{x},r_{f}}\hat{\rho}_{S}\ket{\textbf{y},r_{i}}\equiv\rho_{S}^{r_{i},r_{f}}(\textbf{x},\textbf{y},t)$,
we obtain:

$\displaystyle\frac{d\rho_{S}^{r_{i},r_{f}}}{dt}$
$\displaystyle(\textbf{x},\textbf{y},t)=-\frac{2(2\pi)^{3/2}F_{1}}{64\pi^{2}M^{2}}\int
d\Omega^{\prime}|\mathcal{M}(\Omega^{\prime},\Omega_{0},E_{0})|^{2}$ (36)
$\displaystyle\times\left\\{-e^{-ip_{0}\left(\hat{n}^{\prime}-\hat{n}_{0}\right)\cdot(\textbf{x}-\textbf{y})}+1\right\\}\rho_{S}^{r_{i},r_{f}}(\textbf{x},\textbf{y},t),$

where we have defined
$\rho_{S}^{r_{i},r_{f}}(\textbf{x},\textbf{y},t)\equiv\bra{\textbf{x},r_{f}}\hat{\rho}_{S}\ket{\textbf{y},r_{i}}$.
The term
$(64\pi^{2}M^{2})^{-1}|\mathcal{M}(\Omega^{\prime},\Omega_{0},E_{0})|^{2}$ is
exactly the differential cross-section,
${d\sigma}/{d\Omega^{\prime}}={d\sigma}(\hat{n}_{0},\hat{n}^{{}^{\prime}})/{d\Omega^{\prime}}$,
(see Appendix (A.1)). Note that Eq. (36) has the same form as the result found
in Ref. Kilian:2022kgm , but with a different scattering amplitude, there it
was the weak interaction, and here it is the electromagnetic interaction.

Solution of the Eq. (36) will dictate the decoherence of the matter-wave
interferometer due to the interaction between one environmental particle and
the micro-crystal. It is easy to generalize it to the case when we have $N$
environmental particles. The flux is then given by:

$F=n\bar{v}=\frac{N}{V}\bar{v},$ (37)

where $\bar{v}$ is the average velocity of each environmental particle. In
general, if each particle gives a contribution of the type Eq. (36) to the
decoherence rate, the total contribution will be given by the sum over all
possible momenta $\textbf{p}_{0}$ (and velocities $\textbf{v}_{0}$) weighted
by a distribution of particles $\mu(\textbf{p}_{0})$ in the momentum space,
i.e. a probabilistic distribution such that $\int
d^{3}\textbf{p}_{0}\;\mu(\textbf{p}_{0})=1$. Assuming that the environment is
composed of particles that are isotropically distributed:

$\mu(\textbf{p}_{0})d^{3}\textbf{p}_{0}=\frac{1}{4\pi}S(p_{0})dp_{0}\;d\Omega_{0},$
(38)

where $\int dp_{0}\;S(p_{0})=1$ (such that $\int
d^{3}\textbf{p}_{0}\;\mu(\textbf{p}_{0})=1$). Therefore, we obtain

$\displaystyle\frac{d\rho_{S}}{dt}(\textbf{x},$
$\displaystyle\textbf{y},t)=-2(2\pi)^{\frac{3}{2}}\int
dp_{0}S(p_{0})\;nv(p_{0})\int\frac{d\Omega_{0}d\Omega^{\prime}}{4\pi}$ (39)
$\displaystyle\times\frac{d\sigma}{d\Omega^{\prime}}\left\\{-e^{-ip_{0}\left(\hat{n}^{\prime}-\hat{n}_{0}\right)\cdot(\textbf{x}-\textbf{y})}+1\right\\}\rho_{S}(\textbf{x},\textbf{y},t),$

where we have not written explicitly the spin labels $r_{i}$ and $r_{f}$ for
brevity. Hence, only considered the spatial part of $\hat{\rho}_{S}$. As can
be seen from Eq.(39), the time evolution of the density matrix will be of the
type:

$\begin{split}\frac{d\rho_{S}(\textbf{x},\textbf{y},t)}{dt}&=-\Gamma\cdot\rho_{S}(\textbf{x},\textbf{y},t)\\\
\Rightarrow\rho_{S}(\textbf{x},\textbf{y},t)&=e^{-\Gamma
t}\rho_{S}(\textbf{x},\textbf{y},0),\end{split}$ (40)

with

$\displaystyle\Gamma\equiv 2(2\pi)^{3/2}\int
dp_{0}S(p_{0})\;nv(p_{0})\int\frac{d\Omega_{0}d\Omega^{\prime}}{4\pi}$ (41)
$\displaystyle\times\frac{d\sigma}{d\Omega^{\prime}}(\hat{n}_{0},\hat{n}^{{}^{\prime}})\left\\{-e^{-ip_{0}\left(\hat{n}^{\prime}-\hat{n}_{0}\right)\cdot(\textbf{x}-\textbf{y})}+1\right\\}.$

Eq. (40) shows the role played by $\Gamma$: it suppresses the off-diagonal
density matrix elements, while leaving the diagonal matrix elements unchanged.
The decay factor $\Gamma$ dictates the decoherence rate Schlosshauer:2014pgr .

In the limit $t\rightarrow+\infty$, these off-diagonal elements are completely
suppressed, i.e. they become asymptotically $0$. This means that the density
matrix $\rho_{S}$ becomes diagonal, indicating a classical mixed state. The
environment thus over time acquires information from the system, leading to
the information leaking to the environment, requiring an observer to measure
both the crystal and environment to regain the system’s quantum information.
This is why $\Gamma$ as given in Eq. (41) is called the decoherence rate, it
causes the decoherence of the system over time.

##  III Dipole-Dipole interaction

In this section, we will consider a special case of electromagnetic
interaction; the dipole-dipole interaction. This type of interaction is
relevant also in the case of neutral test masses. Typically crystals such as
diamond possesses dielectric properties. Therefore, we will consider a neutral
diamond type crystal. Diamnond also provides defects such as NV centers, which
helps to create the spatial superposition, see Bose:2017nin ;
PhysRevLett.125.023602 ; Marshman:2021wyk ; Zhou:2022epb . In this section, we
have also converted the expressions from the previous section to the SI units,
as we will discuss the physics of the experimental aspects.

To obtain an explicit expression for the decoherence rate (see Eq. (41)), we
have to specify the differential cross section related to the type of
interaction between the environment and the crystal. Therefore, let us assume
that the environmental particles are gas molecules with a non-zero dipole
moment. The dipole-dipole potential is given by jackson_classical_1999 :

$\begin{split}V(\textbf{r})=\frac{1}{4\pi\epsilon_{0}}\frac{\textbf{d}_{1}\cdot\textbf{d}_{2}-3\left(\textbf{n}\cdot\textbf{d}_{1}\right)\left(\textbf{n}\cdot\textbf{d}_{2}\right)}{|\textbf{r}|^{3}},\end{split}$
(42)

Figure 1: The figure shows a schematic representation of the dipole-dipole
interaction. The crystal has a finite size $R$ and possesses a dipole
$\textbf{d}_{1}$ parallel to the $\hat{\textbf{z}}$-axis, while the
environmental particle has a dipole $\textbf{d}_{2}$ which is randomly
oriented. The vector r denotes the distance between the centers of the two
dipoles, while we can define the orientation of the two dipoles,
$\textbf{d}_{1}$ and $\textbf{d}_{2}$, with respect to this vector as
$\theta_{1}$ and $\theta_{2}$, respectively. The time evolution of the
superposition size is illustrated by the dotted lines.

where $\textbf{d}_{1}$ and $\textbf{d}_{2}$ are the electric dipole of the
crystal and of the environmental particle respectively, r represents the
distance between the centers of the two dipoles, and
$\textbf{n}=\textbf{r}/|\textbf{r}|$ is its associated unit vector. Figure 1
shows a schematic representation of a generic dipole-dipole interaction, where
the crystal is represented to be in a spatial superposition. We have also
introduced the angles $\theta_{1}$ and $\theta_{2}$ between r and respectively
$\textbf{d}_{1}$ and $\textbf{d}_{2}$.

Our goal is to compute the differential cross-section for this type of
potential. A very useful tool for this purpose is the Born approximation ,
which expresses the differential cross section ${d\sigma}/{d\Omega}$ in terms
of the Fourier Transform $\tilde{V}(q)$ of the potential $V(\textbf{r})$:

$\frac{d\sigma}{d\Omega}=\frac{m^{2}}{4\pi^{2}\hbar^{4}}|\tilde{V}(q)|^{2},$
(43)

with $\tilde{V}(q)$ is the Fourier transform of the potential given by Eq.
(42), i.e. $\tilde{V}(\textbf{q})=\int
d^{3}\textbf{x}\;e^{\frac{i}{\hbar}\textbf{q}\cdot\textbf{x}}V(\textbf{r})$.
In particular, defining the vectorial components for
$\textbf{d}_{1}=(0,0,d_{1})$, $\textbf{d}_{2}=(d_{2x},d_{2y},d_{2z})$ and
$\textbf{n}=(\sin(\bar{\theta})\cos(\bar{\phi}),\sin(\bar{\theta})\sin(\bar{\phi}),\cos(\bar{\theta}))$,
we can perform the angular integral over
$d\bar{\Omega}=d\bar{\phi}d\bar{\theta}\;\sin(\bar{\theta})$:

$\begin{split}&\int
d\bar{\phi}d\bar{\theta}\;\sin(\bar{\theta})\frac{e^{\frac{i}{\hbar}qr\;\cos(\bar{\theta})}}{4\pi\epsilon_{0}}\frac{\textbf{d}_{1}\cdot\textbf{d}_{2}-3\left(\textbf{n}\cdot\textbf{d}_{1}\right)\left(\textbf{n}\cdot\textbf{d}_{2}\right)}{r^{3}}\\\
&=-\frac{2d_{1}d_{2z}}{\epsilon_{0}}\frac{\hbar^{3}}{{q^{3}r^{6}}}\bigg{[}\left(\frac{r^{2}}{\hbar^{2}}q^{2}-3\right)\sin\big(\frac{r}{\hbar}q\big{missing})\\\
&\mbox{\quad\quad}\mbox{\quad\quad}\mbox{\quad\quad}\mbox{\quad\quad}\mbox{\quad\quad}+3\frac{r}{\hbar}q\;\cos\big(\frac{r}{\hbar}q\big{missing})\bigg{]}\end{split}$
(44)

where $\textbf{q}=\textbf{p}_{f}-\textbf{p}_{0}$ is the transferred momentum
of the scattered particle.

In a low-energy regime that we have assumed, the environmental particle will
not be able to penetrate the crystal. We therefore assume a minimal distance
of interaction equal to the radius of the crystal, $R$. This gives a lower
limit for the integral over $r$:

$\begin{split}\tilde{V}(q)&=-\frac{2d_{1}d_{2z}\hbar^{3}}{\epsilon_{0}q^{3}}\int_{R}^{+\infty}dr\;r^{2}\\\
&\mbox{\quad\quad}\times\frac{1}{{r^{6}}}\bigg{[}\left(\frac{r^{2}}{\hbar^{2}}q^{2}-3\right)\sin\big(\frac{r}{\hbar}q\big{missing})+3\frac{r}{\hbar}q\;\cos\big(\frac{r}{\hbar}q\big{missing})\bigg{]}\\\
&=\frac{2d_{1}d_{2z}\hbar^{3}}{\epsilon_{0}R^{3}q^{3}}\bigg{[}\sin\big(\frac{R}{\hbar}q\big{missing})-\frac{R}{\hbar}q\;\cos\big(\frac{R}{\hbar}q\big{missing})\bigg{]}.\end{split}$
(45)

The minimum interaction distance $R$ in this case is completely equivalent to
considering a form factor $F(q)$, which is compulsory especially when the
environmental particle is able to resolve the dimensions of the crystal, i.e.
when $\lambda\ll R$. In particular, considering a constant charge distribution
density for the crystal $\rho(\textbf{x})\equiv c$ for $r\leq R$ and
$\rho(\textbf{x})\equiv 0$ otherwise, the form factor will be given by:

$\displaystyle F(q)$ $\displaystyle=\int
d^{3}\textbf{x}\;e^{\frac{i}{\hbar}\textbf{q}\cdot\textbf{x}}\rho(\textbf{x})=\int
d\Omega\int_{0}^{R}dr\;r^{2}e^{\frac{i}{\hbar}\textbf{q}\cdot\textbf{x}}\cdot
c$
$\displaystyle\propto\frac{\sin\big(\frac{R}{\hbar}q\big{missing})-\frac{R}{\hbar}q\;\cos\big(\frac{R}{\hbar}q\big{missing})}{\big{(}\frac{R}{\hbar}q\big{)}^{3}}\,,$
(46)

which is exactly the term that appears inside the parentheses of the right-
hand-side of Eq. (45).

Eq. (45) can be inserted inside the Born formula in order to find the
differential cross section ${d\sigma}/{d\Omega}$. After computing the modulus
squared of $\tilde{V}(\boldsymbol{q})$ (see Eq. (45)) we can average the final
result over the orientation of the environmental dipole $d_{2}$, which in Eq.
(45) appears with the term $d_{2z}=d_{2}\cos(\theta_{2})$. We can
schematically summarize these operations as:
$\tilde{V}(q)\rightarrow|\tilde{V}(q)|^{2}\rightarrow\int
d\phi_{2}d\theta_{2}\;\sin(\theta_{2})|\tilde{V}(q)|^{2}$.

The final expression for the differential cross-section will thus be given by:

$\begin{split}\frac{d\sigma}{d\Omega}=\frac{4\hbar^{2}m^{2}d_{1}^{2}d_{2}^{2}}{3\pi\epsilon_{0}^{2}R^{6}q^{6}}\bigg{[}\sin\big(\frac{R}{\hbar}q\big{missing})-\frac{R}{\hbar}q\;\cos\big(\frac{R}{\hbar}q\big{missing})\bigg{]}^{2},\end{split}$
(47)

where $m$ is the mass of the environmental dipole, $d_{1}$ ($d_{2}$) is the
strength of the crystal (environmental) dipole, $R$ is the radius of the
crystal, and $q$ the momentum transfer from the scattered particle to the
crystal. $\hbar$ is the Planck constant and $\epsilon_{0}$ is the vacuum
permittivity. We use Eq. (47) to find the decoherence rate in the long-
wavelength and short-wavelength approximations.

###  III.1 Short-wavelength limit approximation

In the short-wavelength limit the wavelength of the environmental particle,
$\lambda_{0}$, is small compared to the superposition size, $\Delta x$, of the
spatially superposed crystal, see Fig.1:

$\lambda_{0}\ll\Delta x\,.$

This implies that $p_{0}\Delta x\gg 1$, meaning that when the phase
exponential $e^{-ip_{0}(\hat{n}^{\prime}-\hat{n}_{0})\cdot(\vec{x}-\vec{y})}$
in Eq. (41) is integrated over $p_{0}$, it will oscillate very fast and its
contribution will be negligible. Therefore the approximation of Eq. (41) for
the decoherence rate in the short-wavelength limit becomes:

$\displaystyle\Gamma_{S}=2(2\pi)^{3/2}\int
dp_{0}S(p_{0})\;nv(p_{0})\;\sigma_{\text{CM}}(p_{0}),$ (48)

where

$\sigma_{\text{CM}}=\int\frac{d\Omega_{0}d\Omega^{\prime}}{4\pi}\frac{d\sigma}{d\Omega^{\prime}}(\hat{n}_{0},\hat{n}^{{}^{\prime}}),$

is the total cross-section and the subscript $S$ in $\Gamma_{S}$ stands for
‘Short’, indicating that we are computing the decoherence rate in the short-
wavelength limit.

Before computing explicitly $\sigma_{\text{CM}}$, let us find out where the
angular dependence comes from inside Eq. (47). We know that $q$ is the
transferred momentum: $\vec{q}=\vec{p^{\prime}}_{f}-\vec{p}_{0}$. However, we
have assumed that the environmental particle is significantly lighter than the
crystal and that the environmental dipole possesses lower energy compared to
the mass-energy of the crystal. These approximations were already introduced
in the Section II. In particular, as can be seen from Eq. (II.2), these
assumptions lead to the conservation of energy through Dirac’s delta
$\delta(E_{p^{\prime}_{f}}-E_{p_{0}})$. This means that
$E_{p^{\prime}_{f}}=E_{p_{0}}$ and therefore $p^{\prime}_{f}=p_{0}$, i.e. the
modulus of the momentum of the environmental particle is the same before and
after the scattering process. Only the direction of the final momentum is
different from the initial one, given our assumptions.

This means that the transferred momentum $q$ will have the following
expression:

$\begin{split}q&=\sqrt{p_{f}^{\prime
2}+p_{0}^{2}-2\vec{p}\;^{\prime}_{f}\cdot\vec{p}_{0}}=\sqrt{2p_{0}^{2}-2p_{0}^{2}\cos(\theta^{\prime})}\\\
&=2p_{0}\sin(\theta^{\prime}/2).\end{split}$ (49)

Using Eqs. (47), (49) we compute $\sigma_{\text{CM}}$:

$\begin{split}\sigma_{\text{CM}}&=\int\frac{d\Omega_{0}}{4\pi}\int
d\Omega^{\prime}\frac{4\hbar^{2}m^{2}d_{1}^{2}d_{2}^{2}}{3\pi\epsilon_{0}^{2}R^{6}q^{6}}\bigg{(}\sin\Big(\frac{R}{\hbar}q\Big{missing})-\frac{R}{\hbar}q\;\cos\Big(\frac{R}{\hbar}q\Big{missing})\bigg{)}^{2}\\\
&=\frac{\hbar^{2}m^{2}d_{1}^{2}d_{2}^{2}}{48\pi\epsilon_{0}^{2}R^{6}p_{0}^{6}}2\pi\int_{0}^{\pi}d\theta^{\prime}\;\sin(\theta^{\prime})\frac{1}{\sin^{6}(\theta^{\prime}/2)}\bigg{(}\sin\Big(2\frac{R}{\hbar}p_{0}\sin(\theta^{\prime}/2)\Big{missing})-2\frac{R}{\hbar}p_{0}\sin(\theta^{\prime}/2)\;\cos\Big(2\frac{R}{\hbar}p_{0}\sin(\theta^{\prime}/2)\Big{missing})\bigg{)}^{2}\\\
&=\frac{m^{2}d_{1}^{2}d_{2}^{2}\hbar^{2}}{48\epsilon_{0}^{2}R^{6}p_{0}^{6}}\bigg{[}-1-8\left(\frac{R}{\hbar}p_{0}\right)^{2}+32\left(\frac{R}{\hbar}p_{0}\right)^{4}+\cos\left(4\frac{R}{\hbar}p_{0}\right)+4\frac{R}{\hbar}p_{0}\;\sin\left(4\frac{R}{\hbar}p_{0}\right)\bigg{]}.\end{split}$
(50)

In order to compute $\Gamma_{S}$ using Eq.(48) we choose a distribution
$S(p_{0})$ for the environmental particles. Supposing that the environmental
dipoles form a thermal bath with temperature $T$, the average energy of one
environmental particle will be $\bar{E}\sim k_{B}T$, which leads to an average
momentum of $\bar{p}\sim\sqrt{2mk_{B}T}$. Therefore, we consider for
$S(p_{0})$ the Maxwell-Boltzmann distribution:

$\displaystyle S(p_{0})=4\pi p_{0}^{2}\Big{(}2\pi
mk_{B}T\Big{)}^{-3/2}e^{-\frac{p_{0}^{2}}{2mk_{B}T}},$ (51)

which satisfies the requirement $\int dp_{0}\;S(p_{0})=1$. For very low
temperatures Eq. (51) is has a strong peak around its mean value
$\bar{p}=\sqrt{2mk_{B}T}$. This is because Eq. (51) is a Gaussian function of
the type $S(p_{0})\sim e^{-{p_{0}^{2}}/{\sigma}}$, where $\sigma$ represents
the standard deviation, which intuitively tells us how much the distribution
is spread over the momentum space. In our case we have $\sigma\propto
mk_{B}T$, where $m$ is the environmental particle’s mass which can be
considered to be molecules or atoms or air molecules. Therefore, at low
temperatures (i.e. $T\sim 1\;\text{K}$) we will have that $\sigma\ll 1$ and
the Maxwell-Boltzmann function will be a very narrow distribution centered
around $\bar{p}=\sqrt{2mk_{B}T}$.

We can thus consider all the particles to have approximately the same momentum
$\bar{p}$, i.e. $S(p_{0})=\delta(p_{0}-\bar{p})$.

Putting the expression for the total cross section given by Eq. (50) inside
Eq. (48), we find the decoherence rate due to dipole-dipole interactions in
the short wavelength limit:

$\begin{split}\Gamma_{S}&=2(2\pi)^{\frac{3}{2}}\int
dp_{0}\;\delta(p_{0}-\bar{p})n\frac{p_{0}}{m}\frac{m^{2}d_{1}^{2}d_{2}^{2}\hbar^{2}}{48\epsilon_{0}^{2}R^{6}p_{0}^{6}}\bigg{[}-1-8\left(\frac{R}{\hbar}p_{0}\right)^{2}+32\left(\frac{R}{\hbar}p_{0}\right)^{4}+\cos\left(4\frac{R}{\hbar}p_{0}\right)+4\frac{R}{\hbar}p_{0}\;\sin\left(4\frac{R}{\hbar}p_{0}\right)\bigg{]}\\\
&=(2\pi)^{\frac{3}{2}}\frac{\hbar^{2}md_{1}^{2}d_{2}^{2}n}{24\epsilon_{0}^{2}R^{6}\bar{p}^{5}}\bigg{[}-1-8\left(\frac{R}{\hbar}\bar{p}\right)^{2}+32\left(\frac{R}{\hbar}\bar{p}\right)^{4}+\cos\left(4\frac{R}{\hbar}\bar{p}\right)+4\frac{R}{\hbar}\bar{p}\;\sin\left(4\frac{R}{\hbar}\bar{p}\right)\bigg{]}.\end{split}$
(52)

Here, $m$ is the mass of the environmental dipole with dipole strength
$d_{2}$, while $d_{1}$ is the strength of the crystal’s dipole, $R$ is the
radius of the crystal, $\bar{p}$ denotes the mean momentum magnitude
$\bar{p}=\sqrt{2mk_{B}T}$ and $n$ is the environmental particle’s number
density, i.e. $\big{[}n\big{]}=\text{m}^{-3}$. $\hbar$ is the Planck constant
and $\epsilon_{0}$ is the vacuum permittivity.

###  III.2 Long-wavelength approximation

We will now consider the long-wavelength approximation to Eq. (41), where the
wavelength associated with the environmental particle is much bigger than the
superposition size, see Fig. 1

$\lambda_{0}\gg\Delta x\,.$

As a result, in the long-wavelength approximation
$ip_{0}(\hat{n}^{\prime}-\hat{n}_{0})\cdot(\vec{x}-\vec{y})\ll 1$, and we can
take the Taylor expansion of the exponential that appears in the right-hand-
side of Eq. (41) to obtain Schlosshauer:2014pgr :

$\frac{i}{\hbar}p_{0}\left(\hat{n}^{\prime}-\hat{n}_{0}\right)\cdot(\vec{x}-\vec{y})+\frac{1}{2\hbar^{2}}p_{0}^{2}[\left(\hat{n}^{\prime}-\hat{n}_{0}\right)\cdot(\vec{x}-\vec{y})]^{2}.$
(53)

The first term in the equation above gives an integral of an odd function, due
to the fact that $\hat{n}^{\prime}-\hat{n}_{0}$ is antisymmetric in the
exchange of $\hat{n}$ and $\hat{n}^{\prime}$, while
$\frac{d\sigma}{d\Omega}(\hat{n}_{0},\hat{n}^{\prime})$ is symmetric, giving a
total odd function.

The second term can be simplified by assuming that the particular direction
$\vec{x}-\vec{y}=|\vec{x}-\vec{y}|\;\hat{s}=\Delta x\;\hat{s}$ of the
scattering center (i.e. of the crystal) does not depend on the direction
$\hat{s}$. We can thus average this term over all possible directions
$\hat{s}$, obtaining Schlosshauer:2014pgr :

$\begin{split}\left(\Delta
x\right)^{2}\frac{1}{3}\sum_{s=x,y,z}\left[\hat{s}\cdot\left(\hat{n}^{\prime}-\hat{n}_{0}\right)\right]^{2}=\frac{1}{3}\left(\Delta
x\right)^{2}|\hat{n}^{\prime}-\hat{n}_{0}|^{2}\\\ =\frac{2}{3}\left(\Delta
x\right)^{2}|1-\hat{n}^{\prime}\cdot\hat{n}_{0}|=\frac{2}{3}\left(\Delta
x\right)^{2}(1-\cos(\theta^{\prime})).\end{split}$ (54)

where $\theta^{\prime}$ is the scattering angle.
Performing the angular integral in Eq. (41) gives:

$\begin{split}&\int
d\Omega^{\prime}\frac{d\sigma}{d\Omega^{\prime}}(\hat{n}_{0},\hat{n}^{{}^{\prime}})\frac{2}{3}(1-\cos(\theta^{\prime}))\\\
&\mbox{\quad\quad}=\frac{2\pi}{3}\int
d(\cos(\theta^{\prime}))(1-\cos(\theta^{\prime}))\frac{d\sigma}{d\Omega^{\prime}}(\hat{n}_{0},\hat{n}^{{}^{\prime}})\equiv\sigma_{\text{eff}},\end{split}$
(55)

where we have integrated over the azimutal angle $\int_{0}^{2\pi}d\phi=2\pi$
and we have defined the effective cross section $\sigma_{\text{eff}}$. Using
Eq. (41), the final expression for the decoherence rate $\Gamma$ in the long-
wavelength limit is:

$\Gamma_{L}=2(2\pi)^{3/2}\Delta x^{2}\int
dp_{0}\;S(p_{0})n\frac{p_{0}}{m}\sigma_{\text{eff}}(p_{0})\frac{p_{0}^{2}}{\hbar^{2}}.$
(56)

which is an expression that can also be found in the literature
Schlosshauer:2014pgr ; joos2003decoherence . The subscript $L$ stands for
‘Long’, since we have computed the decoherence rate in the long-wavelength
limit.

Comparing Eq. (56) and Eq. (48) we notice that we can obtain the long-
wavelength limit expression by substituting
$\sigma_{\text{CM}}\longrightarrow\sigma_{\text{eff}}$ inside the short-
wavelength limit formula and by multiplying it by the term $\Delta
x^{2}{p_{0}^{2}}/{\hbar^{2}}$.

In particular, $\sigma_{\text{eff}}$ will be of the same order as
$\sigma_{\text{CM}}$, given that the only difference between the two is a
purely geometrical factor $\left(1-\cos(\theta^{\prime})\right)$ inside Eq.
(55). Therefore, the main difference between the two expressions is entirely
encoded inside the term $\Delta x^{2}{p_{0}^{2}}/{\hbar^{2}}$. When the
physical situation is that of the short-wavelength limit
($\lambda_{0}\sim{\hbar}/{p_{0}}\ll\Delta x$), then $\Delta
x^{2}{p_{0}^{2}}/{\hbar^{2}}\gg 1$, and the following inequality holds:

$\frac{\Gamma_{S}}{\Gamma_{L}}\sim\frac{\hbar^{2}}{\Delta x^{2}p_{0}^{2}}\ll
1\,\Rightarrow\,\Gamma_{S}\ll\Gamma_{L}.$ (57)

This means that when the physical limit is the short-wavelength limit, the
long-wavelength decoherence rate given in Eq. (56) can be used as an upper
bound estimation of the decoherence rate.

We now compute $\sigma_{\text{eff}}$ for the dipole-dipole interaction case.
Plugging Eq. (47) into Eq. (55), we obtain:

$\begin{split}\sigma_{\text{eff}}&=\frac{2}{3}\pi\int_{-1}^{1}d\big{[}\cos(\theta^{\prime})\big{]}\big{(}1-\cos(\theta^{\prime})\big{)}\\\
&\mbox{\quad\quad}\times\frac{4\hbar^{2}m^{2}d_{1}^{2}d_{2}^{2}}{3\pi\epsilon_{0}^{2}R^{6}q^{6}}\bigg{(}\sin\big(\frac{R}{\hbar}q\big{missing})-\frac{R}{\hbar}q\;\cos\big(\frac{R}{\hbar}q\big{missing})\bigg{)}^{2}\\\
&=\frac{2}{3}\pi\beta\int_{0}^{1}\differential{u}\;4u\cdot
2u^{2}\frac{\big{[}\sin\big(au\big{missing})-au\;\cos\big(au\big{missing})\big{]}^{2}}{u^{6}},\end{split}$
(58)

where in the very last line we have substituted $q$ with
$q=2p_{0}\sin(\theta^{\prime}/2)$ as in Eq. (49), and we have defined

$\beta\equiv\frac{\hbar^{2}m^{2}d_{1}^{2}d_{2}^{2}}{48\pi\epsilon_{0}^{2}R^{6}p_{0}^{6}}\,,\mbox{\quad\quad}a\equiv
2{Rp_{0}}/{\hbar}$ (59)

Moreover, we defined $u\equiv\sin(\theta^{\prime}/2)$ and used
$\cos(\theta^{\prime})=1-2\;\sin(\theta^{\prime}/2)\equiv 1-2u^{2}$, to
rewrite the integral:

$\begin{split}\int_{-1}^{1}d\big{[}\cos(\theta^{\prime})\big{]}\big{(}1-\cos(\theta^{\prime})\big{)}&=\int_{0}^{1}\differential{u}\;4u\cdot
2u^{2}.\end{split}$ (60)

The integral inside Eq. (58) can now be computed analytically:

$\displaystyle\sigma_{\text{eff}}$
$\displaystyle=\frac{16}{3}\pi\beta\int_{0}^{1}\differential{u}\frac{\big{[}\sin\big(au\big{missing})-au\;\cos\big(au\big{missing})\big{]}^{2}}{u^{3}}$
(61)
$\displaystyle=\frac{4}{3}\pi\beta\bigg{[}a^{2}\big{(}\ln(4a^{2})-2\;\text{Ci}(2a)+2(\gamma-1)\big{)}$
$\displaystyle\mbox{\quad\quad}\mbox{\quad\quad}\mbox{\quad\quad}+2a\;\sin(2a)+\cos(2a)-1\bigg{]},$
(62)

where $\gamma\simeq 0.577$ is the Euler-Mascheroni constant and

$\text{Ci}(2a)=-\int_{2a}^{+\infty}dt\frac{\cos(t)}{t}\,,$

is a known trigonometric integral.

In the limit $p_{0}\to 0$ (i.e. $a\to 0$) the effective cross section (Eq.
(62)) is well-defined, it goes to zero 121212This can be easily seen from the
expansion of
$\text{Ci}(x)=\gamma+\ln(x)+\sum_{n=1}^{+\infty}\frac{(-x^{2})^{n}}{2n(2n)!}=\gamma+\ln(x)-\frac{x^{2}}{2\cdot
2!}+\frac{x^{4}}{4\cdot 4!}+...$ For $x\rightarrow 0$, one can see that the
only terms in the expansion that survive are $\gamma+\ln(x)$, i.e. for small
$x$ we have that $\text{Ci}(x)-\ln(x)\sim\gamma$, which is the reason why the
overall limit is $0$.. Since $p_{0}\propto T$, this means that the decoherence
rate is well-defined for very small temperatures. Specifically in the limit
$T\to 0$, the decoherence rate given in Eq. (56) becomes $\Gamma_{L}\to 0$.

We compute explicitly Eq.(56) in terms of the newly defined effective cross
section. We can again consider a momentum distribution of the environmental
particles like $S(p_{0})=\delta(p_{0}-\bar{p})$, and further approximate Eq.
(62) using the property $\lim_{z\to\infty}\text{Ci}(z)=0$. This is possible
when, as can be seen from Eq. (62), ${R\bar{p}}/{\hbar}\gg 1$, true in most
experimental setups Bose:2017nin ; Afek:2021bua ; Carney_2019 ; Barker:2022mdz
. The average momentum from atomic masses at low temperatures ($\sim 1\ \text{
K}$) is $\bar{p}=\sqrt{2mk_{B}T}\sim
10^{-25}\text{kg}\;\text{m}\;\text{s}^{-1}$.

For a micron size spherical crystal one thus finds ${R\bar{p}}/{\hbar}\sim
10^{3}\gg 1$, meaning that $\text{Ci}(10^{3})\sim 10^{-4}$. We therefore
approximate $\sigma_{\text{eff}}$ as (see Eq. (62)):

$\begin{split}\sigma_{\text{eff}}(\bar{p})\simeq\frac{4}{3}\pi\beta
a^{2}\ln(4a^{2})=\frac{2m^{2}d_{1}^{2}d_{2}^{2}}{9\epsilon_{0}^{2}R^{4}\bar{p}^{4}}\ln\bigg(4\frac{R}{\hbar}\bar{p}\bigg{missing}).\end{split}$
(63)

Using this expression for $\sigma_{\text{eff}}(p)$ inside Eq. (56) we obtain:

$\Gamma_{L}=(2\pi)^{\frac{3}{2}}\frac{4md_{1}^{2}d_{2}^{2}n\Delta
x^{2}}{9\epsilon_{0}^{2}\hbar^{2}R^{4}\bar{p}}\ln\bigg(4\frac{R}{\hbar}\bar{p}\bigg{missing}),$
(64)

which represents the final formula for the decoherence rate due to dipole-
dipole interactions in the long-wavelength limit.

As mentioned before, $m$ is the mass of the environmental dipole, $d_{1}$
($d_{2}$) is the strength of the crystal (environmental) dipole, $R$ is the
radius of the crystal, $\bar{p}$ denotes the mean momentum magnitude
$\bar{p}=\sqrt{2mk_{B}T}$, $n$ is the environmental particle’s number density,
i.e. $\big{[}n\big{]}=\text{m}^{-3}$, $\hbar$ is the Planck constant and
$\epsilon_{0}$ is the vacuum permittivity. Compared to Eq. (52), Eq. (64) is
dependent on the spatial superposition size of the crystal, $\Delta x$.

##  IV QGEM experiment

In this section, we explore the decoherence of the dipole-dipole interaction
in the context of the QGEM experiment. Although different setups have been
proposed, generally in the QGEM experiment Bose:2017nin , a spatial
superposition is created of size $\Delta x$ and kept adjacent to each other
for a time $\tau\sim 1\;\text{s}$. In the QGEM experiment the neutral test
masses, which we take to be diamond micron size crystals with an embedded spin
in their NV centre, interact only via gravity, see for details Bose:2017nin ;
vandeKamp:2020rqh . The quantum nature of gravity entangles the test masses,
and therefore the quantum nature of gravity can be empirically measured by
witnessing the generation of an entangled state from an unentangled state
Bose:2017nin ; Marshman:2018upe ; Bose:2022uxe ; Danielson:2021egj ;
Carney_2019 ; Christodoulou:2022vte . It is very important to keep track of
all the possible sources of decoherence, i.e. everything that can lead to a
destruction of the spatial superposition. Considering that the superpositions
are held for a time $\tau\sim 1\;\text{s}$, we require the decoherence rate to
not be bigger than $\Gamma_{S}=10^{-2}\text{ Hz}$. Such a value for
$\Gamma_{S}$ leads to a decoherence time $\tau_{d}=10^{2}\text{ s}\gg\tau$,
therefore making sure that the coherence loss due to the interaction with the
environment is small during the realization of the experiment. This value of
the decoherence rate has been deemed to be safe by various other
considerations, see vandeKamp:2020rqh ; Toros:2020dbf ; Tilly:2021qef ;
Schut:2021svd . For a diamond density of $\rho=3.5\times
10^{3}\text{kg}/\text{m}^{3}$, and for the mass of $\sim 10^{-14}\text{ kg}$
is required to witness the entanglement in the QGEM experiment Bose:2017nin .
We also assume that the experimental box with sides of length $L\sim{\cal
O}(1)$cm Toros:2020dbf .

One of the unavoidable sources of decoherence during the experiment arises
from electromagnetic interactions. Specifically, despite creating a vacuum
with extremely low pressure inside the experimental box Toros:2020dbf , there
is still a possibility that some random air molecules may inadvertently
persist within the box. Such molecules are neutral but can possess an electric
dipole $d_{2}$ of the order of $d_{2}\sim 1\;\text{D}$ table . This means that
the interaction between an environmental particle and the crystal in
superposition is possible if the crystal is made of a dielectric medium and/or
if the crystal also has a dipole $d_{1}$ too, see abdul ; Afek:2021bua .

In this section we are going to analyze three different physical situations
that are likely to happen in a QGEM setup.

  * •

In the first case, (subsection IV.1) the environmental particles generate an
electric field inside the crystal, which will induce a dipole moment in the
crystal because of its dielectric properties.

  * •

In the second case (subsection IV.2), the diamond is considered to have a
permanent dipole, as estimated by some experiments Afek:2021bua . This
permanent dipole moment $d_{1}$ it is treated as a free parameter, on which we
put constraints by requiring that the decoherence rate has to be smaller than
$\Gamma_{S}\sim 10^{-2}\text{Hz}$.

  * •

Finally, in the last case (subsection IV.3) the crystal is considered to have
a permanent dipole, which will generate an electric field that induces an
electric dipole moment in the environmental particles. Considering neutral air
molecules (e.g. $\text{N}_{2},\text{O}_{2},\text{Ar}$ and $\text{CO}_{2}$) we
will be able to find their dipole moment $d_{2}$ induced by the crystal and we
will analyze the resulting decoherence rate.

###  IV.1 Crystal’s dipole induced by the environment

The first case that we will analyze is when $d_{1}$ is induced by the
environment. In fact, being a dielectric material, the diamond crystal has a
polarizability $\alpha$ when subjected to an external electric field
$E_{\text{ext}}$, which inside the crystal is perceived as a local field
$E_{\text{loc}}=E_{\text{ext}}/\epsilon_{r}$, with $\epsilon_{r}\sim 5.7$
being the relative dielectric constant of the crystal. In particular, in
isotropic media, the external field will create a local dipole in each atom of
the crystal’s lattice jackson_classical_1999 :

$d_{1}=\alpha\,E_{\text{loc}}.$ (65)

This means that the total contribution will be given by the sum over all
$N^{\prime}$ atoms of the lattice:

$d_{1}=N^{\prime}\alpha E_{\text{loc}}.$ (66)

So in order to find an explicit expression for $d_{1}$, we need the
expressions for $\alpha$ and $E_{\text{loc}}$. For $\alpha$ we have the
Classius-Mossotti relation:

$\frac{n^{\prime}\alpha}{3\epsilon_{0}}=\frac{\epsilon_{r}-1}{\epsilon_{r}+2},$
(67)

where $n^{\prime}=\frac{N^{\prime}}{V}$ is the atomic density inside the
crystal, the volume is given by $V=\frac{4\pi}{3}R^{3}$ if we assume the
crystals to be perfect spheres. The induced electric field will be generated
by the environmental dipole $d_{2}$ jackson_classical_1999 :

$E_{\text{loc}}=\frac{d_{2}}{2\pi\epsilon_{0}\epsilon_{r}}\frac{1}{\bar{r}^{3}},$
(68)

where we have considered the angular dependence inside Eq. (42) to be maximum,
e.g. $\theta_{1}=0$ and $\theta_{2}=\pi$, which results in a factor of $2$
from the squared parenthesis of Eq. (42). Here, $\bar{r}$ represents the
average interaction distance between an environmental particle and the
crystal. In the worst case scenario, this average distance is given by
$\bar{r}\simeq R\sim 10^{-6}\text{m}$ , which is the closest possible distance
between $d_{1}$ and $d_{2}$. We consider the environment to be composed of
atomic dipoles, for example by Helium atoms. This means that we can consider
$d_{2}$ to be table :

$\displaystyle d_{2}$ $\displaystyle\sim e\cdot R_{a}\sim(1.602\times
10^{-19}\text{C})(10^{-10}\text{m})$ (69) $\displaystyle\sim
10^{-29}\text{C}\cdot\text{m}=3\;\text{D},$

where $R_{a}\sim 10^{-10}\text{m}$ represents the average atomic radius and
$e$ is the electron charge. Here we have used as a unit for the dipole the
Debye, which in standard units is $1\;\text{D}=3.336\times
10^{-30}\;\text{C}\cdot\text{m}$. Another fair assumption would be that there
is some water vapour left in the vacuum. Water has a dipole of $6.19\times
10^{-30}\text{C}\cdot\text{m}$lide2004crc , which is very close to the value
for $d_{2}$ given by Eq.(69).

With this expression for $E_{\text{loc}}$, Eqs. (66), (67) and (68) give the
following value for $d_{1}$:

$d_{1}=\frac{3d_{2}}{2\pi\epsilon_{r}}\left(\frac{\epsilon_{r}-1}{\epsilon_{r}+2}\right)\sim
10^{-30}\;\text{C}\cdot\text{m},$ (70)

where we have used the same value for $d_{2}$ used in Eq.(69), i.e. $d_{2}\sim
10^{-29}\text{C}\cdot\text{m}$.

Let us now compute the decoherence rate due to this type of interaction. We
know that the de Broglie wavelength associated with the environmental ions is
$\lambda_{0}={2\pi\hbar}/{p_{0}}$. If we consider the particles to be inside a
box of temperature $T$, the average momenta of one ion will be
$p_{0}=\sqrt{2mk_{B}T}$, which gives:
$\lambda_{0}={2\pi\hbar}/{\sqrt{2mk_{B}T}}\sim 10^{-9}\text{ m}$, where, we
have used $m\sim 10^{-27}\text{ kg}$ and $T\sim 1\;\text{ K}$. Considering a
superposition size of $\Delta x\sim 10^{-5}\text{m}$ (smaller values would
give an entanglement that is too small to measure), and a mass for the ion
similar to the one of the proton $m\sim 10^{-27}\text{ kg}$, we are in the
short-wavelength limit if

$\begin{split}&\lambda_{0}=\frac{2\pi\hbar}{\sqrt{2mk_{B}T}}\ll\Delta x\\\
&\Rightarrow T\gg\frac{4\pi^{2}\hbar^{2}}{2mk_{B}\Delta x^{2}}\sim
10^{-7}\text{K},\end{split}$ (71)

which is always satisfied in a QGEM setup, considering that $T\sim 1\text{K}$.
Therefore, the short-wavelength limit will be the right approximation to
perform if we want to analyze the QGEM experiment and the decoherence rate
will be therefore given by Eq. (72).

We consider the two dipoles $d_{1}$ and $d_{2}$ to have the values as given in
Eq. (70) and Eq. (69), corresponding to the physical situation where the
environmental dipole $d_{2}$ induces a dipole $d_{1}$ inside the crystal
because of its dielectric properties. Furthermore we take $m\sim
10^{-27}\text{ kg}$ (e.g. Helium molecule), $T\sim 1\;\text{ K}$, $\Delta
x\sim 10^{-5}\text{ m}$, $R\sim 10^{-6}\text{ m}$ and $n\sim 10^{8}\text{
m}^{-3}$. This last value can be related also to the pressure $p$, which is
the actual parameter that is controlled in a QGEM setup proposal. We can
consider $n$ and $p$ to be correlated through the perfect gas law $p=nk_{B}T$,
which gives the corresponding value for the pressure of the order of $p\sim
10^{-15}\;\text{ Pa}$.

In a QGEM setup, we have that $R\sim 10^{-6}\,\mathrm{m}$ and $\bar{p}\sim
10^{-25}\,\mathrm{kg\,m\,s^{-1}}$, giving ${R\bar{p}}/{\hbar}\sim 10^{3}\gg
1$. This means that, in the final expression for the short-wavelength
approximation of the decoherence rate given in Eq. (52), the dominant term
inside the parenthesis is $32\left({R\bar{p}}/{\hbar}\right)^{4}$, while the
others can be considered negligible compared to this one. This gives the
following approximation for the decoherence rate:

$\Gamma_{S}\simeq(2\pi)^{\frac{3}{2}}\frac{2d_{1}^{2}d_{2}^{2}n}{3\epsilon_{0}^{2}\hbar^{2}R^{2}}\sqrt{\frac{2m}{k_{B}T}}.$
(72)

Filling in all the values for the variables that appear in the final
decoherence rate as discussed above, Eq. (72) gives:

$\Gamma_{S}\sim 10^{-9}\;\text{Hz}.$ (73)

This means that, if the value for $\Gamma_{S}$ found in Eq. (73) holds, the
off-diagonal elements of $\rho_{S}$ will be suppressed in a time
$t_{\text{supp}}$ of the order of $t_{\text{supp}}\sim 10^{9}\text{s}$. This
is a very high value, especially compared to the time $\tau\sim 1\;\text{s}$
during which the crystal is kept in superposition in the QGEM proposal.
Therefore, assuming that the environmental dipole $d_{2}$ generates a
crystal’s dipole $d_{1}$ given by Eq. (70), the dipole-dipole interaction
analyzed in this work should not give problems for the realization of the
experiment.

Figure 2: This figure illustrates the relationship between the decoherence
rate $\Gamma_{S}$, calculated in the short-wavelength limit using Eq.(72), and
the temperature $T$. In particular, the scenario under investigation
corresponds to the scenario where the environmental dipole $d_{2}$ (assumed to
be on the order of $d_{2}\sim 1\;\text{D}$ table ) induces an electric dipole
within the micro-crystal, $d_{1}$ due to its dielectric properties, as is
analyzed in Eq.(70). With an environmental particle’s mass of $m\sim
10^{-27}\text{kg}$, $T\sim 1\;\text{K}$, $\Delta x\sim 10^{-5}\text{m}$ and
the crystal’s radius $R\sim 10^{-6}\text{m}$. For a range of temperatures
$T\in\\{10^{-1}\text{K},10\;\text{K}\\}$, the figures shows
$-\log_{10}({\Gamma_{S}}/{\text{Hz}})\in\\{5,9\\}$ which corresponds to a
decoherence rate $\Gamma_{S}\in\\{10^{-9}\text{Hz},10^{-5}\text{Hz}\\}$.
Figure 3: This figure is illustrates the relationship between the crystal’s
permanent dipole $d_{1}$, and the environmental dipole $d_{2}$ for a given
decoherence rate $\Gamma_{S}$. These two variables are related through
Eq.(72). The various plots with different colours correspond to different
values for the decoherence rate $\Gamma_{S}\in\left\\{10^{-5}\text{H
z},10^{-2}\text{ Hz}\right\\}$. The values for all the other parameters that
appear in Eq.(72) are $m\sim 10^{-27}\text{kg}$, $T\sim 1\;\text{K}$, $\Delta
x\sim 10^{-5}\text{m}$ and $R\sim 10^{-6}\text{m}$. The graph shows that if
$d_{2}$ is considered to be of the order of $d_{2}\sim 3.3\times
10^{-30}\;\text{C}\cdot\text{m}$ table , then $d_{1}$ will be of the order of
$d_{1}<10^{-26}\;\text{C}\cdot\text{m}=10^{-1}\text{e}\cdot\mu\text{m}$ if we
require $\Gamma_{S}<10^{-2}$ Hz.

The general behaviour of the decoherence rate is shown in figure 2. Figure 2
shows the behaviour of $-\log_{10}(\frac{\Gamma_{S}}{\text{Hz}})$ as a
function of the ambient temperature $T$, where the different plots correspond
to different values for the pressure
$p\in\\{10^{-15}\text{Pa},10^{-12}\text{Pa}\\}$ (or, equivalently, for the
number density $n$). For a range of temperatures
$T\in\\{10^{-1}\text{K},10\;\text{K}\\}$, we find
$-\log_{10}(\frac{\Gamma_{S}}{\text{Hz}})\in\\{5,9\\}$, which corresponds to
the decoherence rate in the range:
$\Gamma_{S}\in\\{10^{-9}\text{Hz},10^{-5}\text{Hz}\\}$.

###  IV.2 Constraint for the crystal’s dipole

The short wavelength limit decoherence rate given in Eq. (52) is also used to
study the decoherence from the dipole interaction between the permanent dipole
of the crystal and the permanent dipole of the environmental particle. By
requiring a maximal decoherence rate we put constraints on the crystal’s
allowed dipole moment, $d_{1}$.

Fig. 3 shows the maximum allowed value of the crystal’s dipole, $d_{1}$, as a
function of the environmental dipole, $d_{2}$, for different orders of
magnitude of the decoherence rate
$\Gamma_{S}\in\\{10^{-5}\text{Hz},10^{-2}\text{ Hz}\\}$.

We require the decoherence rate $\Gamma_{S}$ to be smaller than $10^{-2}\text{
Hz}$, as discussed at the beginning of this section. The environmental dipole
should be associated in a realistic experiment to molecules present, for
example, in the air. The values for the electric dipole of this type of
molecules center around $d_{2}\sim 1\;\text{D}\simeq 3.336\times
10^{-30}\text{C}\cdot\text{m}$ table , see Eq. (69). With this order for
magnitude of $d_{2}$, figure 3 shows that the crystal’s dipole $d_{1}$ should
be of $d_{1}<10^{-26}\;\text{C}\cdot\text{
m}=10^{-1}\text{e}\cdot\mu\text{m}$.

For microspheres with diameters of the order $\sim 10\,\mu\text{m}$ the dipole
moment has been experimentally estimated to be of the order $\sim
10^{-23}\text{C}\cdot\text{m}$ Afek:2021bua . This situation is different from
the one considered in this paper, where the radius of the micro-crystal in
superposition is considered to be of the order of $\sim 1\,\mu\text{m}$, i.e.
ten times smaller. Recent studies Rivic  on measurement of the scaling of the
dipole moment suggest that the electric dipole moment $d$ should scale with
the number of atoms $N$ of the crystal. Therefore, considering a micro-crystal
with a constant number density $n$ of atoms, the total number of atoms $N$
increases with the volume $N=nV=n({4\pi}/{3})R^{3}$. This suggests that the
dipole will scale as $d_{1}\propto R^{3}$. Assuming that the measurements done
in Afek:2021bua  can be adapted to the situation considered in this paper by
scaling $d$ with the radius to estimate the permanent dipole magnitude for our
micro-crystals, we find $d_{1}\sim 10^{-26}\text{C}\cdot\text{m}$, which is
exactly the magnitude of the upper bound for the crystal’s dipole found in
this work, see Fig. 3.

###  IV.3 Induced environmental dipoles

particle |  $\alpha^{\prime}$ (Å3) |  $d_{2}\,(\text{C}\cdot\text{m})$
---|---|---
N2 |  $1.710$ |  $3.425\times 10^{-35}$
O2 |  $1.562$ |  $3.13\times 10^{-35}$
Ar |  $1.664$ |  $3.335\times 10^{-35}$
CO2 |  $2.507$ |  $5.02\times 10^{-35}$
Table 1: The polarizability $\alpha^{\prime}$ for the most common air
particles: dinitrogen (N2), carbondioxide (CO2), Argon (Ag) and dioxygen (O2),
in CGS units pollistx . Converting in SI units gives the polarizability
$\alpha=4\pi\epsilon_{0}\alpha^{\prime}$. The induced dipole is found from the
permanent dipole electric field of the crystal ($E$, see Eq. (74)) and the
polarizability ($\alpha$), see Eq. (65):
$\boldsymbol{d}_{2}=\alpha\boldsymbol{E}$. The magnitude of the induced dipole
is given in the third column for a crystal diamond dipole moment
$d_{1}=10^{-23}\text{C}\cdot\text{m}$ Afek:2021bua . Figure 4: This figure
shows the decoherence rate $\Gamma_{S}$ from the dipole-dipole interaction
between the permanent dipole in the crystal ($d_{1}$) and the dipole it
induces in several types of polarizable environmental particle, as function of
the crystal’s dipole $d_{1}$. The decoherence rate is calculated in the short-
wavelength limit, see Eq.(72). The crystal’s permanent dipole $d_{1}$ is
assumed to be in the range of
$d_{1}\in\big{\\{}10^{-26}\text{C}\cdot\text{m},10^{-23}\text{C}\cdot\text{m}\big{\\}}\text{D}$
(see section IV.2). The polarizability of the most common air compounds:
dinitrogen (N2), carbondioxide (CO2), Argon (Ag) and dioxygen (O2), are given
in table 1. Since their polarizabilities are very similar, the lines seem to
overlap. In the enlarged figure in the corner one sees that N2 causes the most
decoherence, then CO2 and Ar which overlap approximately, and then O2. With
$m\sim 10^{-27}\text{kg}$, $T\sim 1\;\text{K}$, $n\sim 10^{8}\text{m}^{-3}$
(or $p\sim 10^{-15}\text{Pa}$), $\Delta x\sim 10^{-5}\text{m}$ and $R\sim
10^{-6}\text{m}$. The graph shows a decoherence rate
$\Gamma_{S}\in\\{10^{-17}\text{ Hz},10^{-7}\text{ Hz}\\}$.

If we assume that the diamond crystal has a measured permanent dipole moment
of $d_{1}=10^{-23}\text{C}\cdot\text{m}$ (see section IV.2,Afek:2021bua ;
Rivic , it could then induce dipoles in the environmental particles as well,
resulting in another source of decoherence from the dipole-dipole interaction.
To illustrate this, let us consider the content of the air, which consists
mostly of nitrogen ($\sim 78\%$) and oxygen ($\sim 21\%$), with some argon
($\sim 0.9\%$) and carbon-dioxide ($0.04\%$), and very small traces of other
molecules atmosphere . Additionally one could assume some water vapour is
present in the vacuum chamber. Although, water has a permanent dipole (and is
included in the previous discussion of environmental dipoles), the other
components do not. However, the electromagnetic field produced by the
permanent dipole in the diamond can induce a dipole within the environmental
particles, depending on their polarisability. The induced dipole moment is
found from the polarizability and the electric field from the permanent dipole
moment of the diamond. In particular, taking an average distance
$\bar{r}=R\sim 10^{-6}\text{m}$ (see section IV.1), the electric field for
this average distance is:

$E=\frac{d_{1}}{2\pi\epsilon_{0}}\frac{1}{R^{3}}$ (74)

which we find to be: $E\sim 1.8\times 10^{5}{\text{N}}/{\text{C}}$. The
induced dipoles for the most common air particles are given in table 1. Since
the taken dipole value $d_{1}=10^{-23}\text{C}\cdot\text{m}$ is for larger
spheres then our micro-spheres, this is probably an overestimate since there
are indications that dipole moments scale with volume Rivic .

Fig.4 shows the behaviour of $\Gamma_{S}$ for different values of the
crystal’s dipole $d_{1}$. The decoherence rate arises from the dipole-dipole
interaction, where the environmental dipole $d_{2}$ is induced by the dipole
of the crystal, denoted as $d_{1}$. For each element presented in table 1, it
is possible to compute the decoherence rate $\Gamma_{S}$ using Eq. (72). It is
evident from the figure that when a crystal’s dipole of the magnitude
$d_{1}\in\big{\\{}10^{-26}\text{C}\cdot\text{m},10^{-23}\text{C}\cdot\text{m}\big{\\}}$
induces a dipole moment in the air molecules, the resulting decoherence rate
is approximately
$\Gamma_{S}\in\big{\\{}10^{-17}\text{Hz},10^{-7}\text{Hz}\big{\\}}$.
Therefore, the final decoherence time will have a minimum magnitude of the
order of $10^{7}\text{s}$, which does not present as a problem in realization
the QGEM experiment within $t\sim 1$ s.

##  V Discussion

In this paper, we have analyzed the decoherence rate in the matter-wave
interferometer due to the electromagnetic interactions. In particular, we have
analyzed a special case of dipole-dipole interaction, which plays an important
role since generally both the environmental particles (e.g. air molecules)
table ; atmosphere ; pollistx  and the test mass micro-crystal Afek:2021bua
can possess an electric dipole moment, or can have an induced dipole from an
external electric field.

In particular, we have used the Born-Markov Master Equation
Schlosshauer:2014pgr ; joos2003decoherence , a well-known tool for
investigating the dynamics of quantum systems within a large environment. This
model provides a valuable means of deriving a dynamical equation for the
density matrix of a quantum system that finds itself immersed in an
environment. By employing the Born approximation, which considers the
environment to be much larger than the micro-crystal, and incorporating the
Markovian assumption, which disregards memory effects within the environment
after its interaction with the system, we acquire a non-unitary time evolution
equation for the crystal’s density matrix.

In this work, we have delved further into the application of this model,
focusing on a generic QED interaction Hamiltonian between two fermions within
the framework of the Born-Markov master equation. This Hamiltonian,
constructed in terms of destruction/construction operators for both the
crystal and the environment, is given by Eq. (18). Using this expression for
the QED Hamiltonian interaction inside the Born-Markov master equation, we
were able to find Eq. (41), which gives the decoherence rate for the
suppression of the off-diagonal elements of the micro-crystal’s density
matrix. It is worth noting that this final result has the same structure found
in the so-called Scattering model, a well know model for decoherence used in
the literature Schlosshauer:2014pgr .

Further exploring our results, we have directed the focus toward analyzing the
dipole-dipole interactions. The first step was to derive the differential
cross section for such interactions, as we have done in Eq. (47). Using this
expression for the cross section, we have successfully obtained an explicit
expression for the decoherence rate, both in the short-wavelenght limit (see
Eq. (52)) and the long-wavelength limit (see Eq. (64)).

In section IV, we have finally applied our results to the QGEM proposal
Bose:2017nin ; vandeKamp:2020rqh ; Schut:2021svd . In particular, using the
parameters employed within the QGEM experiment, such as temperature, pressure,
and the dipole of the micro-crystal, we were able to find an explicit
expression for the decoherence rate in the short wavelength limit (see Eq.
(72)). The main source of decoherence was found to be from the interaction
between the permanent dipole of the crystal and the permanent dipole of the
environmental particles (taken to be $d_{2}\sim 1\;\text{D}$ table ). The
decoherence from the induced dipoles considered here is negligible compared to
the permanent dipole scenario.

Using typical numerical values for the QGEM proposal Bose:2017nin ;
Barker:2022mdz , we were able to find an upper bound for the crystal’s dipole,
ensuring a decoherence rate smaller than 0.01 Hz. This requirement arises from
the necessity of preserving the spatial superposition of the massive particle
for a minimum duration of 1 second, accordingly to the QGEM proposal. By
guaranteeing a decoherence rate of $0.01$ Hz or less, we can confidently
ensure the preservation of the crucial superposition throughout the
experiment. The upper bound for the crystal’s dipole determined through this
work is on the order of $d_{1}=10^{-26}\text{C}\cdot\text{m}$, which
represents a relatively small value when compared to certain measurements
conducted within laboratory settings Afek:2021bua .

This research shows that in order to perform the QGEM experiment we need to
measure accurately the crystal’s permanent dipole for the relevant size and
mass. If the fixed dipole is found to be larger than the upper bound found
here, given a certain decoherence rate, then one needs to take measures to
mitigate the decoherence due to the dipole moment. Here we do not discuss the
strategies to ameliorate this effect but merely point toward a new challenge
that the QGEM experiment might have to address these challenges. We will leave
that for future investigation.

## Acknowledgements

MS is supported by the Fundamentals of the Universe research program at the
University of Groningen. MT acknowledges funding by the Leverhulme Trust (RPG-
2020-197). S.B. would like to acknowledge EPSRC grants (EP/N031105/1,
EP/S000267/1, and EP/X009467/1) and grant ST/W006227/1. AM’s research is
funded by the Netherlands Organisation for Science and Research (NWO) grant
number 680-91-119.

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##  Appendix A Appendix

###  A.1 Differential cross section for a $2\longrightarrow 2$ process

In this appendix we are going to show the details of some QED calculations. In
particular, we are going to find the expression for the matrix element
$|\mathcal{M}|^{2}$ and its relation with the differential cross section
$\frac{d\sigma}{d\Omega}$. We will again use the natural units system for
simplicity, i.e. $\hbar=c=1$.

Let us start from the definition of cross section Schwartz:2014sze :

$d\sigma=\frac{1}{T\Phi}dP,$ (75)

where $T$ is the total time during which interactions happen, $\Phi$ is the
flux of particles (e.g. if we are in the LAB frame then $\Phi$ is the flux of
the incoming projectile particles) and $dP$ is the (quantum) probability that
one interaction happens.
If we consider the COM frame, the flux $\Phi$ will be given by:

$\Phi=\frac{|\vec{v_{1}}-\vec{v_{2}}|}{V},$ (76)

where $\vec{v_{1}}$ and $\vec{v_{2}}$ are the velocities of the two initial
particles; the minus sign is because they run into each other during the
collision.
Let us now compute $dP$. During scattering processes, the operator involved is
the Scattering matrix $\hat{S}$ and its matrix elements give us the transition
probability from an initial state $\ket{i}$ to a final one $\ket{f}$:

$dP=\frac{|\bra{f}\hat{S}\ket{i}|^{2}}{\innerproduct{i}{i}\innerproduct{f}{f}}\prod_{j}\frac{V}{(2\pi)^{3}}d^{3}p_{j}.$
(77)

This formula represents the probability of the interaction inside an
infinitesimal volume of the momentum space
$d\Pi=\prod_{j}\frac{V}{(2\pi)^{3}}d^{3}p_{j}$, where $j$ represents the
number of final states.
Because in QFT we have $\ket{p}=\sqrt{2E_{p}}\hat{a}_{p}^{\dagger}\ket{0}$,
the inner products in the denominator of (77) gives:

$\innerproduct{i}{i}=\prod_{i}(2\pi)^{3}2E_{i}\delta^{(3)}(0).$ (78)

In a finite volume, we have:

$(2\pi)^{3}\delta^{(3)}(\vec{p}=0)=\int_{V}d^{3}x\;e^{i\vec{p}\cdot\vec{x}}=\int_{V}d^{3}x=V.$
(79)

Similarly in $4D$:

$\delta^{(4)}(0)=\frac{TV}{(2\pi)^{4}}.$ (80)

This means that:

$\begin{cases}\innerproduct{i}{i}=(2E_{1}V)(2E_{2}V)\\\
\innerproduct{f}{f}=\prod_{j}2E_{j}V\end{cases}.$ (81)

Now, the transferred matrix $\mathcal{T}$ is related to $S$ through:

$S=\mathbf{1}+i\mathcal{T}=\mathbf{1}+i(2\pi)^{4}\delta^{(4)}(\Sigma
p)\mathcal{M}.$ (82)

Thus the non-trivial part of the $\hat{S}$ matrix element is:

$|\bra{f}\hat{S}\ket{i}|^{2}=\delta^{(4)}(0)\delta^{(4)}(\Sigma
p)(2\pi)^{8}|\bra{f}\mathcal{M}\ket{i}|^{2}=TV\delta^{(4)}(\Sigma
p)(2\pi)^{4}|\mathcal{M}|^{2}.$ (83)

Plugging everything back in (77), we obtain:

$dP=\frac{T}{V}\frac{1}{2E_{1}2E_{2}}|\mathcal{M}|^{2}\prod_{j}\frac{d^{3}p_{j}}{(2\pi)^{3}2E_{j}}(2\pi)^{4}\delta^{(4)}(\Sigma
p).$ (84)

Finally, we have an expression for the cross section in the COM:

$d\sigma=\frac{1}{2E_{1}2E_{2}|\vec{v}_{1}-\vec{v}_{2}|}|\mathcal{M}|^{2}d\Pi_{lips},$
(85)

where
$d\Pi_{lips}=\prod_{j}\frac{d^{3}p_{j}}{(2\pi)^{3}2E_{j}}(2\pi)^{4}\delta^{(4)}(\Sigma
p)$.
In the special case where we have 2 final states, such that
$\vec{p}_{1}=-\vec{p}_{2}$, $\vec{p}_{3}=-\vec{p}_{4}$ and
$E_{1}+E_{1}=E_{3}+E_{4}=E_{CM}$, $d\Pi_{lips}$ becomes:

$d\Pi_{lips}=\frac{d^{3}p_{3}}{(2\pi)^{3}2E_{3}}\frac{d^{3}p_{4}}{(2\pi)^{3}2E_{4}}(2\pi)^{4}\delta^{(4)}(\Sigma
p).$ (86)

Integrating over $\vec{p_{4}}$ we obtain:

$d\Pi_{lips}=\frac{1}{16\pi^{2}}d\Omega\int
dp_{f}\frac{p_{f}^{2}}{E_{3}E_{4}}\delta(E_{3}+E_{4}-E_{CM})=\frac{1}{16\pi^{2}}d\Omega\frac{p_{f}}{E_{CM}}\theta(E_{CM}-m_{3}-m_{4}),$
(87)

where $p_{f}=|\vec{p_{3}}|=|\vec{p_{4}}|$ and
$p_{i}=|\vec{p_{1}}|=|\vec{p_{2}}|$.
We can now rewrite $|\vec{v}_{1}-\vec{v}_{2}|$ as:

$|\frac{p_{i}}{E_{1}}+\frac{p_{i}}{E_{2}}|=p_{i}\frac{E_{CM}}{E_{1}E_{2}},$
(88)

in order to obtain the final expression for the differential cross section:

$\frac{d\sigma}{d\Omega}=\frac{1}{64\pi^{2}E_{CM}^{2}}\frac{p_{f}}{p_{i}}|\mathcal{M}|^{2}\theta(E_{CM}-m_{3}-m_{4}).$
(89)

Applying (89) in the case where the target is much heavier than the projectile
($M\gg m$), we can write $E_{CM}\simeq M$ and also $p_{i}=p_{f}$. In this way,
considering the case where $E_{CM}>m_{3}+m_{4}$, we obtain the final
expression for the differential cross section:

$\frac{d\sigma}{d\Omega}=\frac{1}{64\pi^{2}E_{CM}^{2}}|\mathcal{M}|^{2}.$ (90)

###  A.2 Number density factor

In this Appendix, we will give meaning to the factor
$\pi^{-3/2}\tilde{\sigma}^{3}$. In particular, let us see the connection
between $\tilde{\sigma}$ and the uncertainty in space along one direction
$\sigma_{x}=\sqrt{<x^{2}>_{\psi}-<x>^{2}_{\psi}}$ (i.e. along $x$), with
$\psi(x)$ being the wave-function in physical space. From (25), we can see
that:

$\begin{split}\psi(x)=\int_{-\infty}^{+\infty}dp_{x}e^{-ip_{x}x}\tilde{\psi}_{p_{x}}(p)\\\
=\int_{-\infty}^{+\infty}dp_{x}e^{-ip_{x}x}\frac{(2\pi)^{1/2}}{(\pi\tilde{\sigma}^{2})^{1/4}}e^{-\frac{p_{x}^{2}}{2\tilde{\sigma}^{2}}}\\\
=2\pi^{3/4}\tilde{\sigma}^{1/2}e^{-\frac{\tilde{\sigma}^{2}}{2}x^{2}}.\end{split}$
(91)

Now we can compute $\sigma_{x}$:

$\begin{split}\sigma_{x}=\sqrt{<x^{2}>_{\psi}-<x>^{2}_{\psi}}\\\
=\int_{-\infty}^{+\infty}dx\;x^{2}|\psi(x)|^{2}=\frac{\sqrt{2}\pi}{\tilde{\sigma}}.\end{split}$
(92)

But we know also that the uncertainty in space is physically due to the fact
that the environmental particles, as all the other particles involved in the
experiment, are confined in a volume $V=L^{3}$. This means that:

$V=L^{3}=\sigma_{x}^{3}=\frac{2^{3/2}\pi^{3}}{\tilde{\sigma}^{3}},$ (93)

which leads finally to:

$\pi^{-3/2}\tilde{\sigma}^{3}=\frac{(2\pi)^{3/2}}{V}.$ (94)