# Dephasing due to electromagnetic interactions in spatial qubits

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  Herre Bosma  Van
Swinderen Institute for Particle Physics and Gravity, University of Groningen,
9747 AG Groningen, the Netherlands  MengZhi Wu  Van Swinderen Institute for
Particle Physics and Gravity, 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

(September 3, 2024)

###### Abstract

Matter-wave interferometers with micro-particles are excellent quantum sensors
as they can be sensitive to a minute quantum phase information, which a
classical detector cannot. Two such adjacent micro-particles in the
interferometers can be entangled solely via the quantum nature of gravity
known as the quantum gravity-induced entanglement of masses (QGEM) protocol.
The micro-particles can also be entangled via EM interactions. Therefore, it
is essential to estimate the decoherence, noise/dephasing rate for such
interferometers. In this paper, we will focus on a particular source of an
electromagnetic interaction. We will treat this interaction as a noise which
will lead to dephasing. We assume that our matter-wave interferometer has a
residual charge which can interact with a neighbouring ion in the ambience,
e.g., inside the experimental capsule. This will provide dephasing of the
matter-wave interferometer due to the Coulomb interaction with external
charges and the charge-dipole interaction with external dielectrics or
dipoles. Similarly, we will consider neutral micro-particles, which can
interact with charged and/or neutral particles in the ambience via induced
dipole-charge, permanent dipole-charge, and dipole-dipole interactions. All
these interactions constitute electromagnetically driven dephasing to a single
and a twin interferometer. We will discuss their relevance for the QGEM
experiment and provide insight into the noise of an entangled state for
charged micro-particles kept adjacently with an implication for the C-NOT
gate.

##  I Introduction

One of the key features of quantum mechanics is the matter-wave duality
exhibited in a matter-wave interferometer de1923waves . Matter-wave
interferometery has played a central role in many experimental breakthroughs
in quantum mechanics thomson1927diffraction ; davisson1927diffraction ;
davisson1928reflection , and also exhibits the main idea of a macroscopic
spatial quantum superposition Schrodinger:1935zz . Furthermore, if two such
interferometers can be placed adjacent to each other, they can test the
quantum entanglement feature einstein1935can ; bell1964einstein , that cannot
be mimicked by a classical theory Horodecki:2009zz .

Matter-wave interferometry has also been used to detect Earth’s
gravitationally-induced phase via interference with neutrons and atoms
colella1975observation ; nesvizhevsky2002quantum ; fixler2007atom ;
asenbaum2017phase ; overstreet2022observation . Moreover, it has been
suggested to detect gravitational waves Marshman:2018upe , quantum sensors for
neutrons Kilian:2022kgm , space debris Wu:2022rdv , and probe the physics
beyond the standard model such as fifth force and axion Barker:2022mdz .

Recently, it has been suggested that the next generation of matter-wave
interferometers with micro-particles will be sensitive enough to probe the
quantum nature of gravity via witnessing entanglement, known as the quantum
gravity-induced entanglement of masses (QGEM) Bose:2017nin ; ICTS , see also
Ref. Marletto:2017kzi . The protocol is based on genuine quantum observables,
which test whether gravity is a classical or a quantum entity. The
entanglement between the two matter wave interferometers can be confirmed in
the former case. At the same time, if gravity is inherently classical, then
there will be no entanglement Bose:2017nin ; Marshman:2019sne ; Bose:2022uxe ,
because classical gravity cannot entangle the two quantum systems as
formalized by the local operation and classical communication (LOCC) theorem
Bennett:1996gf ; Bose:2017nin ; Bose:2022uxe .

The critical observation is that as long as we follow 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 .

One can go one step further and test the quantum analogue of the light bending
experiment in gravity in an optomechanical setup where we can test the quantum
gravitational entanglement between matter and photons Biswas:2022qto ; we can
also test the quantum version of the equivalence principle Bose:2022czr ;
Chakraborty:2023kel , test for massive graviton Elahi:2023ozf , non-local
aspects of quantum gravity Vinckers:2023grv , and verify the quantum nature of
gravity in the process of measurement Hanif:2023fto .

However, many experimental challenges need to be overcome before we can test
all these new fundamental physics with the help of matter-wave
interferometers. 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, from the effects of quantum electrodynamics Fragolino:2023agd ,
gravity gradient noise and seismic noise Toros:2020dbf .

Besides testing the quantum nature of gravity, such entangled pairs provide
the basis for a quantum computer as they form the controlled-NOT (CNOT) gate,
also known as the Molmer-Sorensen gate in the context of an ion trap
Soerensen:1998mp .

Given so many applications, it is essential to understand various causes of
decoherence and dephasing. We will not exhaust all the sources of dephasing,
but we will consider particular ones. In particular, we will treat the
external source as a noise which will lead to dephasing. Note that to witness
entanglement we must ensure that the total decoherence, dephasing/external
noise is below the entanglement phase for the case of the QGEM experiment and
for the CNOT gate.

First, we will consider a single matter-wave interferometer, we will imagine
an acceleration noise caused by a single particle passing by the
interferometer and how it will dephase the matter-wave interferometer via
electromagnetic interactions. Then, we will consider if two such
interferometers were kept adjacent, such a source of acceleration noise will
also contribute towards the total decoherence budget. If the matter-wave
interferometer is created by a neutral micro-particle then the source of
acceleration will be due to dipole (induced and permanent)-charge interaction
or dipole-dipole interaction. If the micro-particle possesses any residual
charge, such a charged interferometer can interact with another charged
particle or a neutral particle. The common aspect of all these interactions is
that the moving particle will always create a slight jitter in the paths of
the spatial interferometer, and will induce a noise surmounting dephasing. We
will analyse these cases systematically and provide the acceleration noise for
each scenarios.

We will apply the technique of computing the relative acceleration noise by
employing the techniques similar to that of the Newtonian noise, see
Saulson:1984 ; Toros:2020dbf ; Wu:2022rdv , instead of a traditional
computation of decoherence Schlosshauer:2014pgr ; Schlosshauer:2019ewh . The
reason is that both Coulomb and Newtonian interactions are long-range
interactions, and provide $1/r$ potential, where $r$ is the distance between
the interferometer and the source, in both the cases the total cross section
is infinite unless a cutoff for small scattering angles $\theta$ is applied
griffiths_introduction_2018 . Since the computations of decoherence rate
depend on the total cross-section, see Schlosshauer:2014pgr , therefore, for
long-range interactions it is an effective technique to employ the noise model
to estimate the dephasing rate or the total dephasing. In this paper we will
compute the latter.

Further, note that there is also a slight difference between the relative
acceleration between the electromagnetic (EM) and the gravitational jitter.
The general theory of relativity possesses an equivalence principle, e.g., the
equivalence between the gravitational and the inertial mass, which allows that
the gravitational acceleration felt by a geodesic is always proportional to
the Riemann contribution, which translates to a quadratic order in position
corrections at the leading order in the acceleration noise, see Saulson:1984 ;
Toros:2020dbf ; Wu:2022rdv , while Maxwell’s equations do not possess such
equivalence, hence the leading order acceleration noise will be proportional
to the linear order in the position correction at the leading order.

We will begin the paper with a generic discussion on relative acceleration
noise, which will apply to all our cases of interest. Then, we will discuss
four scenarios. First, charge-charge interaction between the charged
interferometer and an external charge, then the charge-dipole interaction
between the charged interferometer and and induced or permanent external
dipole (sec. IV). Then we will consider a neutral interferometer particle with
a permanent or induced dipole that interacts first with an external charge
(dipole-charge interaction) and then with an external dipole (dipole-dipole
interaction), see sec. V.

##  II Relative Acceleration Noise

This section will describe the relative acceleration noise (RAN) computation
due to moving ions and dipoles interacting via the EM interaction of a matter-
wave interferometer. We will consider two distinct cases, electrically charged
and neutral micro-spheres, creating the spatial superposition for the matter-
wave interferometer. We will assume that the Schrödinger Cat state can be
created via the Stern-Gerlach (SG) setup as it happens in the case of spin-
embedded systems such as in nitrogen-vacancy (NV) spin, see Bose:2017nin . We
will create spin-up and down states, which we will discuss below. The other
possibility will be to create a spatial superposition in an ion trap, see
Wineland:1992 ; Wineland:1995 ; kielpinski2001recent . In either case, the
matter-wave interferometer can be considered a spatial qubit, where
interactions with an environment can induce a relative phase between the
interferometer’s two arms (or the spatial superposition states of the spatial
qubit). Our analysis will also be useful for the EM counterpart of the QGEM
experiment, envisaged in Barker:2022mdz . It is believed that the EM
counterpart of the QGEM can be performed first, since the EM interaction
strength is many orders of magnitude larger than that of the gravitational
case, and also one would not require a large spatial splitting to witness
entanglement due to the Coulomb interaction. In any case, a phase offset
between the two superposition instances dephases the wavefunction. It reduces
the contrast (i.e. the overlap between the wavepackets that is needed for the
measurement), and therefore, it is pertinent to pin down this noise. In fact,
we can reverse the question, and can ask how much EM-induced noise our single
matter-wave interferometer can detect, in which case our single matter-wave
interferometer will be a quantum sensor sensitive to the EM channels, in
analogous to the gravity gradient sensor developed inWu:2022rdv . We will
introduce the setup and relevant interactions, then briefly introduce the
noise analysis.

###  II.1 General noise analysis

The difference in the phase picked up between the two arms of an
interferometer is determined by the difference in their Lagrangians,
specifically determined by the action of the two spatial trajectories. Taking
the superposition to be along one dimension, e.g., in the $\hat{x}$-direction,
without any loss of generality, it follows that the difference in phase-shift
is given by Storey:1994oka :

$\displaystyle\delta\phi$ $\displaystyle=\phi_{R}-\phi_{L}$ (1)
$\displaystyle=\frac{1}{\hbar}\int_{t_{i}}^{t_{f}}L_{R}\left[x_{R},\dot{x_{R}}\right]-L_{L}\left[x_{L},\dot{x_{L}}\right]dt,$

where $\phi_{i}$ and $L_{i}$ are the phases and the Lagrangians for $i=L,R$,
the left and the right arm of the superposition, respectively. Here we
consider a single interferometer, we will consider the case of adjacent
interferometers later. The bounds of the integral are equal to the time of
splitting and recombination of the two trajectories to complete an one-loop
interferometer. The time when the interferometer is alive is given by
$\tau=t_{f}-t_{i}$, where $f,i$ stands for the final and the initial time for
the interferometer.

Here, we consider the non-relativistic regime of a free-falling test mass
split into the superposition by an acceleration, $\lambda_{j}$, in the
$\hat{x}$-direction 111The acceleration of the two superposition states is due
to the SG setup in the context of NV centred micro-particles Bose:2017nin ;
Marshman:2021wyk ; WoodPRA22_GM . In the case of ions, the force exerted to
create the superposition will be due to the photon kicks Wineland:1992 ;
Wineland:1995 . We can also envisage creating quantum superposition for the
charged micro-spheres with the help of SG setup, see Henkel_2019 ..

We can envisage a spin-magnetic field interaction responsible for creating the
superposition, e.g. using an SG apparatus, similar to what has been applied to
the atomic case Folman2013 ; folman2019 ; Margalit:2020qcy ; Keil2021 ;
Amit_2019 , and charged case Henkel_2019 . Any external movement interacting
via Coulomb or dipole interactions in the vicinity of the matter-wave
interferometer will cause a tiny jitter/acceleration noise, which we denote
here by $a_{\text{noise, EM}}$. The subscript EM represents EM-type external
interaction. Noise from other types of interactions, such as the gravitational
interaction, in the context of nano-particle interferometer has been discussed
before, see Refs. Grossardt:2020def ; Toros:2020dbf ; Wu:2022rdv .

The Lagrangian of the two arms of the interferometer can be expressed as:

$L_{j}=\frac{1}{2}mv_{j}^{2}-m\lambda_{j}(t)x_{j}-m\,a_{\operatorname{noise,EM}}(t)x_{j}\,.$
(2)

We wish to close the trajectories (recombine the spatial qubit into a spin
qubit) such that the wavefunctions of both trajectories nearly overlap. If
they do not, there will be a substantial loss of visibility. If the spread of
the wavepackets is $\sigma$ then one would require
$|X_{L}(t_{f})-X_{R}(t_{f})|\ll\sigma$ for visibility Schwinger ; Scully ;
Englert .

Any noise in the trajectories can be modelled by the fluctuations in the
Lagrangian, as given by the last term in Eq. (2). We assume that
$a_{\rm{noise,EM}}$ is independent of the force responsible for creating the
superposition, but that it is time-dependent during $\tau=t_{f}-t_{i}$. Hence,
the fluctuation in the phase shift due to the EM interaction will be given by:

$\delta\phi=\frac{m}{\hbar}\int_{t_{i}}^{t_{f}}a_{\operatorname{noise,EM}}(t)\left(x_{R}-x_{L}\right)dt\,.$
(3)

We wish to treat $\delta\phi(t)$ as a statistical quantity. The measurable
noise in any experiment is the statistical average of any stochastic entity,
defined by $\mathbb{E}[\delta\phi]$. The averaging can be obtained over time
using a single realization of the noise for a time-varying ergodic noise. For
example, the average of a time-varying stochastic quantity $\delta\phi(t)$ can
be expressed as:

$\mathbb{E}[\delta\phi]=\frac{1}{T}\int_{0}^{T}\delta\phi(t)dt\,,$ (4)

where $T$ is the time scale much larger than any time scale characterizing the
noise itself, e.g. $T\gg\tau$, where $\tau$ denotes the experimental time,
$\tau=t_{f}-t_{i}$. We will use $T\sim{\cal O}(b/v)\gg\tau\sim 1/\omega$,
where $\omega$ is the frequency of the interferometer. This analysis is very
similar to the case of Saulson:1984 ; Toros:2020dbf ; Wu:2022rdv . An
estimation of $T\sim b/v$ was also made in Ref. Saulson:1984 .

In any experiment, the mean value of the noise can be assumed to be zero, e.g.

$\mathbb{E}[a_{\text{noise, EM}}]=\mathbb{E}[\delta\phi]=0.$

This is because the zero point, or the baseline of the phase, has to be
calibrated before the experiment starts, so the contribution of the mean value
of every noise will be considered in the offset of the baseline in our current
analysis. The autocorrelation of the acceleration noise measured at different
times, $\mathbb{E}[a_{\text{noise, EM}}(t_{1}),a_{\text{noise, EM}}(t_{1})]$,
is usually non-zero and is related to the power spectrum density (PSD) of the
noise, denoted by $S_{aa}(\omega)$. The definition of the PSD of the noise
$a_{\text{noise, EM}}$ is

$S_{aa}=\lim\limits_{T\to\infty}\frac{|a_{\text{noise, EM}}(\omega)|^{2}}{T},$
(5)

where $a_{\text{noise, EM}}(\omega)$ is the Fourier transform of
$a_{\text{noise, EM}}(t)$ on the time domain $[0,T]$. According to the Wiener-
Khinchin theorem, the autocorrelation function of a random process is given by
the PSD of that process Wiener:1930 ; Khintchine1934 :

$\displaystyle\mathbb{E}\big{[}a_{\operatorname{noise,EM}}(t_{1})$
$\displaystyle a_{\operatorname{noise,EM}}(t_{2})\big{]}$ (6)
$\displaystyle=\frac{1}{2\pi}\int_{\omega_{\rm
min}}^{\infty}S_{aa}(\omega)e^{i\omega(t_{1}-t_{2})}d\omega\,,$

where the minimum frequency $\omega_{\operatorname{min}}=2\pi/(t_{f}-t_{i})$
is the frequency resolution in the real experiment. In particular, signals and
noises are sampled as a discrete finite data series, so the frequency-domain
analysis has to be based on the discrete Fourier transform, which has such a
frequency resolution Chatfield:2016 .

Using the relation in Eq. 6 and the difference in phase-shift form Eq. 3, the
variance $\mathbb{E}[(\delta\phi)^{2}]$ can be written in the Fourier space
by:

$\Gamma_{\operatorname{n}}=\mathbb{E}\left[(\delta\phi)^{2}\right]=\frac{1}{2\pi}\left(\frac{m}{\hbar}\right)^{2}\int_{\omega_{\rm
min}}^{\infty}d\omega S_{\operatorname{aa}}(\omega)F(\omega)\,.$ (7)

where $\Gamma_{\operatorname{n}}$ in Eq. (7) is the variance in the noise,
which is an observable entity, and $F(\omega)$ is known as the transfer
function, see Ref.Wu:2022rdv . As shown below, $\Gamma_{\rm n}$ will be useful
to estimate the dephasing for the two spatial qubits, which will add up to any
other sources of decoherence caused by the extrenal/internal degrees of
freedom. Note that The $\Gamma_{n}$ is a dimensionless entity.

The transfer function is equal to the absolute value squared of the Fourier
transform of the difference in the trajectories:

$\displaystyle F(\omega)$
$\displaystyle=\left|\int_{t_{i}}^{t_{f}}dt\left(x_{L}-x_{R}\right)e^{i\omega
t}\right|^{2}$ (8)
$\displaystyle=\left|-\frac{1}{\omega^{2}}\int_{t_{i}}^{t_{f}}\frac{d^{2}}{dt^{2}}\left(x_{L}-x_{R}\right)e^{i\omega
t}dt\right|^{2},$

the second equation follows from integration by parts and the equality of both
the position and velocity at the path’s endpoints.

###  II.2 Example of Stern-Gerlach interferometer

The dephasing in Eq. (7) is dependent on the specific trajectory of the two
arms of the matter-wave interferometer (via the transfer function
$F(\omega)$). Here, we assume a SG type interferometer that is often proposed
in the context of entanglement-based quantum-gravity experiments, see
Bose:2017nin ; Marshman:2019sne ; Zhou:2022jug ; Zhou:2022frl ; Zhou:2022epb ;
Japha:2022xyg .

We specify the trajectory of two arms of the interferometer, see Fig. 1. Here
we assume a symmetric path for the interferometer as a prototype model. We
consider a simplistic scenario, assuming that the acceleration of the spin
state $j$, denoted $\lambda_{j}$, is proportional to the gradient of the
magnetic field, given by:

$\lambda_{j}=s_{j}\frac{g\mu_{B}}{m}|\nabla B|$ (9)

where $s_{j}$ is the spin of the state, $g$ is the Lande g-factor of the
electron ($g=2$), $\mu_{B}=9\cdot 10^{-24}\operatorname{J\,T}^{-1}$ is the
Bohr-magneton, $m$ is the mass of the test-mass and $|\nabla B|$ is the
magnitude of the gradient of the magnetic field. To create the path in Fig. 1,
one controls the spin so that the acceleration of the right arms is positive
from $[0,t_{a}]$ and $[3t_{a}+t_{e},4t_{a}+t_{e}]$ and negative from
$[t_{a},2t_{a}]$ and $[2t_{a}+t_{e},3t_{a}+t_{e}]$, while it is zero for the
time range $[2t_{a},2t_{a}+t_{e}]$. For the left arm, the spin and hence the
acceleration will be equal in magnitude but in the opposite direction. We also
assume $x_{j}(0)=0$ and $\dot{x}_{j}(0)=0$ for solving the equations of the
motion for the trajectory, given by: $\ddot{x}_{j}(t)=\lambda_{j}(t)$. Micro-
particles can be electrically charged by ultraviolet irradiation Frimmer_2017
. For small charge $ne$, the Lorentz force $\sim nevB/m$ acting on the micro-
particle is negligible due to its large mass $m$, see Henkel_2019 .

Figure 1: Schematic illustration of the paths of the arms of the
interferometer (adapted from Wu:2022rdv ). Based on this figure, the transfer
function in Eq. 10 can be calculated. For the left arm, the acceleration is in
the negative direction from $[0,t_{a}]$ and $[3t_{a}+t_{e},4t_{a}+t_{e}]$, in
the positive direction from $[t_{a},2t_{a}]$ and
$[2t_{a}+t_{e},3t_{a}+t_{e}]$, while it is zero for the time range
$[2t_{a},2t_{a}+t_{e}]$. Meanwhile, the acceleration is the opposite for the
right arm.

With the help of the equations of motion,
$\frac{d^{2}}{dx^{2}}x_{j}(t)=\lambda_{j}$, the transfer function
corresponding to the trajectory illustrated in Fig. 1 can be computed:

$\displaystyle F(\omega)$
$\displaystyle=\frac{1}{\omega^{2}}\left|\int_{t_{i}}^{t_{f}}\left(\lambda_{L}-\lambda_{R}\right)e^{i\omega
t}dt\right|^{2}=\left|\frac{2}{\omega^{2}}\int_{t_{i}}^{t_{f}}\lambda_{s}e^{i\omega
t}dt\right|^{2}$ (10)
$\displaystyle=\frac{32\lambda_{s}^{2}}{\omega^{6}}\sin^{4}\left(\frac{t_{a}\omega}{2}\right)\sin^{2}\left(\frac{\omega}{2}(2t_{a}+t_{e})\right)\,.$

Since the means by which the superposition is created is immaterial, for the
calculation of the transfer function it is more fitting to define the transfer
function in terms of the maximal superposition size created after two
acceleration times, $\Delta x=\lambda t_{a}^{2}$:

$F(\omega)=\frac{32(\Delta
x)^{2}}{\omega^{6}t_{a}^{4}}\sin^{4}\left(\frac{t_{a}\omega}{2}\right)\sin^{2}\left(\frac{\omega}{2}(2t_{a}+t_{e})\right).$
(11)

By finding the PSD, $S_{aa}$, due to different types of interaction and
combining it with the specifics of the trajectory encoded in the transfer
function in Eq. (11), we can find the noise via Eq. (7).

##  III EM induced acceleration noise

Figure 2: The schematic motion of the external particle with the position of
closest approach $(x_{0},y_{0},z_{0})$ and with a velocity
$(v_{x},v_{y},v_{z})$ with respect to the interferometer. We will work within
the assumption that $v\ll c$, where $c$ is the velocity of light. The
magnitude of the position vector is given by
$|\vec{r}|=\sqrt{b^{2}+(vt^{2})}$, where b is the distance of the closest
approach, i.e. the impact parameter. The projection angles are defined by
$\cos(\alpha)=x_{0}/b$ and $\cos(\beta)=v_{x}/v$. The dipole moments and the
relevant angles are also indicated. The centre of mass position is used for
the definitions since we assume $\Delta x<<b$.

We will assume that the external particle is moving with a constant velocity
$\vec{v}$ with an impact parameter $b$ from the matter-wave interferometer
(see Fig. 2). We will follow Refs. Toros:2020dbf ; Wu:2022rdv  to compute the
acceleration noise. The position of the external particle with respect to the
matter-wave interferometer can be conveniently described by:

$r(t)=\sqrt{b^{2}+(vt)^{2}}.$ (12)

We can set our coordinate system such that (see Fig. 2):

$\cos(\alpha)=x_{0}/b,\mbox{\quad\quad}\cos(\beta)=v_{x}/v.$ (13)

The moving external particle can cause a random acceleration noise on the
interferometer due to the EM interactions, which will contribute to a phase
fluctuation of the interferometer. We also assume the micro-particle and the
moving particle to be perfect spheres. Specifically, we consider two cases for
the interactions.

Charged interferometer: If the interferometer is charged there can be two EM
interactions between the charged interferometer and the external particle:

  * •

A charge-charge interaction between the external charge, $q_{\text{ext}}$, and
the charge of the interferometer, denoted here by $q_{\text{int}}$. This
interaction is given by the Coulomb potential

$V_{\text{cc}}(r)=\frac{1}{4\pi\epsilon_{0}}\frac{q_{\text{int}}q_{\text{ext}}}{r}\,,$
(14)

where both systems are considered as point charges.

  * •

A charge-dipole interaction between the charge of the interferometer,
$q_{\text{int}}$, and the dipole of the external particle,
$\boldsymbol{d}_{\text{ext}}$. This interaction is given by the potential, see
griffiths2013introduction ; jackson_classical_1999 :

$V_{\text{cd}}(r)=\frac{q_{\text{int}}}{4\pi\epsilon_{0}}\frac{\textbf{d}_{\text{ext}}\cdot\hat{\textbf{r}}}{r^{2}}\,.$
(15)

The external dipole can be either induced or permanent.

Neutral interferometer: If the interferometer is neutral, there can be two
types of EM interactions between the interferometer and the external particle.

  * •

A dipole-charge interaction between the dipole of the interferometer,
$\boldsymbol{d}_{\text{int}}$ (which can be either permanent or induced by the
EM field of the external particle), and the charge of the external particle,
$q_{\text{ext}}$. This interaction is given by the potential:
griffiths2013introduction

$V_{\text{dc}}(r)=\frac{q_{\text{ext}}}{4\pi\epsilon_{0}}\frac{\textbf{d}_{\text{int}}\cdot\hat{\textbf{r}}}{r^{2}}\,,$
(16)

where the external particle is assumed to be a point charge.

  * •

A dipole-dipole interaction, where the dipole of the interferometer,
$\boldsymbol{d}_{\text{int}}$, and the dipole of the external dipole,
$\boldsymbol{d}_{\text{ext}}$, are assumed to be permanent. This interaction
is given by the potential: griffiths2013introduction ; jackson_classical_1999

$\displaystyle V_{\text{dd}}(r)=\frac{1}{4\pi\epsilon_{0}}\bigg{[}$
$\displaystyle\frac{\textbf{d}_{\text{int}}\cdot\textbf{d}_{\text{ext}}}{r^{3}}$
$\displaystyle\,\,-\frac{3(\textbf{d}_{\text{int}}\cdot\hat{\textbf{r}})(\textbf{d}_{\text{ext}}\cdot\hat{\textbf{r}})}{r^{3}}\bigg{]}\,,$
(17)

where $\hat{\boldsymbol{r}}$ is the unit vector pointing in the direction from
the interferometer centre of mass to the external particle centre of mass.

These interaction potentials will cause a random noise on the interferometer
$a_{\text{noise, EM}}$, which will contribute a phase fluctuation that can be
detectable by sensing it if we were to use the matter-wave interferometer as a
quantum sensor, or this acceleration noise will lead to dephasing the
experimental outcome in a QGEM-type experiment for instance. We now give a
brief outline of all the accelerations. For each of the interactions discussed
above, eqs. (14)–(17), the corresponding acceleration is found via
$\boldsymbol{F}=m\,\boldsymbol{a}$, with $\boldsymbol{F}=-\partial
V/\partial\boldsymbol{r}$, where $m$ is the mass of the interferometer.

$\displaystyle\boldsymbol{a}_{\text{cc}}$
$\displaystyle=\frac{q_{\text{int}}q_{\text{ext}}}{4\pi
m\epsilon_{0}}\frac{\hat{\boldsymbol{r}}(t)}{r^{2}(t)},$ (18)
$\displaystyle\boldsymbol{a}_{\text{cd}}$
$\displaystyle=\frac{q_{\text{int}}}{2\pi
m\epsilon_{0}}\frac{\mathbf{d}_{\text{ext}}\cdot\hat{\mathbf{r}}(t)}{r^{3}(t)}\,\hat{\boldsymbol{r}}(t)\,,$
(19) $\displaystyle\boldsymbol{a}_{\text{dc}}$
$\displaystyle=\frac{q_{\text{ext}}}{2\pi
m\epsilon_{0}}\frac{\mathbf{d}_{\text{int}}\cdot\hat{\mathbf{r}}(t)}{r^{3}(t)}\,\hat{\boldsymbol{r}}(t)\,,$
(20) $\displaystyle\boldsymbol{a}_{\text{dd}}$
$\displaystyle=\frac{3\,\hat{\boldsymbol{r}}}{4\pi\epsilon_{0}m}\left[\frac{\boldsymbol{d}_{\text{ext}}\cdot\boldsymbol{d}_{\text{int}}}{r^{4}(t)}-3\frac{(\boldsymbol{d}_{\text{ext}}\cdot\hat{\boldsymbol{r}})(\boldsymbol{d}_{\text{int}}\cdot\hat{\boldsymbol{r}})}{r^{4}(t)}\right]\,.$
(21)

The vector $\hat{\boldsymbol{r}}$ is time-dependent and given by
$\hat{\boldsymbol{r}}(t)=\boldsymbol{r}(t)/r(t)$, with
$r(t)=\absolutevalue{\boldsymbol{r}(t)}$ (as defined in eq. (12)). The dipole
can be either permanent or induced, which we distinguish with the notation
$\boldsymbol{d}(i)$ and $\boldsymbol{d}(p)$, respectively.

Since the interferometer is created in one dimension, e.g. $\hat{x}$-direction
in our case, the only relevant acceleration is in the $\hat{x}$-direction, and
we are specifically interested in $a_{x}(t)$. We can find the $x$-component of
the acceleration via $\hat{\boldsymbol{r}}_{x}=r_{x}/r(t)$ and using the
properties of the setup (see Fig. 2), such as $r_{x}(t)=x_{0}+v_{x}t$ and
$\absolutevalue{\boldsymbol{r}(t)}=\sqrt{b^{2}+(vt)^{2}}$, we can express the
acceleration in terms of the velocity $v$, impact parameter $b$, time $t$ and
projection angles as defined in eq. (13). This gives for example of the
charge-charge interaction:

$a_{x}^{\text{cc}}(t)=\frac{q_{\text{int}}q_{\text{ext}}}{4\pi\epsilon_{0}mb^{2}}\frac{\cos(\alpha)+(vt/b)\cos(\beta)}{(1+(vt/b)^{2})^{3/2}}\,.$
(22)

The acceleration is found separately now for every interaction (14)–(17) (Sec.
IV, V) and some assumptions are made on the orientation of the dipole vectors
for simplification.

Taking the Fourier transform to find $a_{x}(\omega)$ on the time domain
$[0,T]$, we can use eq. (5) to find the noise
$S(\omega)=\absolutevalue{a_{x}(\omega)}^{2}/T$. Combining the power spectrum
of the noise with the transfer function gives the dephasing:

$\Gamma_{\text{n}}=\frac{1}{2\pi}\frac{m^{2}}{\hbar^{2}}\int_{\omega_{\text{min}}}^{\infty}\differential{\omega}S_{aa}(\omega)F(\omega)\,.$
(23)

The results for the noise are given in Sec. IV, V and the detailed
calculations are given in Appendix A.

##  IV Charged interferometer

If there is some charge on the interferometric crystal, then charge-charge
interactions and charge-external dipole interactions will cause the noise in
the system. We consider the charge-charge interaction and the charge-dipole
interaction between the interferometer and the external particle, where we
consider separately the dipole of the external particle to be permanent or
induced 222These interactions depend on the relative sign of the charges and
dipoles (whether they are attractive or repulsive). In this section, we do not
specify the relative sign as we are interested in finding the dephasing and we
note that this is proportional to the acceleration squared. The dephasing is
therefore independent of the sign of the interaction potential.

###  IV.1 Dephasing due to internal charge and external charge interaction

First of all, we will assume the simplest case of a charged matter-wave
interferometer: the matter-wave interferometer has a charge $q_{\text{int}}$,
which is interacting with ambient charge $q_{\text{ext}}$ that is moving with
a constant velocity and has the closest approach $b$ from the interferometer’s
centre of mass. As presented as an example in sec. III (see eq. (24)), the
acceleration due to the charge-charge interaction in the $\hat{x}$-direction
is given by:

$a_{x}^{\text{cc}}(t)=\frac{q_{\text{int}}q_{\text{ext}}}{4\pi\epsilon_{0}mb^{2}}\frac{\cos(\alpha)+(vt/b)\cos(\beta)}{(1+(vt/b)^{2})^{3/2}}.$
(24)

Finding $S_{aa}$ from $a_{x}(\omega)$ and putting it in Eq. 7 yields the
dephasing:

$\displaystyle\Gamma_{\operatorname{n}}^{\text{cc}}=\frac{q_{\text{int}}^{2}q_{\text{ext}}^{2}(\Delta
x)^{2}}{\pi^{3}\hbar^{2}\epsilon_{0}^{2}Tv^{2}b^{2}t_{a}^{4}}\int_{\omega_{\operatorname{min}}}^{\infty}\omega^{-4}\operatorname{sin}^{2}\left(\frac{\omega}{2}(2t_{a}+t_{e})\right)$
(25)
$\displaystyle{\operatorname{sin}}^{4}\left(\frac{t_{a}\omega}{2}\right)\left[\cos^{2}(\alpha)K_{1}^{2}\left(\frac{b\omega}{v}\right)+\cos^{2}(\beta)K_{0}^{2}\left(\frac{b\omega}{v}\right)\right]d\omega.$

with $K_{\nu}$ is the modified Bessel function (see appendix A). The dephasing
depends on the charges $q_{\text{int}}$ and $q_{\text{ext}}$, the size of the
spatial superposition and the time during which it is created, the impact
parameter and the velocity of the external particle. Furthermore, the time $T$
which is the time over which the phase shift is averaged and the minimum
frequency $\omega_{\text{min}}$ are determined by the experimental setup,
impact parameter $b$ and the velocity $v$ of the external particle. The
integral in Eq. (25) is solved numerically, and the resulting dephasing is
plotted in Sec. VI (figure 3(a)) for a specific interferometer scheme.

###  IV.2 Dephasing due to internal charge and external dipole interaction

In this section, we will consider a charged interferometer particle
interacting with an external dipole. The dipole could arise due to any ambient
gas particle left inside the vacuum chamber. We will only consider the
acceleration noise due to one such external particle. The water vapour that
can be left in the vacuum chamber consists of the water molecules which carry
a permanent dipole moment of $\absolutevalue{\boldsymbol{d}}=6.19\times
10^{-30}\,$\mathrm{C}\text{\,}\mathrm{m}$$ lide2004crc . Left-over air
molecules in the vacuum chamber, such as dinitrogen, carbon dioxide, argon and
dioxygen are polarisable (they have a polarisability of $1.5-2.5$ Å3
atmosphere ; pollistx ) and could thus have an induced dipole moment from the
charge of the interferometer. The acceleration due to the charge-dipole
interaction is given in eq. (19), the $x$-component of the acceleration is
dependent on whether the dipole is induced or permanent.

####  IV.2.1 Permanent external dipole

If the external particle has a permanent dipole, such as is the case of water
vapour, then the dipole moment magnitude can be taken to be the experimentally
determined value, e.g. $\boldsymbol{d}_{\text{ext}}^{\text{(p)}}=6.19\times
10^{-30}\,$\mathrm{C}\text{\,}\mathrm{m}$$ in the case of water vapour (the
superscript $p$ standing for permanent). Taking the worst-case scenario, we
assume that at the point of closest approach $b$, the dipole vector of the
external particle is aligned with the vector $r(t)$. Assuming that the
particle is moving very slowly, we take the worst-case scenario of the dipole
vector and the vector $\boldsymbol{r}(t)$ aligning during the experimental
time $\tau=t_{f}-t_{i}$. The acceleration in the $\hat{x}$-direction can be
found to be:

$a_{x}^{\text{cd(p)}}(t)=\frac{q\absolutevalue{\mathbf{d}_{ext}}}{2\pi
mb^{3}\epsilon_{0}}\frac{\cos(\alpha)+(vt/b)\cos(\beta)}{(1+(vt/b)^{2})^{2}}\,.$
(26)

Its Fourier transform and the resulting PDF of the noise are given in Appendix
A. The dephasing is then given by $\Gamma_{\rm n}=\int_{\omega_{\rm
min}}^{\infty}S_{aa}(\omega)F(\omega)\differential{\omega}$, the full integral
will be:

$\displaystyle\Gamma_{\operatorname{n}}^{\text{cd(p)}}=$
$\displaystyle\frac{q^{2}|\mathbf{d}_{\operatorname{ext}}|^{2}(\Delta
x)^{2}}{2\pi^{2}\hbar^{2}\epsilon_{0}^{2}Tv^{3}b^{3}t_{a}^{4}}\int_{\omega_{min}}^{\infty}\omega^{-3}\operatorname{sin}^{2}\left(\frac{\omega}{2}(2t_{a}+t_{e})\right)$
(27)
$\displaystyle\operatorname{sin}^{4}\left(\frac{t_{a}\omega}{2}\right)\Biggr{[}\cos^{2}(\alpha)K_{3/2}^{2}\left(\frac{b\omega}{v}\right)$
$\displaystyle\mbox{\quad\quad}\mbox{\quad\quad}\mbox{\quad\quad}\mbox{\quad\quad}+\cos^{2}(\beta)K_{1/2}^{2}\left(\frac{b\omega}{v}\right)\Biggr{]}d\omega\,,$

Where, $K_{\nu}$ is the modified Bessel function (see appendix A). The results
of the integration with the parameters of a specific interferometer setup is
given in Sec. VI (figure 3(b)).

####  IV.2.2 Induced external dipole

If the dipole moment in the external particle with polarizability
$\alpha_{\text{pol}}$ is induced due to the electric field $\boldsymbol{E}$
from the charged interferometer, then the result is slightly different. The
induced dipole by a charge $q_{\text{int}}$ is given by:

$\boldsymbol{d}_{\text{ext}}^{\text{(i)}}=\alpha_{\text{pol}}\boldsymbol{E}=\alpha_{\text{pol}}\frac{q_{\text{int}}}{4\pi\epsilon_{0}r^{2}}\hat{\boldsymbol{r}}\,,$
(28)

the superscript (i) standing for induced. The acceleration in the
$\hat{x}$-direction thus becomes:

$\displaystyle a_{x}^{\text{cd(i)}}(t)$
$\displaystyle=\frac{q_{\text{int}}^{2}\alpha_{\text{pol}}}{8\pi^{2}mb^{5}\epsilon_{0}^{2}}\frac{\cos(\alpha)+(vt/b)\cos(\beta)}{(1+(vt/b)^{2})^{2}}\frac{1}{r^{2}(t)},$
$\displaystyle=\frac{q_{\text{int}}^{2}\alpha}{8\pi^{2}mb^{5}\epsilon_{0}^{2}}\frac{\cos(\alpha)+(vt/b)\cos(\beta)}{(1+(vt/b)^{2})^{3}}.$
(29)

The Fourier transform and PSD are given in Appendix A. The dephasing is then
found to be:

$\displaystyle\Gamma_{\operatorname{n}}^{\text{cd(i)}}=\frac{q^{4}\alpha^{2}_{\text{pol}}(\Delta
x)^{2}}{512\pi^{4}\hbar^{2}\epsilon_{0}^{4}v^{5}b^{5}Tt_{a}^{4}}$
$\displaystyle\int_{\omega_{\operatorname{min}}}^{\infty}d\omega\frac{1}{\omega}\sin^{4}\left(\frac{t_{a}\omega}{2}\right)$
(30) $\displaystyle\sin^{2}\left(\frac{\omega}{2}(2t_{a}+t_{e})\right)$
$\displaystyle\Biggr{[}\cos^{2}(\alpha)K^{2}_{5/2}\left(\frac{b\omega}{v}\right)$
$\displaystyle+\cos^{2}(\beta)K^{2}_{3/2}\left(\frac{b\omega}{v}\right)\Biggr{]}.$

$K_{\nu}$ is the modified Bessel function (see appendix A). Again, this
expression can be solved numerically in order to obtain the dephasing. The
results of the integration with the parameters of a specific interferometer
setup are given in Sec. VI (figure 3(c)).

Eq. (38) differs from the permanent dipole result in Eq. (35) since the
induced dipole is assumed to be induced by the interferometer charge, as a
result, Eq. (38) depends on the interferometer charge and the polarisability
of the external particle, $\alpha_{\text{pol}}$. The magnitude of the induced
dipole is distance-dependent, resulting in a different $v,b$-dependence of Eq.
(38) compared to the permanent dipole dephasing result.

##  V Neutral interferometer

In this section, we will assume that the micro-particles in the interferometer
have a dipole $d_{\text{int}}$ (either permanent or induced) and that it is
interacting with an external charged particle (of charge $q_{\text{ext}}$). We
assume that the particle is moving slowly within the chamber.

We consider the dipole-charge and dipole-dipole interaction between the micro-
particle in the interferometer with an external dipole/charge, where we
consider separately the permanent and induced dipole moments of the micro-
particle.

Although the noise/dephasing results obtained in this section are generic, we
will apply them to find the dephasing in the QGEM experiment which consists of
a neutral matter-wave interferometer in a diamond-type crystal of mass $\sim
10^{-15}\,$\mathrm{kg}$$. We briefly discuss the two different dipoles of the
micro-particle in this context.

  * •

Permanent dipole: It is possible that the microcrystal may have a permanent
dipole due to the impurities on the surface or in the volume. The dipole
moment of silica type material, ($\text{SiO}_{2}$), was experimentally
measured in Afek:2021bua , which showed that the material exhibits no clear
correlation between mass and dipole moment. That being said, the micro-
particle spheres of size $10\mu\text{m}$, $15\mu\text{m}$ and $20\mu\text{m}$
had permanent dipoles varying from $\sim 1000-5000e\mu\text{m}$, $\sim
200-8000e\mu\text{m}$ and $\sim 8500\pm 1500e\mu\text{m}$ Afek:2021bua . This
shows that there is at least an order of magnitude uncertainty in the
permanent dipole magnitude. The study showed that this material’s permanent
dipole moments exhibit a volume scaling, leaving the question open whether in
general, the permanent dipole moment scales with the volume. For the test-mass
of radius $r=0.5\,\mu$m (corresponding to $\sim 10^{-15}\,$\mathrm{kg}$$ for a
spherical diamond test mass), we take the dipole moment to be
$|\textbf{d}_{\text{int}}^{(p)}|\sim 1e\,$\mathrm{\SIUnitSymbolMicro m}$$,
thus assuming a volume scaling, and using the experimental data in
Afek:2021bua  as a benchmark.

  * •

Induced dipole: The diamond-type crystal is a dielectric material, the crystal
has a polarizability $\alpha$ (in SI-units
$\mathrm{C}\text{\,}{\mathrm{m}}^{2}\text{\,}{\mathrm{V}}^{-1}$). In
particular, in isotropic media, a local electric field will produce a local
dipole in each atom of the crystal’s lattice jackson_classical_1999 :

$\boldsymbol{d}_{\text{int}}^{(i)}=N\alpha_{\text{pol}}\,\boldsymbol{E}_{\text{loc}}.$
(31)

The local polarizability (which we denote $\alpha_{\text{pol}}$) of the atoms
is related to the polarisability of the medium via the Classius-Mossotti
relation:

$\frac{n\,\alpha_{\text{pol}}}{3\epsilon_{0}}=\frac{\epsilon_{r}-1}{\epsilon_{r}+2}\,,$
(32)

with $n$ the number density of atoms, $n=3N/4\pi R^{3}$ for spherical masses
($N$ the number of atoms, $R$ the radius). As a result, the dipole for a
spherical diamond crystal due to an external point charge $q_{\text{ext}}$ is
given by:

$\boldsymbol{d}_{\text{int}}^{(i)}=\frac{\epsilon_{r}-1}{\epsilon_{r}+2}\frac{q_{\text{ext}}R^{3}}{r^{2}}\hat{\boldsymbol{r}}$
(33)

with $\epsilon_{r}$ relative permittivity of the medium.

###  V.1 Dephasing due to internal dipole and charge interaction

We now consider the dephasing due to an internal dipole interacting with an
external charge. The acceleration was given in eq. (20), the $x$-component of
which depends on whether the dipole is permanent or induced.

####  V.1.1 Permanent dipole of a micro-particle

Suppose the interferometer’s micro-particle has a permanent dipole moment,
similar to that of silica-type material Afek:2021bua . In that case, the
dipole moment magnitude can be taken to be the experimentally determined
value, e.g. $\sim e\,$\mathrm{C}\text{\,}\mathrm{m}$$ for micro-spheres,
assuming a volume scaling as discussed previously. Furthermore, if we assume
full control of the interferometer particle, we can take the direction of the
intrinsic dipole moment to be aligned with the $\hat{z}$-axis. As detailed in
Appendix A, in this scenario the component of the acceleration in the
$\hat{x}$-direction is:

$\displaystyle
a_{x}^{\text{d(p)c}}(t)=\frac{q_{\text{ext}}|\mathbf{d}_{\text{int}}|}{2\pi
mb^{3}\epsilon_{0}}$
$\displaystyle\frac{\cos(\alpha)+(vt/b)\cos(\beta)}{(1+(vt/b)^{2})^{5/2}}$
(34) $\displaystyle\bigg{[}\cos(\theta_{0})+(vt/b)\cos(\gamma)\bigg{]},$

where $\gamma$ is the angle between the velocity vector and the $\hat{z}$-axis
and $\theta_{0}$ is the angle between the vector $b$ and the $\hat{z}$-axis
(the angles $\gamma$, $\theta_{0}$ are similar to the projection angles
$\alpha$, $\beta$, respectively, in figure 2, but $\gamma$, $\theta_{0}$ are
projection angles on the $\hat{z}$-axis rather than on the $\hat{x}$-axis,
they are discussed in more detail in Appendix A).

The Fourier transform $a(\omega)$ gives an expression for the PSD,
$S_{aa}(\omega)$, and from these expressions we find the dephasing parameter,
see Appendix A. Taking the worst-case dephasing based on the angles
$\alpha,\beta,\gamma,\theta_{0}$ (see Appendix B, where we optimise these
angles to see the largest dephasing), the dephasing is given as:

$\displaystyle\Gamma_{\operatorname{n}}^{\text{d(p)c}}=$
$\displaystyle\frac{q_{\text{ext}}^{2}|\mathbf{d}_{\text{int}}|^{2}(\Delta
x)^{2}}{9\pi^{3}\epsilon_{0}^{2}\hbar^{2}b^{2}v^{4}t_{a}^{4}}\int^{\infty}_{\omega_{\text{min}}}\frac{1}{\omega^{2}}\sin^{4}\left(\frac{t_{a}\omega}{2}\right)$
(35)
$\displaystyle\Bigg{[}K_{2}^{2}\left(\frac{b\omega}{v}\right)+\left(2+\frac{b^{2}\omega^{2}}{v^{2}}\right)K_{1}^{2}\left(\frac{b\omega}{v}\right)$
$\displaystyle+K_{0}^{2}\left(\frac{b\omega}{v}\right)-2\frac{b\omega}{v}K_{0}\left(\frac{b\omega}{v}\right)K_{1}\left(\frac{b\omega}{v}\right)\Bigg{]}$
$\displaystyle\mbox{\quad\quad}\sin^{2}\left(\frac{\omega}{2}(2t_{a}+t_{e})\right)\differential{\omega}$

which can be solved numerically and where $K_{\nu}$ is the modified Bessel
function (see Appendix A). The results of the integration with the parameters
of a specific interferometer setup are given in Sec. VI (figure 4(a)).

####  V.1.2 Induced dipole of a micro-particle

If the dipole is induced by the electric field of an external charge
$q_{\text{ext}}$, its magnitude is given by jackson_classical_1999 ;
griffiths2013introduction :

$\boldsymbol{d}_{\text{int}}^{\text{(i)}}=\frac{\epsilon_{r}-1}{\epsilon_{r}+2}\frac{q_{\text{ext}}R^{3}}{r^{2}(t)}\hat{\boldsymbol{r}}\,$
(36)

with the relative permittivity of diamond $\epsilon_{r}\sim 5.7$ for diamond
and $R$ the radius of the spherical diamond (the superscript (i) indicates the
induced dipole). The acceleration in the $\hat{x}$-direction then becomes:

$\displaystyle a_{x}^{\text{d(i)c}}(t)$
$\displaystyle=\frac{\epsilon_{r}-1}{\epsilon_{r}+2}\frac{q_{\text{ext}}^{2}R^{3}}{2\pi\epsilon_{0}mb^{5}}\frac{\cos(\alpha)+(vt/b)\cos(\beta)}{(1+(vt/b)^{2})^{3}}$
(37)

The dephasing parameter is derived in detail in Appendix A, and is given by:

$\displaystyle\Gamma_{\operatorname{n}}^{\text{d(i)c}}=$
$\displaystyle\left(\frac{\epsilon_{r}-1}{\epsilon_{r}+2}\right)^{2}\frac{q_{\text{ext}}^{4}R^{6}(\Delta
x)^{2}}{32\pi^{2}\epsilon_{0}^{2}\hbar^{2}v^{5}b^{5}Tt_{a}^{4}}\int_{\omega_{\text{min}}}^{\infty}\frac{1}{\omega}$
(38)
$\displaystyle\Biggr{[}\cos^{2}(\alpha)K^{2}_{5/2}\left(\frac{b\omega}{v}\right)+\cos^{2}(\beta)K^{2}_{3/2}\left(\frac{b\omega}{v}\right)\Biggr{]}\,,$
$\displaystyle\mbox{\quad\quad}\sin^{4}\left(\frac{t_{a}\omega}{2}\right)\sin^{2}\left(\frac{\omega}{2}(2t_{a}+t_{e})\right)\differential{\omega}.$

$K_{\nu}$ is the modified Bessel function (see appendix A). Again, this
expression can be solved numerically in order to obtain the dephasing. The
results of the integration with the parameters of a specific interferometer
setup are given in Sec. VI (figure 4(b)). Eq. (38) differs from the permanent
dipole result in Eq. (35) since the induced dipole is assumed to be induced by
the external charge, as a result, Eq. (38) depends on the external charge and
the relative permittivity and radius of the interferometer. The magnitude of
the induced dipole is distance-dependent, resulting in a different
$v,b$-dependence of Eq. (38) compared to the permanent dipole dephasing result
of Eq. (35).

###  V.2 Dephasing due to internal dipole and external dipole

The acceleration from the dipole-dipole interaction was given in Eq. (21). For
large $r$, we can see that this interaction is dominated by the other
interactions since it goes as $1/r^{4}$. Sticking with the assumptions made in
previous sections, that $\boldsymbol{d}_{\text{ext}}$ aligns with the
$\hat{z}$-direction, and that for a short interaction time $\tau$ and low
velocity particles, $\boldsymbol{d}_{\text{int}}$ approximately aligns with
$\hat{\boldsymbol{r}}(t)$, the acceleration in the $\hat{x}$-direction is:

$\displaystyle\boldsymbol{a}_{x}^{\text{dd}}(t)=\frac{6\absolutevalue{\boldsymbol{d}_{\text{ext}}}\absolutevalue{\boldsymbol{d}_{\text{int}}}\cos(\theta_{0})}{4\pi\epsilon_{0}mb^{4}}$
$\displaystyle\frac{\cos(\alpha)+(vt/b)\cos(\beta)}{(1+(vt/b)^{2})^{5/2}}$
(39)

More details are given in Appendix A, where the Fourier transform of the
acceleration and the resulting PSD is given. From the PSD and the transfer
function, the dephasing is found to be:

$\displaystyle\Gamma_{\operatorname{n}}^{\text{dd}}=$
$\displaystyle\frac{4\absolutevalue{\boldsymbol{d}_{\text{ext}}}^{2}\absolutevalue{\boldsymbol{d}_{\text{int}}}^{2}(\Delta
x)^{2}\cos^{2}(\theta_{0})}{\pi^{3}\hbar^{2}\epsilon_{0}^{2}b^{4}v^{4}Tt_{a}^{4}}\int_{\omega_{\text{min}}}^{\infty}\frac{1}{\omega^{2}}$
(40)
$\displaystyle\Bigg{[}\cos^{2}(\alpha)K_{2}^{2}\left(\frac{b\omega}{v}\right)+\cos^{2}(\beta)K_{1}^{2}\left(\frac{b\omega}{v}\right)\Bigg{]}$
$\displaystyle\sin^{4}\left(\frac{t_{a}\omega}{2}\right)\sin^{2}\left(\frac{\omega}{2}(2t_{a}+t_{e})\right)\differential{\omega},$

see Appendix A for the detailed derivation. However, the result here is an
approximation given the assumptions made on this setup’s dipole moments and
velocities. Ref. Fragolino:2023agd  provides a more general expression of the
decoherence due to dipole-dipole interactions for spatial interferometers. The
results of the integration with the parameters of a specific interferometer
setup is given in Sec. VI (figure 4(c)).

##  VI Dephasing Results and quantum sensing

Figure 3: Dephasing as a function of the velocity of the external particle due
to (a) the charge-charge interaction (cc), (b) the charge-permanent dipole
interaction (cd(p)) and (c) the charge-induced dipole interaction (cd(i)), for
a different number of interferometer charges. With projection angles
$\alpha=\beta=0$ and $T=b/v$. In Fig.(a) $b=1\,$\mathrm{m}$$ and the external
charge is that of an electron, in Fig.(b) $b=0.1\,$\mathrm{m}$$ and the
external dipole moment $d_{\text{ext}}$ is that of a water molecule, in
Fig.(c) $b=0.01\,$\mathrm{m}$$ and the external particle’s polarizability is
that of the dinitrogen. Fig.(d) shows the different dephasing for a charged
interferometer as a function of the impact parameter, for
$v=0.03\,$\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$, $q_{\text{int}}=e$ and the
interaction-specific parameters discussed above. This plot (d) shows the
relevance of a charged micro-particle interferometer as a quantum sensor. The
interferometer can detect $q_{\text{ext}}=e$ charge moving with a velocity
$v=0.03{\rm m/s}$ in the vicinity of $100$ cm with a detectable phase of
${\cal O}(1)$ rad. In all the plots we have ensured $T\sim b/v>\tau\sim 3$
sec.

First, we discuss the dephasing for the charged micro-particle, discussed in
Secs. IV. They are plotted in Fig. 3. We specify here the experimental
parameters and summarise the assumptions made on the dipole moment orientation
and evolution. The dephasing that is plotted is given in Eqs. (25), (27) and
(30) corresponding to dephasing in a charged interferometer. We will also
discuss the possibility of the charged interferometer to be an ideal quantum
sensor to detect another electromagnetically charged particle moving in the
vicinity.

The dephasing has been integrated in the frequency range from
$\omega_{\text{min}}=2\pi/(t_{f}-t_{i})$ to $\omega=\infty$. The experimental
time is set to be roughly $\tau\sim 3$ s (see figure 1). This frequency cut-
off is chosen due to the frequency resolution of the discrete Fourier
transform for the experimental setup. for QGEM specific example we will take
$\tau\sim 1$ s, and for CNOT gate we will take $\tau\sim 1{\rm\mu s}$.

Here, the time window for the PSD is chosen as the total experiment time
$\tau=t_{f}-t_{i}=4t_{a}+t_{e}=3$ s. These parameters are chosen such that no
acceleration jitter can be averaged beyond the experimental time. We have also
optimised the projection angles, see Figs. 9-9, and the discussion in Appendix
B. We have chosen the projection angles such that they maximise the dephasing.
This will enable us to gauge the maximum acceleration noise which the matter-
wave interferometer can tolerate due to EM interactions. Also, it will help us
do the total noise budget for the QGEM-type experiment Bose:2017nin , and with
the charged micro-particles Barker:2022mdz .

We have also plotted the dephasing given in Eqs. (35), (38) and (40) relating
to the neutral micro-particle, which we will discuss below. While plotting
this dephasing for the neutral micro-particle, we have made the following
additional assumptions for the dipoles to obtain approximately the upper
bounds on the dephasing. We summarize them below:

  * •

In Eq. (27) the external dipole is assumed to align approximately with the
vector $\hat{\boldsymbol{r}}(t)$ during the time of the experiment $\tau$.

  * •

In Eq. (35) the interferometer dipole is assumed to align with the
$\hat{z}$-axis.

  * •

In Eq. (40) we have assumed both the above assumptions.

Other relevant parameters which we have set are the maximum superposition size
of $\Delta x=20\,\mu$m and $t_{a}=0.50$ s and $t_{e}=1.00$ s with total time
$\tau=3$ s. Furthermore, we have taken the time scale over which the noise is
averaged, $T\sim b/v>3$ s. We have also varied $T$ in Fig.4(a) for the neutral
case for illustration.

  * •

Dephasing of a charged micro-particle: As it is clear from the plots for the
case of a charged micro-particle interferometer, the dephasing depends on the
particle’s velocity $v$, and minimum impact parameter, which we have taken to
be $b=1$m. It is expected that the ambient particle has a small velocity $v\ll
c$ for our analysis to be valid. One can also see that in Fig. 3(a) the
dephasing grows as the velocity increases for a fixed $b$. The charge-charge
case dominates the dephasing over other cases. This can be seen from Fig.
3(d). The Coulomb interaction is by far the most dominant source of dephasing
for the charged interferometer, see Figs. 3(a-d). In fact, the charged
interferometer can be treated as an excellent quantum sensor, sensing a
charged ion moving with a velocity range of order $v\sim{\cal O}(10^{-1}{\rm
m/s}$ in the vicinity of $b\sim 1$m from the matter-wave interferometer.
Measuring the phase of order $10^{-4}$ rad can be technically feasible.

Figure 4: Dephasing due to Fig.(a) the permanent dipole-charge interaction
(d(p)c), Fig.(b) the induced dipole-charge interaction (d(i)c) and Fig.(c) the
dipole-dipole interaction (dd) as a function of the velocity of the external
particle. With projection angles $\alpha=\beta=\gamma=\theta_{0}=\pi/2$ for
Fig. (a) and Fig.(c), and $\alpha=\beta=0$ for Fig.(b), see Appendix B. In
Fig.(a) $b=1\,$\mathrm{m}$$,
$d_{\text{int}}=0.1e\,$\mathrm{C}\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$q_{\text{ext}}=e$. Different lines for the noise time scale $T_{n}$ are
shown. In Fig.(b) $b=0.1\,$\mathrm{m}$$, the external dipole moment
$d_{\text{ext}}$ is that of a water molecule, and different lines for the
interferometric particle’s radius are plotted. In Fig.(c)
$d_{\text{int}}=0.1e\,$\mathrm{C}\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
the external particle is taken to be a water molecule, the dephasing is shown
for different impact parameters. In Fig.(d), we summarise the dephasing from
neutral micro-particle’s interactions as a function of the superposition width
$\Delta x$, for $v=0.03{\rm m/s}$, $b=0.1\,$\mathrm{m}$$, and the interaction-
specific parameters discussed above for the charge and dipoles. We have
ensured that $T=b/v>\tau\sim 3$ s.

The dephasing due to charge-dipole interaction is shown in Fig. 3(b),(c). Fig
(b) shows the charged micro-particle interacting with a molecule with a dipole
interaction, where the external dipole is intrinsic to the external particle,
for example, in a water molecule. The figure is plotted for a permanent dipole
moment $\absolutevalue{\boldsymbol{d}_{\text{ext}}}\sim
6.17*10^{-30}\,$\mathrm{C}\text{\,}\mathrm{m}$$ of a water molecule
lide2004crc . The dephasing is plotted as a function of the velocity of the
external water molecule. For smaller velocities, the dephasing is small, but
as the velocity increases, the dephasing increases and it also increases with
the charge of the micro-particle in the interferometer. The dephasing is small
compared to the other interactions, especially considering that the figure has
been plotted for a smaller impact parameter $b=0.01\,$\mathrm{m}$$. We can see
that the dephasing is very small compared to the dephasing due to the Coulomb
interaction.

Fig. 3(c) shows the dephasing due to the charged micro-particle and a dipole
interaction where the external dipole is induced, for example, in the
polarized air molecules. In this figure, we have taken the polarisability of
di-nitrogen , $(1.710$Å${}^{3})$, which is most frequently present in the air.
Again, the dephasing is negligible.

Now, we summarise our results in fig. 3(d), which shows the different
dephasings charge-charge (cc), charge-permanent dipole (cd(p)) and charge-
induced dipole (cd(i)) as a function of the impact parameter $b$ for $v\sim
0.03{\rm m/s}$. The plot shows that the impact parameter $b$ influences
dephasing in a fixed velocity scenario. Here again, note that the charged
interferometer can be used as quantum sensing device. The charge-charge
scenario presents a motif for a charged quantum sensor. The matter-wave
interferometer can detect an external ion with charge $e$ at a distance of
$100$ cm moving with a velocity of $0.03{\rm m/s}$ with an ${\cal O}(1)$ rad
phase.

  * •

Dephasing of a neutral micro-particle: In the case of a neutral
interferometer, the results are shown in Fig. 4. Fig. 4(a) shows the dephasing
due to the interaction of an intrinsic dipole of a micro-particle aligned at
the $\hat{z}$-axis interacting with an external charge, we have taken the
external particle to have the charge of an electron. Following the analysis in
Appendix B, where we have chosen the angles to maximise the dephasing, the
projection angles are considered $\alpha=\beta=\pi/2$. Furthermore, the impact
parameter $b=1\,$\mathrm{m}$$. In this plot, we have also shown different
values of the $T\sim b/v$ parameter to show how the dephasing behaves with
varying interaction times. As the velocity increases for a fixed impact
parameter $b=1$m, the dephasing increases. Nevertheless, the dephasing remains
very small for our analysis to be valid $T=b/v>\tau=3$s.

Fig. 4(b) shows the dephasing due to the interaction between an external
charge and the dipole it induces within the diamond microsphere. We have
varied the mass of the microdiamond by changing the radius from ($R\sim
1-0.01{\rm\mu m}$, and the dielectric constant of the diamond is taken to be
$\epsilon_{r}=5.7$). The figure is very similar to (a), but the dephasing is
significantly tiny, even though the plot is made for a smaller impact
parameter, $b=0.1\,$\mathrm{m}$$. The dephasing would be further suppressed
for smaller radii of the interferometer mass. In this case, the maximal
projection angles were $\alpha=\beta=0$, which would maximise the dephasing;
see Appendix B.

The dephasing due to the dipole-dipole interaction is given in fig. 4(c).
Again, the interferometer is assumed to have a dipole aligned with the
$\hat{z}$-axis of magnitude $\absolutevalue{\boldsymbol{d}_{\text{int}}}\sim
0.1e\,$\mathrm{C}\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The external dipole
is assumed to be that of water,
$\absolutevalue{\boldsymbol{d}_{\text{ext}}}\sim 6.17\times
10^{-30}\,$\mathrm{C}\text{\,}\mathrm{m}$$. Following the analysis in Appendix
B the projection angles are taken to be $\alpha=\beta=\pi/2$, which gives the
maximum dephasing, e.g. worst-case scenario, if $vt/b<1$. Different lines are
shown for different impact parameters; one can see the dephasing is tiny.

It is worth pointing out that Ref. Fragolino:2023agd  also studied the
decoherence due to the dipole-dipole interactions in approximately the same
parameter space. Specifically, they found the decoherence rate in a Born
approximation. The decoherence rate due to dipole-dipole contribution turned
out to be the dominant effect than the relative acceleration noise we have
found here in Fig. 4(c). The direct comparison may not be valid, as the
authors considered an ensemble of gas particle and constrained the number
density and the pressure of the container.

Finally, figure 4(d) gives an overview of the dephasings as described in figs
(a)-(c) as a function of the superposition width, showing that a smaller
superposition width reduces the dephasing. The most dominant source of
dephasing is from the permanent dipole interacting with an external (electron)
charge for $b=0.1\,$\mathrm{m}$$ and $v=0.03{\rm m/s}$, which ensures $T\sim
b/v>\tau$. The dephasing in all the cases is negligible.

##  VII Applications

We have discussed above the dephasing expressions for the charged and the
neutral interferometers due to external charges and dipoles. We now discuss
both the setups and the applications. The neutral interferometer is discussed
in the context of the QGEM proposal for testing the quantum nature of gravity
by witnessing the spin entanglement Bose:2017nin . We will also discuss the
charged interferometer in the context of an ion-based quantum computer and
study the dephasing due to the external charge and dipoles for a setup similar
to the QGEM case. Still, the entanglement is due to the interaction of the
photon between the two ions in adjacent ion traps.

###  VII.1 QGEM Protocol

Figure 5: We consider a QGEM microdiamond setup in spatial superpositions of
two microdiamonds of $10\,$\mathrm{\SIUnitSymbolMicro m}$$, separated by a
distance $d\sim{\cal O}(50){\rm\mu m}\ll b$, see Schut:2023hsy . The
experimental time for the interferometer is taken to be
$\tau=1\,$\mathrm{s}$$. The noise is found from the sum of all the neutral
interferometer interactions discussed in Figs. 4(a-d). We take similar
properties of the dipoles (induced and permanent) as that of Figs. 4(a-d). The
dephasing is shown for different velocities of the external particle.

In the context of the QGEM protocol, details on which can be found here
Bose:2017nin , we consider two neutral interferometers that are set up in a
way such their spatial superposition directions are parallel Chevalier:2020uvv
; Tilly:2021qef ; Schut:2021svd ; Schut:2023eux ; Schut:2023hsy . The creation
of single spatial interferometer is assumed to be as illustrated in figure 1.
The initial state of the interferometers is given by:

$\ket{\psi(0)}=\frac{1}{2}\left(\ket{\uparrow}_{1}+\ket{\downarrow}_{1}\right)\otimes\left(\ket{\uparrow}_{2}+\ket{\downarrow}_{2}\right)\,,$
(41)

which is a separable state. A quantum gravitational interaction between the
two massive interferometers can cause an entangling phase (while a classical
gravitational interaction does not cause this phase, see Bose:2017nin ;
Marshman:2019sne ; Bose:2022uxe ; Danielson:2021egj ; Carney_2019 ;
Carney:2021vvt ; Biswas:2022qto ; Christodoulou:2022vte ; Vinckers:2023grv ;
Elahi:2023ozf ; Chakraborty:2023kel ). In this case, we have two adjacent
interferometers separated by a distance $d$ and we assume the same
superposition size of $\Delta x$. For the QGEM experiment, we take the total
time $\tau=1$ sec for the gravity-induced entanglement to grow. We consider
the interferometric setup to be very similar to that described in Fig. 1. We
wish to take now the external noise induced by the EM interaction, identical
to the case we discussed earlier in the context of a neutral micro-particles
interferometer.

Taking into account the dephasing of the two interferometers, the final
wavefunction at a time $t$ is described by:

$\displaystyle\ket{\psi(\tau)}=\frac{e^{i\phi}e^{-\Gamma_{\text{n}}}}{2}\big{(}$
$\displaystyle
e^{\Gamma_{\text{n}}}\ket{\uparrow}_{1}\ket{\uparrow}_{2}+e^{i\Delta\phi}\ket{\uparrow}_{1}\ket{\downarrow}_{2}$
$\displaystyle+e^{i\Delta\phi}\ket{\downarrow}_{1}\ket{\uparrow}_{2}+e^{-\Gamma_{\text{n}}}\ket{\downarrow}_{1}\ket{\downarrow}_{2}\big{)}$
(42)

with

$\displaystyle\phi$
$\displaystyle=\frac{\tau}{\hbar}\frac{Gm^{2}}{d}\,,\mbox{\quad\quad}\Delta\phi=\frac{\tau}{\hbar}\int_{0}^{\tau}\frac{Gm^{2}}{\sqrt{d^{2}+\Delta
x^{2}(t)}}\,\differential{t}-\phi$ (43)

where the interferometer mass is $m\sim 10^{-14}-10^{-15}$kg to get an order
one entanglement phase, see Bose:2017nin ; Schut:2023hsy  and the
interferometers are separated a distance $d$, and $\Gamma_{\text{n}}$ is
defined by the results found in the previous sections for the neutral micro-
particle. The dephasing is approximately the same for both interferometers if
the impact parameter $b\gg d$. We can find the density matrix from the final
wavefunction with $\rho=\ket{\psi}\bra{\psi}$.

To witness the entanglement due to the gravitational interaction, we use the
positive partial transpose (PPT) witness, which, in the case of two qubits,
provides a sufficient and necessary condition for entanglement based on the
Peres-Horodecki criterion. The PPT witness is defined as
$\mathcal{W}=\ket{\lambda_{-}}\bra{\lambda_{-}}^{T_{2}}$, where
$\ket{\lambda_{-}}$ is the eigenvector corresponding to the lowest eigenvalue,
$\lambda_{-}$, of the partially transposed density matrix and the superscript
$T_{2}$ denotes the partial transpose over system $2$. The expectation value
of the witness is given by Chevalier:2020uvv ; Tilly:2021qef ; Schut:2021svd :

$\langle\mathcal{W}\rangle=\Tr(\mathcal{W}\rho)=\lambda_{-}$ (44)

For the wavefunction in eq. (42) the expectation value is found to be:

$\langle\mathcal{W}\rangle=-\frac{1}{2}\sin(\absolutevalue{\Delta\phi})e^{-2\Gamma_{\text{n}}}\,,$
(45)

see also Ref. Schut:2023hsy  for more details. The entanglement detection
condition is $\langle\mathcal{W}\rangle<0$. If $\langle\mathcal{W}\rangle\geq
0$, there is no entanglement. Performing a short-time expansion of the
witness, these inequalities can approximate the maximum allowed value of the
dephasing rate.

$-\frac{\absolutevalue{\Delta\phi}}{2}\left(1-2\Gamma_{\text{n}}\right)<0$
(46)

Showing that if $\Gamma_{\text{n}}>1/2$, then $\langle\mathcal{W}\rangle>0$
and there is no entanglement. Hence, the total dephasing due to the EM
interaction must be $\Gamma_{n}<0.5$ for the gravity-induced entanglement to
be detectable.

We now accumulate all the sources of dephasing due to the EM interaction, such
as arising from the Figs. 4(a-d), e.g. induced dipole and charge, permanent
dipole and charge, and dipole-dipole interactions. In our case, we consider
the diamond to have a permanent dipole of the order of $0.1{\rm e\mu m}$.

The cumulative effect of all the sources of dephasing for the QGEM experiment
can be shown in Fig. 5. The Fig. 5 shows the sum of all the neutral dephasings
as shown in Figs. 4(a-d), applied to the QGEM protocol. The transfer function
of the interferometer is assumed to be as in Eq. (11), where the phases
$t_{e}$, $t_{a}$ are scaled-down compared to previous figures such that the
total experimental time is $1\,$\mathrm{s}$$. For the external particle, we
have taken it to be the same properties as in fig. 4(a-d), but now plotted the
impact parameter $b$ for different velocities, naturally, as the velocity
increases of the external particle and as the impact parameter decreases the
dephasing increases, as expected. Nevertheless, the dephasing is so tiny that
it will hardly be a problem for the QGEM experiment. In this analysis, we have
considered the superposition size to be $\Delta x=10{\rm\mu m}$. The inter-
separation distance between the two interferometers does not play a
significant role since $d\sim{\cal O}(50){\rm\mu m}\ll b\sim 100{\rm cm}-1{\rm
m}$ in our case.

###  VII.2 CNOT Gate

Figure 6: For two ion traps with two traps each consisting of $10$ Rb atoms
with $10e$ charges in $10^{5}\,$\mathrm{Hz}$$ traps. The superposition’s
experimental time is $\tau=1\,$\mathrm{\SIUnitSymbolMicro s}$$. The dephasing
is found from the dominant charge-charge interactions with the external
particle having the charge of an electron. The dephasing is shown for
different velocities of the external particle. The transfer function of the
interferometer is given by Eq. (11).

In quantum computers, one often considers trapped charged particles with a
Gaussian state Wineland:1992 ; Wineland:1995 . Here, we consider a CNOT gate
consisting of two trapped ions separated by a distance $d$. Their initial
wavefunction will be very similar to Eq. 41. The entanglement between the two
trapped ions builds up due to the EM interaction mediated via photons,
essentially the Coulomb interaction, which creates a maximal entanglement
state, e.g., a CNOT gate. The final wavefunction of such an entangled system
will be given by Eq. 42, with an entangled phase given by:

$\displaystyle\phi$
$\displaystyle=\frac{\tau}{\hbar}\frac{\kappa_{e}q_{1}q_{2}}{d}\,,\Delta\phi=\frac{\tau}{\hbar}\int_{0}^{\tau}\frac{\kappa_{e}q_{1}q_{2}}{\sqrt{d^{2}+\Delta
x^{2}(t)}}\,\differential{t}-\phi\,,$ (47)

Where $\kappa_{e}=(4\pi\epsilon_{0})^{-1}$, and $q_{1}$ and $q_{2}$ are
internal charges of the ions in their respective traps. We also consider the
limit when $d\gg\Delta x$, so our spatial superpositions become a Gaussian
wavefunction. We can estimate the width of the Gaussian wavepacket by the trap
frequency and the mass of the trapped ion, i.e.,

$\delta x=\sqrt{\frac{\hbar}{2m\omega}}$ (48)

where $m$ is the mass of the ion, and $\omega$ is the trapping frequency.

In fig. 6, we show the dephasing from the charge-charge interaction only. We
assume that an external charge is flying by with an impact parameter $b$ and a
constant velocity $v$. We assume that the ion trap consists of $10$ Rb-atoms
with one remnant charge, hence $q_{1}=q_{2}=10e$ for the sake of illustration
in a trapping frequency of $10^{5}$Hz. The superposition size it will create
is given by $\delta x\sim 0.18{\rm\mu m}$. As we will see, the entanglement
builds up quite swiftly by the Coulomb interaction, which is many orders of
magnitude larger than the gravitational interaction strength. In figure 6 the
total experimental time we require is $\tau=1\,$\mathrm{\SIUnitSymbolMicro
s}$$. For illustration, we took a similar transfer function, which is
responsible for creating the spatial superposition in Fig. 1. In our case, we
have taken the external charge as that of an electron with different values of
$b$ and $v$ to illustrate the dephasing. Note that the dephasing increases if
the impact parameter is small, and notably, a large constant velocity of a
moving ion can create a significantly larger dephasing of order ${\cal
O}(10^{-3})$ rad.

##  VIII Conclusion

In this paper, we computed the dephasing of the matter-wave interferometer due
to long range interaction. The traditional methods of computing decoherence
via utilising the cross-section does not work for long-range interactions. The
total cross-section becomes infinite for both Coulomb and Newtonian
interactions unless we impose a small cut-off angle for scattering. For long
range interactions the gravitational wave community came up with a clever
strategy of computing the gravitational acceleration noise arising from the
movements of the external masses via analysing the PSD and the transfer
function of the interferometer, see Saulson:1984 . We had employed this
technique to compute the gravitational jitter for the matter-wave
interferometer in Toros:2020dbf  and in Wu:2022rdv  to detect the space debris
by measuring the gravity gradient noise.

In this paper, we have adopted similar techniques to compute the noise arising
from the EM interactions. We considered two distinct scenarios, one where the
micro-particle creating the superposition is charged, and the other where the
micro-particle is neutral. In the former case, the charged interferometer can
interact via Coulomb interaction with an external charged particle. The
external particle’s velocity is assumed to be constant, and the acceleration
it will impart on both the arms of the interferometer will cause a jitter,
which would effectively dephase the interferometer. The charged interferometer
can also interact with a constantly moving dipole. The ambient particle can
have a permanent dipole or an induced dipole. We considered both the cases
separately, See Figs. 3(a-d). The largest dephasing occurs due to the Coulomb
interaction, which sets the exclusion zone for an external charged particle to
be moving with a constant velocity.

In the neutral case, we have again four possibilities, a micro-particle is
typically considered to be diamond in our case, which has a dielectric
property. The microdiamond can then have a permanent and an induced dipole,
hence it can interact with an external charge, and an external dipolar
particle. In all these cases, if the external particle is moving with a
constant velocity it will impart a jitter in the trajectories due to the
relative accelerations of the arms of the matterwave interferometer, and hence
will induce a dephasing.

The relative acceleration noise can occur if the two neutral micro-particles
are kept in an adjacent superpositions, as in the case of the QGEM experiment.
This will allow the dephasing in the entanglement phase, which rqeuires to be
below $\Gamma_{n}<0.5$ for a particular witness we are considering in the
experimental simulation, see Chevalier:2020uvv ; Tilly:2021qef ; Schut:2021svd
. However, we have found that the QGEM type experiment is safe from the
external jitters due to the EM interactions, the dephasing is tiny.

The relative acceleration noise can also affect the ion trap based spatial
qubits, which are essentially form a CNOT gate. We observed that our
methodology can be successfully employed to characterize the quantum noise for
such a setup, we found that large velocity and small impact parameter, e.g.
the distance between the external charged particle and the two ion traps, can
cause a dephasing which might create a detectable phase, hence will add up to
the decoherence budget. For a specific setup we considered the dephasing is
$10^{-3}$rad.

Indeed, our analysis has its own limitations. We have only considered a single
particle moving close to the matter-wave interferometer with a constant
velocity. In our case we can only model the noise if the velocity is non-
relativistic. There are two fold improvements we can bring in; we can compute
the dephasing for an ensemble of external particles each responsible for
dephasing the interferometer(s). The collective behaviour may shed light on
how many ambient particles can we tolerate in the vicinity of the experiment.
Moreover, we can further improve our techniques by considering the
relativistic corrections for the external particles.

Despite these weakness, we have provided an interesting way of computing the
quantum noise in the matter-wave interferometer due to an external particles
interacting solely via the EM long range interaction. Exploiting these
techniques will be important for the success of fundamental problems such as
the entanglement witness of the qunatum nature of gravity in a lab.
Furthermore, our analysis will also shed some light on dephasing ion based
quantum gates. Our analysis also provides an interesting sensing capability
for a charged interferometer, which is sensitive to the Coulomb intercation,
and can measure ${\cal O}(1)$ rad phase for $\Delta x\sim 10{\rm\mu m}$,
internal and external charge of the ion to that of an electron moving with a
velocity of $0.03{\rm m/s}$ in the vicinity of $100$ cm. Indeed, this case can
be a possibility if we could ramp up the charges on the charged
interferometer. A $10^{4}e$ would immediately allow the sensor to probe an ion
moving with a velocity of $10^{-2}{\rm m/s}$ in the vicinity of $10$ m for the
experimental time of $3$ sec. Such a sensor can be helpful to measure stray
charges in the vicinity of the matter-wave interferometer.

##  IX Acknowledgemnets

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.

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##  Appendix A Estimation of dephasing

This section shows the expressions for the Fourier transform of the
acceleration, $a_{x}(\omega)$ and the corresponding PSD for the acceleration
noise $S_{aa}(\omega)$, form which the dephasing due to the acceleration noise
can be derived via

$\Gamma_{\rm n}=\frac{1}{2\pi}\frac{m^{2}}{\hbar^{2}}\int_{\omega_{\rm
min}}^{\infty}S_{aa}(\omega)F(\omega)\differential{\omega}\,.$ (49)

with the transfer function $F(\omega)$ determined by the trajectory, see Eq.
(11). Since,

$\displaystyle a(\omega)$
$\displaystyle=\int_{-\infty}^{\infty}a(t)e^{-i\omega t}\differential{t}$ (50)
$\displaystyle a(t)$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}a(\omega)e^{i\omega
t}\differential{\omega}$ (51)

where the $1/2\pi$ comes from the redefinition of $\omega\to\omega/2\pi$, as a
consequence the minimal frequency is given by
$\omega_{\text{min}}=2\pi/(t_{f}-t_{i})$. The Fourier transform is of a
function $f(vt/b)$ as defined below is given by:

$\displaystyle f(vt/b)$
$\displaystyle=\frac{(vt/b)^{2n}}{((vt/b)^{2}+c^{2})^{\nu+1/2}}$ (52)
$\displaystyle f(b\omega/v)$
$\displaystyle=\frac{(-1)^{n}c^{\nu}\sqrt{\pi}}{2^{\nu}\Gamma(\nu+1/2)}\partialderivative[2n]{(b\omega/v)^{\nu}K_{\nu}(c\,b\omega/v)}{(b\omega/v)}$

with $\Gamma(x)$ the Euler-Gamma-function 333 With relevant values
$\Gamma(3/2)=\sqrt{\pi}/2$, $\Gamma(2)=1$, $\Gamma(5/2)=3\sqrt{\pi}/4$ and
$\Gamma(3)=2$.  and $K_{\nu}$ the modified Bessel function of the second kind
with order $\nu$. The above Fourier transform holds only if $\real(c)>0$ and
$\real(\nu+1/2)>n$. The modified Bessel functions have the property that:

$\displaystyle\partialderivative{K_{\nu}(x)}{x}$
$\displaystyle=-\frac{1}{2}(K_{\nu-1}(x)+K_{\nu+1}(x))$ (53) $\displaystyle
K_{\nu-1}(x)$ $\displaystyle=K_{\nu+1}(x)-\frac{2\nu}{x}K_{\nu}(x)$

The modified Bessel functions with half-integers have the analytic formula:

$\displaystyle K_{1/2}(u)$
$\displaystyle=\sqrt{\frac{\pi}{2u}}\mathrm{e}^{-u}$ (54) $\displaystyle
K_{3/2}(u)$
$\displaystyle=\sqrt{\frac{\pi}{2u}}\left(1+\frac{1}{u}\right)\mathrm{e}^{-u}$
$\displaystyle K_{5/2}(u)$
$\displaystyle=\sqrt{\frac{\pi}{2u}}\left(1+\frac{3}{u}+\frac{3}{u^{2}}\right)\mathrm{e}^{-u}$

The integer modified Bessel functions of the second kind do not have an
analytical formula, but can be expressed in the integral form:

$\displaystyle K_{0}(u)$
$\displaystyle=\int_{0}^{\infty}\frac{\cos(tu)}{\sqrt{1+t^{2}}}\differential{t}$
(55) $\displaystyle K_{1}(u)$
$\displaystyle=\frac{1}{u}\int_{0}^{\infty}\frac{\cos(tu)}{(1+t^{2})^{3/2}}\differential{t}$
$\displaystyle K_{2}(u)$
$\displaystyle=\frac{3}{u^{2}}\int_{0}^{\infty}\frac{\cos(tu)}{(1+t^{2})^{5/2}}\differential{t}$

###  A.1 Charge-charge interaction

As presented in Eq. (24), the acceleration due to the charge-charge
interaction in the $\hat{x}$-direction is:

$a_{x}^{\text{cc}}(t)=\frac{q_{\text{int}}q_{\text{ext}}}{4\pi\epsilon_{0}mb^{2}}\frac{\cos(\alpha)+(vt/b)\cos(\beta)}{(1+(vt/b)^{2})^{3/2}}.$
(56)

Transforming the acceleration $a_{x}^{\text{cc}}(t)$ into the frequency
domain, we use Eq. (52) (with $c=1,\nu=1,n=0,1/2$) to obtain:

$\displaystyle
a_{x}^{\text{cc}}(\omega)=\frac{q_{\text{int}}q_{\text{ext}}\omega}{4\pi\epsilon_{0}mvb}\bigg{[}$
$\displaystyle\cos(\alpha)K_{1}\left(\frac{b\omega}{v}\right)$ (57)
$\displaystyle\,\,-i\cos(\beta)K_{0}\left(\frac{b\omega}{v}\right)\bigg{]}\,,$

where $K_{\nu}(x)$ is a modified Bessel function, and we used its properties
in Eq. (53). Note that after Fourier transform, the unit of $a_{x}(\omega)$ is
$\rm m/(s^{2}\cdot Hz)$, which is because $a_{x}(t)=\int
a_{x}(\omega)e^{i\omega t}d\omega$ has a unit $\rm m/s^{2}$ and $d\omega$ has
a unit Hz.

Substituting Eq. 57 for the acceleration due to the Coulomb interaction into
Eq. 6 for the PSD results in the desired PSD of the acceleration noise from
the Coulomb interaction:

$\displaystyle
S_{aa}(\omega)=\frac{q_{\text{int}}^{2}q_{\text{ext}}^{2}\omega^{2}}{16\pi^{2}Tm^{2}v^{2}b^{2}\epsilon_{0}^{2}}$
$\displaystyle\biggr{[}\cos^{2}(\alpha)K_{1}^{2}\left(\frac{b\omega}{v}\right)$
(58)
$\displaystyle+\cos^{2}(\beta)K_{0}^{2}\left(\frac{b\omega}{v}\right)\biggr{]}.$

Note that the unit of $S_{aa}(\omega)=|a_{x}(\omega)|^{2}/T$ is $\rm
m^{2}/(s^{4}\cdot Hz)$. Using Eq. (49) gives the dephasing in Eq. (25).

###  A.2 Charge-dipole interaction

####  A.2.1 Permanent dipole

As presented in Eq. (26), the acceleration due to the charge-permanent dipole
interaction in the $\hat{x}$-direction is:

$a_{x}^{\text{cd(p)}}(t)=\frac{q\absolutevalue{\mathbf{d}_{ext}}}{2\pi
mb^{3}\epsilon_{0}}\frac{\cos(\alpha)+(vt/b)\cos(\beta)}{(1+(vt/b)^{2})^{2}}\,.$
(59)

Transforming the acceleration $a_{x}^{\text{cd(p)}}(t)$ into the frequency
domain, we use Eq. (52) (with $c=1,\nu=3/2,n=0,1/2$) to obtain:

$\displaystyle
a_{x}({\omega})=\frac{q|\mathbf{d}_{ext}|\omega^{3/2}}{4\pi\epsilon_{0}mv^{3/2}b^{3/2}}\sqrt{\frac{\pi}{2}}$
$\displaystyle\Bigg{[}\cos(\alpha)K_{3/2}\left(\frac{b\omega}{v}\right)$ (60)
$\displaystyle-i\cos(\beta)K_{1/2}\left(\frac{b\omega}{v}\right)\Bigg{]}\,.$

where $K_{\nu}(x)$ is a modified Bessel function, and we used its properties
in Eq. (53).

Now substituting this expression into definition of the PSD of the
acceleration noise results in:

$\displaystyle
S_{aa}(\omega)=\frac{q^{2}|\mathbf{d}_{ext}|^{2}\omega^{3}}{32\pi
Tm^{2}v^{3}b^{3}\epsilon_{0}^{2}}$
$\displaystyle\Biggr{[}\cos^{2}(\alpha)K_{3/2}^{2}\left(\frac{b\omega}{v}\right)$
(61)
$\displaystyle+\cos^{2}(\beta)K_{1/2}^{2}\left(\frac{b\omega}{v}\right)\Biggr{]}.$

The resulting dephasing is given in Eq. (27).

####  A.2.2 Induced dipole

As presented in Eq. (29), the acceleration due to the charge-permanent dipole
interaction in the $\hat{x}$-direction is:

$a_{x}^{\text{cd(i)}}(t)=\frac{q^{2}_{\text{int}}\alpha}{8\pi^{2}mb^{5}\epsilon_{0}^{2}}\frac{\cos(\alpha)+(vt/b)\cos(\beta)}{(1+(vt/b)^{2})^{3}}.$

Performing a Fourier transform given in Eq. (52) (with $c=1,\nu=5/2,n=0,1/2$)
to find the acceleration in the frequency domain gives:

$\displaystyle
a_{x}(\omega)=\frac{q^{2}\alpha_{\text{pol}}\omega^{5/2}}{64\pi^{2}mb^{5/2}v^{5/2}\epsilon_{0}^{2}}$
$\displaystyle\sqrt{\frac{\pi}{2}}\Biggr{[}\cos(\alpha)K_{5/2}\left(\frac{b\omega}{v}\right)$
(62)
$\displaystyle-i\cos(\beta)K_{3/2}\left(\frac{b\omega}{v}\right)\Biggr{]}.$

Substituting this expression into definition of the PSD of the acceleration
noise results in a slightly modified expression compared to the permanent
dipole case:

$\displaystyle
S_{aa}(\omega)=\frac{q^{4}\alpha^{2}_{\text{pol}}\omega^{5}}{8192\pi^{3}m^{2}v^{5}b^{5}\epsilon_{0}^{4}T}\Biggr{[}$
$\displaystyle\cos^{2}(\alpha)K^{2}_{5/2}\left(b\omega/v\right)$ (63)
$\displaystyle+\cos^{2}(\beta)K^{2}_{3/2}(b\omega/v)\Biggr{]}.$

The resulting dephasing is given in Eq. (30).

###  A.3 Dipole-charge interaction

####  A.3.1 Permanent dipole

As presented in Eq. (34) the acceleration in the $\hat{x}$-direction due to
the permanent dipole interacting with an external charge is:

$\displaystyle
a_{x}^{\text{d(p)c}}(t)=\frac{q_{\text{ext}}|\mathbf{d}_{\text{int}}|}{2\pi
mb^{3}\epsilon_{0}}$
$\displaystyle\frac{\cos(\alpha)+(vt/b)\cos(\beta)}{(1+(vt/b)^{2})^{5/2}}$
(64) $\displaystyle\bigg{[}\cos(\theta_{0})+(vt/b)\cos(\gamma)\bigg{]},$

The second line comes from the resolution of the dot product as:

$\displaystyle\boldsymbol{d}_{\text{int}}\cdot\hat{\boldsymbol{r}}(t)$
$\displaystyle=\absolutevalue{\boldsymbol{d}_{\text{int}}}\frac{b_{z}+v_{z}t}{r(t)}\,,$
(65)
$\displaystyle=\absolutevalue{\boldsymbol{d}_{\text{int}}}\frac{\cos(\theta_{0})+(vt/b)\cos(\gamma)}{\sqrt{1+(vt/b)^{2}}}\,,$
(66)

where we have assumed that the direction of the dipole vector of the
interferometer is constant, i.e. $\boldsymbol{d}_{\text{int}}$ is not time-
dependent, and that it points in the $\hat{z}$-direction. For example, because
the microdiamond sphere is magnetically trapped Marshman:2023nkh  and its
rotational degrees of freedom are cooled to near its ground state Deli__2020 ,
achieving good control of the system.

In the second line above, Eq. (66) we have written the dot product in terms of
the angle $\theta_{0}$, which is the angle between the $\hat{z}$-axis and the
vector $b$, and the angle $\gamma$ which gives the $\hat{z}$-component of the
velocity:

$b_{z}=b\cos(\theta_{0})\,,\mbox{\quad\quad}v_{z}=v\cos(\gamma)\,.$ (67)

The Fourier transform of the acceleration $a_{x}^{\text{d(p)c}}(t)$ is found
from Eq. (52) (with $c=1,\nu=2,n=0,1/2,1$) and gives:

$\displaystyle a_{x}({\omega})=$
$\displaystyle\Bigg{[}\cos(\alpha)\bigg{(}K_{2}\left(\frac{b\omega}{v}\right)\cos(\theta_{0})-iK_{1}\left(\frac{b\omega}{v}\right)\cos(\gamma)\bigg{)}$
(68)
$\displaystyle-i\cos(\beta)\bigg{(}K_{1}\left(\frac{b\omega}{v}\right)\cos(\theta_{0})-i\bigg{[}\frac{b\omega}{v}K_{1}\left(\frac{b\omega}{v}\right)$
$\displaystyle-
K_{0}\left(\frac{b\omega}{v}\right)\bigg{]}\cos(\gamma)\bigg{)}\Bigg{]}\frac{q_{\text{ext}}|\mathbf{d}_{\text{int}}|\omega^{2}}{6\pi\epsilon_{0}mbv^{2}}\,.$

Where we used Eqs. (52) to find the Fourier transform and Eqs. (53) to
simplify the expression.

Now substituting this expression into the Eq. (5) of the PSD of the
acceleration noise results in:

$\displaystyle S_{aa}(\omega)=$
$\displaystyle\Bigg{[}\cos^{2}(\alpha)\bigg{(}K_{2}^{2}\left(\frac{b\omega}{v}\right)\cos^{2}(\theta_{0})+K_{1}^{2}\left(\frac{b\omega}{v}\right)\cos^{2}(\gamma)\bigg{)}$
(69)
$\displaystyle+\cos^{2}(\beta)\bigg{(}K_{1}^{2}\left(\frac{\omega}{bv}\right)\cos^{2}(\theta_{0})+\bigg{[}\frac{b\omega}{v}K_{1}\left(\frac{b\omega}{v}\right)$
$\displaystyle-
K_{0}\left(\frac{b\omega}{v}\right)\bigg{]}^{2}\cos^{2}(\gamma)\bigg{)}\Bigg{]}\frac{q_{\text{ext}}^{2}|\mathbf{d}_{\text{int}}|^{2}\omega^{4}}{36\pi^{2}\epsilon_{0}^{2}m^{2}b^{2}v^{4}T}\,.$

The resulting dephasing from Eq. (49) and (11) is given by:

$\displaystyle\Gamma_{\operatorname{n}}^{\text{d(p)c}}=$
$\displaystyle\frac{4q_{\text{ext}}^{2}|\mathbf{d}_{\text{int}}|^{2}(\Delta
x)^{2}}{9\pi^{3}\epsilon_{0}^{2}\hbar^{2}b^{2}v^{4}t_{a}^{4}}\int^{\infty}_{\omega_{\text{min}}}\frac{1}{\omega^{2}}\sin^{4}\left(\frac{t_{a}\omega}{2}\right)$
(70)
$\displaystyle\Bigg{[}\cos^{2}(\alpha)\bigg{(}K_{2}^{2}\left(\frac{b\omega}{v}\right)\cos^{2}(\theta_{0})+K_{1}^{2}\left(\frac{b\omega}{v}\right)\cos^{2}(\gamma)\bigg{)}$
$\displaystyle+\cos^{2}(\beta)\bigg{(}K_{1}^{2}\left(\frac{b\omega}{v}\right)\cos^{2}(\theta_{0})+\bigg{[}\frac{b\omega}{v}K_{1}\left(\frac{b\omega}{v}\right)$
$\displaystyle-
K_{0}\left(\frac{b\omega}{v}\right)\bigg{]}^{2}\cos^{2}(\gamma)\bigg{)}\Bigg{]}\sin^{2}\left(\frac{\omega}{2}(2t_{a}+t_{e})\right)\differential{\omega}$

Which has been presented in Eq. (35) in a simplified form.

####  A.3.2 Induced dipole

Eq. (37) gave the acceleration in the $\hat{x}$-direction for the interaction
between an external charge and the dipole it induces in the interferometer:

$\displaystyle a_{x}^{\text{d(i)c}}(t)$
$\displaystyle=\frac{\epsilon_{r}-1}{\epsilon_{r}+2}\frac{q_{\text{ext}}^{2}R^{3}}{2\pi\epsilon_{0}mb^{5}}\frac{\cos(\alpha)+(vt/b)\cos(\beta)}{(1+(vt/b)^{2})^{3}}$
(71)

Performing a Fourier transform to find the acceleration in the frequency
domain gives (from Eq. (52) with $c=1,\nu=5/2,n=0,1/2$):

$\displaystyle a_{x}(\omega)=$
$\displaystyle\frac{\epsilon_{r}-1}{\epsilon_{r}+2}\frac{q_{\text{ext}}^{2}R^{3}\omega^{5/2}}{16\pi
mb^{5/2}v^{5/2}\epsilon_{0}}\sqrt{\frac{\pi}{2}}$
$\displaystyle\Biggr{[}\cos(\alpha)K_{5/2}\left(\frac{b\omega}{v}\right)-i\cos(\beta)K_{3/2}\left(\frac{b\omega}{v}\right)\Biggr{]}.$
(72)

Substituting this expression into the PSD of the acceleration noise results in
a slightly modified expression compared to the permanent dipole case:

$\displaystyle S_{aa}=$
$\displaystyle\left(\frac{\epsilon_{r}-1}{\epsilon_{r}+2}\right)^{2}\frac{q_{\text{ext}}^{4}R^{6}\omega^{5}}{512\pi\epsilon_{0}^{2}m^{2}v^{5}b^{5}T}$
(73)
$\displaystyle\Biggr{[}\cos^{2}(\alpha)K^{2}_{5/2}\left(\frac{b\omega}{v}\right)+\cos^{2}(\beta)K^{2}_{3/2}\left(\frac{b\omega}{v}\right)\Biggr{]}\,,$

from which Eq. (38) is derived.

###  A.4 Dipole-dipole interaction

The acceleration due to dipole-dipole interactions is given by
griffiths2013introduction :

$\boldsymbol{a}_{\text{dd}}=\frac{3\,\hat{\boldsymbol{r}}}{4\pi\epsilon_{0}m}\left[\frac{\boldsymbol{d}_{\text{ext}}\cdot\boldsymbol{d}_{\text{int}}}{r^{4}(t)}-3\frac{(\boldsymbol{d}_{\text{ext}}\cdot\hat{\boldsymbol{r}})(\boldsymbol{d}_{\text{int}}\cdot\hat{\boldsymbol{r}})}{r^{4}(t)}\right]$
(74)

In order to resolve the dot products, we recall from previous sections the
assumption we have made, that

  1. 1.

The external particle has low velocity, meaning that
$\boldsymbol{d}_{\text{ext}}\cdot\hat{\boldsymbol{r}}$ is approximately
constant in time. Furthermore, in the worst-case scenario
$\boldsymbol{d}_{\text{ext}}\cdot\hat{\boldsymbol{r}}=\absolutevalue{\boldsymbol{d}_{\text{ext}}}$.

  2. 2.

The dipole of the interferometer particle is aligned with the $\hat{z}$-axis.
Such that $\boldsymbol{d}_{\text{int}}\cdot\hat{\boldsymbol{r}}$ is as in Eq.
(66). Assuming that the external particle has small velocity,
$\boldsymbol{d}_{\text{int}}\cdot\hat{\boldsymbol{r}}\approx\absolutevalue{\boldsymbol{d}_{\text{int}}}\cos(\theta_{0})$.

From the orientation of the dipoles in these assumptions, it follows that:

  1. 3.

The angle between the interferometer and external dipoles is $\theta_{0}$,
such that
$\boldsymbol{d}_{\text{int}}\cdot\boldsymbol{d}_{\text{ext}}=\absolutevalue{\boldsymbol{d}_{\text{int}}}\absolutevalue{\boldsymbol{d}_{\text{ext}}}\cos(\theta_{0})$.

In this scenario, the acceleration in the $\hat{x}$-direction becomes:

$\displaystyle\boldsymbol{a}^{\text{dd}}_{x}(t)$
$\displaystyle=-\frac{6}{4\pi\epsilon_{0}m}\frac{\absolutevalue{\boldsymbol{d}_{\text{ext}}}\absolutevalue{\boldsymbol{d}_{\text{int}}}\cos(\theta_{0})}{r^{4}(t)}\frac{\boldsymbol{r}_{x}}{r}$
$\displaystyle=-\frac{6\absolutevalue{\boldsymbol{d}_{\text{ext}}}\absolutevalue{\boldsymbol{d}_{\text{int}}}\cos(\theta_{0})}{4\pi\epsilon_{0}mb^{4}}\frac{\cos(\alpha)+(vt/b)\cos(\beta)}{(1+(vt/b)^{2})^{5/2}}$

as presented in Eq. (39). Using the Fourier transform in Eq. (52) and the
properties given in Eq. (53) gives:

$\displaystyle a_{x}(\omega)=$
$\displaystyle\frac{2\absolutevalue{\boldsymbol{d}_{\text{ext}}}\absolutevalue{\boldsymbol{d}_{\text{int}}}\cos(\theta_{0})}{4\pi\epsilon_{0}m}\frac{\omega^{2}}{b^{2}v^{2}}$
$\displaystyle\Bigg{[}\cos(\alpha)K_{2}\left(\frac{b\omega}{v}\right)-i\cos(\beta)K_{1}\left(\frac{b\omega}{v}\right)\Bigg{]}$
(75)

which gives the PSD of the acceleration noise:

$\displaystyle S_{aa}(\omega)=$
$\displaystyle\frac{\absolutevalue{\boldsymbol{d}_{\text{ext}}}^{2}\absolutevalue{\boldsymbol{d}_{\text{int}}}^{2}\cos^{2}(\theta_{0})}{4\pi^{2}\epsilon_{0}^{2}m^{2}T}\frac{\omega^{4}}{b^{4}v^{4}}$
$\displaystyle\Bigg{[}\cos^{2}(\alpha)K_{2}^{2}\left(\frac{b\omega}{v}\right)+\cos^{2}(\beta)K_{1}^{2}\left(\frac{b\omega}{v}\right)\Bigg{]}$
(76)

Which, together with the transfer function, gives the dephasing due to the
dipole-dipole interaction, as given in Eq. (40).

##  Appendix B Projection Angles analysis

To maximize the dephasing, we take the initial conditions as given by the
projections angels $\alpha$ and $\beta$ (respectively, b and $v$ in the
$\hat{x}$-direction) and in the case of an interferometer dipole $\theta_{0}$
and $\gamma$ (respectively, b and $v$ in the $\hat{z}$-direction), such that
the absolute value of the acceleration in the $\hat{x}$-direction is maximal.

By maximising the acceleration, the noise is also maximised since the transfer
function $S_{aa}(t)\propto a(t)^{2}$ and $\Gamma_{\text{n}}\propto
S_{aa}(\omega)F(\omega)$, where the transfer function $F(\omega)$ is
independent of the projection angles. Physically it is clear that a larger
acceleration causes a larger dephasing.

###  B.1 Charged interferometer

Figure 7: For $vt/b=0.1$ the absolute value of the acceleration (Eq. (77) with
$a=3$) as a function of the projection angles $\alpha,\beta\in[0,\pi]$.

Figure 8: For $vt/b=1$ the absolute value of the acceleration (Eq. (77) with
$a=3$) as a function of the projection angles $\alpha,\beta\in[0,\pi]$.

Figure 9: For $vt/b=10$ the absolute value of the acceleration (Eq. (77) with
$a=3$) as a function of the projection angles $\alpha,\beta\in[0,\pi]$.

For a charged interferometer, the relevant angles that determine the movement
of the external particle are $\alpha$ and $\beta$, see Eq. (13). To find an
upper bound on the dephasing, these angles are maximised. The figs. 9-9 show
the acceleration

$a(t)=\absolutevalue{C\frac{\cos(\alpha)+(vt/b)\cos(\beta)}{(1+(vt/b)^{2})^{a}}}$
(77)

as a function of the angles $\alpha$ and $\beta$. The magnitude of the
acceleration is not given since it depends on the constant $C$, which is
determined by the type of interaction (charged-charged, charged-permanent
dipole or charged-induced dipole). The value $a$ is also specified by the type
of interaction, $a=3/2,2,3$, see Eqs. (24), (26), (29).

The figures show that for $vt>b$ by at least one order of magnitude the
acceleration is maximised for $\beta=0,\pi$ and $\alpha$ arbitrary. If $vt<b$
by at least one order of magnitude then the plot rotates ninety degrees and
the maximum value of the acceleration is given by $\alpha=0,\pi$ and $\beta$
arbitrary. For $vt\sim b$ the maximisation is given by $\alpha=\beta=0$ or
$\alpha=\beta=\pi$. The Figs. 9-9 are approximately constant for $a=3/2,2,3$.

Therefore a simple way to choose the angles such that the absolute
acceleration from the interaction is maximised is to choose $\alpha=\beta=0$
(or $\pi$) 444Note that for the external dipole in section IV, we have assumed
that the particle has low velocity such that
$\absolutevalue{\boldsymbol{r}\cdot\boldsymbol{d}}(t)\approx\absolutevalue{\boldsymbol{r}\cdot\boldsymbol{d}}$.
So we have already made the assumptions on the magnitude of the velocity..

###  B.2 Neutral interferometer

Figure 10: For $vt/b=0.1$ the absolute value of the acceleration (Eq. (78)) as
a function of the projection angles $\alpha,\beta\in[0,\pi]$.

Figure 11: For $vt/b=1$ the absolute value of the acceleration (Eq. (78)) as a
function of the projection angles $\alpha,\beta\in[0,\pi]$.

Figure 12: For $vt/b=10$ the absolute value of the acceleration (Eq. (78)) as
a function of the projection angles $\alpha,\beta\in[0,\pi]$.

Figure 13: For $vt/b=0.1$ the absolute value of the acceleration for the
dipole case (Eq. (78) without the $\cos(\gamma)$-term) as a function of the
projection angles $\alpha,\beta\in[0,\pi]$.

For a neutral interferometer there are the additional angles $\gamma$ and
$\theta_{0}$ that give the projection in the $\hat{z}$-axis, see Eq. (67) 555
Note that for the induced dipole - charge interaction, these angles are not
relevant and the result for the projection angles for this interaction are the
same as discussed in the previous section for the charged interferometer case.
. They are relevant because the interferometer dipole has been chosen to align
with the $\hat{z}$-axis.

Generically, we write the acceleration in the $\hat{x}$-direction from the
dipole-charge and dipole-dipole interactions as:

$a(t)=\absolutevalue{C\frac{\cos(\alpha)+(vt/b)\cos(\beta)}{(1+(vt/b)^{2})^{5/2}}\left(\cos(\theta_{0})+(vt/b)\cos(\gamma)\right)}$
(78)

with $C$ a constant defined by the interaction. The angles $\alpha$ and
$\theta_{0}$ are related in the sense that if $\alpha=0$, it means that
$b=x_{0}$, and therefore $\theta_{0}=\pi/2$ such that $b_{z}=0$. Similarly the
angles $\beta$ and $\gamma$ are related such that if $\beta=0$, $v=v_{x}$ and
thus $\gamma=\pi/2$, such that $v_{z}=0$. Therefore, we define the dependency
as

$\cos(\theta_{0})=\sqrt{1-\cos^{2}(\alpha)},\,\,\,\cos(\gamma)=\sqrt{1-\cos^{2}(\beta)}.$
(79)

This follows from Pythagoras theorem $v^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}$,
where for maximisation of the acceleration, we have set $v_{y}=0$.

The resulting plots that show the acceleration of Eq. (78) as a function of
$\alpha$, $\beta$ for arbitrary $C$ are shown in Figs. 13-13. The figures are
somewhat similar to Figs. 9-9, but one can see the extra dependence on the
projection angles. From the figures, we can conclude that in order to maximise
the acceleration in a generic way one can take the projection angles
$\alpha=\beta=\pi/4,3\pi/4$, which means that
$\theta_{0}=\gamma=\pi/4,3\pi/4$.

In the case of a dipole-dipole interaction, the assumption that the particle
has low velocity such that $\boldsymbol{r}(t)\cdot\boldsymbol{d}$ is
approximately constant in time removes the dependence on $\gamma$. The
resulting plot looks very similar to Fig. 13 and shows that the acceleration
is maximal for $\alpha=\pi/4$, $\beta=0$ and for $\alpha=3\pi/4$ and
$\beta=\pi$, see Fig. 13.