# Spatial Qubit Entanglement Witness for Quantum Natured Gravity

Bin Yi  Department of Physics and Astronomy, University College London, Gower
Street, WC1E 6BT London, United Kingdom.  Urbasi Sinha  Raman Research
Institute, C. V. Raman Avenue, Sadashivanagar, Bengaluru, Karnataka 560080,
India  Dipankar Home  Center for Astroparticle Physics and Space Science
(CAPSS), Bose Institute, Kolkata 700 091, India  Anupam Mazumdar  University
of Groningen, PO Box 72, 9700 Groningen, Netherlands  Van Swinderen Institute,
University of Groningen, 9747 AG Groningen, Netherlands  Sougato Bose
Department of Physics and Astronomy, University College London, Gower Street,
WC1E 6BT London, United Kingdom.

###### Abstract

Evidencing the quantum nature of gravity through the entanglement of two
masses has recently been proposed. Proposals using qubits to witness this
entanglement can afford to bring two masses close enough so that the complete
$1/r$ interaction is at play (as opposed to its second order Taylor
expansion), and micron sized masses separated by 10-100 microns (with/without
electromagnetic screening) suffice to provide a $\sim 0.01-1$ Hz rate of
growth of entanglement. Yet the only viable method proposed for obtaining
qubit witnesses so far has been to employ spins embedded in the masses, whose
correlations are used to witness the entanglement developed between masses
during interferometry. This comes with the dual challenge of incorporating
spin coherence preserving methodologies into the protocol, as well as a
demanding precision of control fields for the accurate completion of spin
aided (Stern-Gerlach) interferometry. Here we show that if superpositions of
distinct spatially localized states of each mass can be created, whatever the
means, simple position correlation measurements alone can yield a spatial
qubit witness of entanglement between the masses. We find that a significant
squeezing at a specific stage of the protocol is the principal new requirement
(in addition to the need to maintain spatial quantum coherence) for its
viability.

##  I Introduction

The quantumness of gravity is an open question due to a lack of empirical
evidence. A lot of research is performed in the setting of semi-classical
gravity, in which matter and non-gravitational fields are treated quantum
mechanically, while gravity is treated classically. A substantial community
argues that gravity can be classical as quantum physics break down at
macroscopic level, where gravitational effects become evident[1, 2, 3, 4].
Various proposals have been made to avoid quantizing gravity, at the cost of
introducing extra stochastic terms[5, 6] – but these are not ruled out by any
current experiment such as measuring forces precisely. Therefore, testing the
quantum nature of gravity experimentally is an open problem. Even if its
quantumness is accepted from the point of view several existing successful
theories [7, 8], its verification is still open.

In 2017, Bose and collaborators proposed a protocol to test whether the nature
of gravity is quantum[9] (see also recorded online talk of 2016, where the
same protocol is presented [10]). Two spatial superposition state of masses
can not entangle via classical channel [11] (see also [12]). Locality in
quantum field theory circumvents the non-local interaction between the two
superposed test masses[13]. Entanglement between the masses can only be
generated through local operations and quantum communication (in fact, quantum
communication is necessary for operator valued interactions, [14] which in
turn is necessary for the coherent interaction that generates entanglement).
On the other hand, once quantum communication is proven through a witnessing
of gravitationally generated entanglement, this unambiguously certifies the
presence of off-shell/virtual graviton as this is the only way to get a
continuous deterministic operator valued interaction[13, 14]. Alternatively,
the witnessed entanglement can also be regarded as evidencing the quantum
superposition of geometries inherent in superposition states of each mass [15,
16].Moreover, from logical arguments, the quantum nature of the Newtonian
component of the gravitational field automatically has a bearing on the
quantum nature of other components [17, 18, 19, 20].

The original proposal exploits spin-embedded masses[9]. Stern-Gerlach
interferometers (SGIs) are used to prepare the spatial superpositions depend
on embedded spins so that the motional and spin degree of freedom is
entangled. The test masses then freely evolve subject to gravitational
interaction. At the end of the proposed protocol, the Stern-Gerlach apparatus
is exploited again to bring the superposition components back to the center.
Spin correlations between the test masses then evidence the entanglement
generated through the gravitational interaction during propagation.
Alternatively, it has also been proposed to witness the gravitational
entanglement growth between two initial delocalized gaussian states by
position and momentum correlations [21, 22, 23] (see also [24, 19, 20] for
similar entanglement via gravity between light and matter or between optical
fields). However, a two-qubit witness for entanglement is applicable to a
situation where masses are brought as close to each other as their
delocalization (spatial superposition scale), so that the entanglement itself
has a faster growth rate ($\sim 1$ Hz) even for smaller (micron sized) masses.

It is challenging to complete SGIs as an exact overlap has to be attained in
both position and momentum of the wavepackets in the two interferometric arms
[25, 26], although this has been achieved very recently for small atomic
interferometers [27]. Moreover, the spin can bring in an extra avenue of
decoherence requiring extra dynamical decoupling procedures which could
potentially complicate the interferometry even more requiring further
dynamical decouplings [28, 29]. Although exceptionally long spin coherence
have been shown, and these are the subject of qubits in quantum computers, it
may be worthwhile to look for witnessing of the entanglement without spins.
This may even be the case where we actually use spin dependent forces (Stern-
Gerlach) to create the initial superposition. However, we do not worry about
also completing the interferometry by appropriate matching of forces in the
two arms. On the other hand, without spins, the advantage of two-qubit
witnesses for quantum entanglement seems to disappear. We thus here look at
the potential of using position measurements themselves to infer states of
“spatial qubits"(which have been called Young qubits in photonic systems [30,
31, 32, 33]) and use that as witness for evidencing gravitational
entanglement. We find that aside the usual coherence requirements to be
satisfied for maintaining the quantum superpositions [9, 34], which is
unavoidable in any scheme, a spatial qubit witnessing of gravitational
entanglement is possible if a challenging squeezing requirement can be met.

The spatial qubit methodology [35] encodes qubit in the spatial degree of
freedom of a freely propagating test mass. The readout of the information
stored in the qubit can be implemented by placing sets of spatial detectors at
appropriate positions. Correlations generated between two such spatial qubits
during evolution can then be tested by spatial detection. One advantage of
this approach is its simplicity: free propagation followed by spatial
detection. An interferometer with beam splitting elements such as Mach-
Zhender, looks highly unfeasible because of the large mass that has to tunnel
through such a system. Of course, a Stern-Gerlach interfrometer is possible,
but this requires a spin., as well as exquisite control in completing the
interferometry. Both these requirements are completely avoided if spatial
qubits are used. However, it requires the application of an additional
squeezing operator in general. In this paper, we apply this methodology to
witness quantum natured gravity.

##  II Spin entanglement witness setup

Ref.[9] presents an unambiguous witness of quantum natured gravity. The scheme
consists of two spin embedded test masses initially in spatially localized
states, say, in respective traps. The test masses pass through a set of Stern-
Gerlach (SG) interferometers so that the spatial degree of freedom entangles
with the spin degree of freedom to prepare

$|\psi_{0}\rangle=|L,\uparrow\rangle_{j}+|R,\downarrow\rangle_{j}(j=1,2)$ (1)

with states $|L\rangle$ and $|R\rangle$ being separated by $d$, and the
distance between the midpoint of each superposition ($\sim$ initial separation
between the masses) being $D$. Next, let the coherent superposition state
propagate for a time $\tau$ by switching off the magnetic field of the SG
apparatus. If gravity were quantum, gravitational interaction can induce,
through relative phases among the superposition components, an entanglement
between the masses (classical gravity as mediator would not give an operator
valued interaction [14], and hence will be unable to entangle the two masses).
The final step refocuses the SG apparatus to bring the spatial superposition
back to the center so that the final spin state reads

$\displaystyle\frac{e^{i\phi}}{\sqrt{2}}$
$\displaystyle[|\uparrow\rangle_{1}\frac{1}{\sqrt{2}}(|\uparrow\rangle_{2}+e^{i\Delta\phi_{\uparrow\downarrow}}|\downarrow\rangle_{2})$
(2)
$\displaystyle+|\downarrow\rangle_{1}\frac{1}{\sqrt{2}}(e^{i\Delta\phi_{\downarrow\uparrow}}|\uparrow\rangle_{2}+|\downarrow\rangle_{2})]$

where $\phi=\frac{Gm_{1}m_{2}}{\hbar
D}\tau,\Delta\phi_{\uparrow\downarrow}=\frac{Gm_{1}m_{2}}{\hbar(D+d)}\tau-\phi,\Delta\phi_{\downarrow\uparrow}=\frac{Gm_{1}m_{2}}{\hbar(D-d)}\tau-\phi$.
This is generically an entangled state as soon as
$\frac{1}{\sqrt{2}}(|\uparrow\rangle_{2}+e^{i\Delta\phi_{\uparrow\downarrow}}|\downarrow\rangle_{2})\neq\frac{1}{\sqrt{2}}(e^{i\Delta\phi_{\downarrow\uparrow}}|\uparrow\rangle_{2}+|\downarrow\rangle_{2})$.
By measuring spin correlations, one can then verify the entanglement induced
during the propagation time $\tau$. That can only arise from the exchange of
quantum coherent mediators. If gravity is the only interaction present, one
can then conclude that gravity is quantum.

To produce observable relative phase, the original proposal considers massive
objects with $m\sim 10^{-14}kg$. The required mass is restricted by the
minimum distance between the two masses $D-d$. At micro-meter scale, Casimir-
Polder force, which is another source of interaction between neutral objects,
becomes dominant over gravitational interaction. A minimum distance $D-d\sim
100\mu m$ is necessary to ignore influence from Casimir-Polder interaction.
Later, a revised scheme based on Casimir screening [34] relaxes the parameters
to $D\sim 47\mu m$, $d\sim 23\mu m$ by placing a conducting plate, which acts
as a Faraday cage, between the test masses to screen the mutual
electromagnetic interaction. On the other hand, to test the spin entanglement
witness also requires a delicate balancing of magnetic field gradient to bring
the superposition components back to the center. In the revised scheme, the
test mass $m\sim 10^{-15}$ kg and the total accumulated phase during
interaction time $\tau\sim 1s$ is of order of $0.01$rad.

##  III  Massive spatial qubit methodology for Witnesssing Gravitational
Entanglement

In this paper, we investigate the viability of testing the quantum nature of
gravity with recently developed methodology: massive spatial qubit. This
approach treats freely evolving spatially superposed masses as qubits, which
does not call for spins. Therefore, it may relax the criteria for witnessing
the quantum aspect of gravity. The schematic diagram is shown in Fig.1.

Figure 1: Testing quantumness of gravity with massive spatial qubits This
interaction induces entanglement which can be deduced from spatial qubit Pauli
measurements. To evaluate the entanglement witness $W$, we conduct spatial
qubit measurements on both masses and evaluate the correlations. For example,
in the figure, Pauli-X, Pauli-Y measurements on test mass $1$ and Pauli-Z
measurements on test mass $2$ are depicted. An additional conducting plate is
inserted between the masses for electromagnetic shielding

Two test masses $m_{1},m_{2}$, each prepared in spatial superposition of two
well separated Gaussian states $|L\rangle$ and $|R\rangle$, are placed
adjacent to each other then freely evolve, under their own Hamiltonian, as
well as undergo a joint phase evolution under their mutual gravitational
interaction as shown in Fig1. The propagation of each individual wavepacket is
modeled as the spreading of a Gaussian wavepacket due to free evolution. Each
spatial superposition $\frac{1}{\sqrt{2}}(|L\rangle+|R\rangle)$ of test mass
can be treated as a state of a qubit with the two qubit states identified with
$|L\rangle$ and $|R\rangle$.

###  Witnessing the quantum nature of gravity:

If quantum gravitational interaction is the only source of interaction,
relative phase induced among the superposition components then reads:

$\displaystyle|\psi(t=0)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|L\rangle_{1}+|R\rangle_{1})\frac{1}{\sqrt{2}}(|L\rangle_{2}+|R\rangle_{2})$
(3) $\displaystyle\rightarrow|\psi(t=\tau)\rangle$ $\displaystyle=$
$\displaystyle\frac{e^{i\phi}}{\sqrt{2}}[|L\rangle_{1}\frac{1}{\sqrt{2}}(|L\rangle_{2}+e^{i\Delta\phi_{LR}}|R\rangle_{2})$
(4) $\displaystyle+$
$\displaystyle|R\rangle_{1}\frac{1}{\sqrt{2}}(e^{i\Delta\phi_{RL}}|L\rangle_{2}+|R\rangle_{2})]$

It is true that the creation of a superposition of two localized states (a
highly non-Gaussian state) is a pre-requisite for being able to use spatial
qubits (we will not delve into that issue here in detail, but mention some
possibilities in the next section). Here we concentrate on how to read-out
spatial qubits in absence of (in this context impractical) beam-splitter
elements, and thereby witness the entanglement of two masses without resorting
to spins. Pauli measurements, which readout the encoded information, can be
performed by placing spatial detectors at particular locations. Any
entanglement induced during the propagation process can be witnessed by the
correlations between two spatial qubits. All of Pauli-x,y and z measurements
are involved in obtaining these correlations. Pauli-z measurement
$\sigma_{z}=|L\rangle\langle L|-|R\rangle\langle R|$ reads the probability
amplitude of the encoded spatial qubit state, which has to be performed before
$|L\rangle$ and $|R\rangle$ have spread so much as to be confused with each
other (i.e., spread to about $\sim d$). On the other hand, Pauli-x
measurements project on to states $\frac{1}{\sqrt{2}}(|L\rangle\pm|R\rangle)$
and Pauli-y measurements project on to states$\frac{1}{\sqrt{2}}(|L\rangle\pm
i|R\rangle)$, which are only discernible in the interference plane as shown in
Fig.1, which implies that they are performed after each of $|L\rangle$ and
$|R\rangle$ have spread to a length $d$ so that they can overlap. At the
interference plane, the probability of detecting the test mass at position $x$
is given by:

$P(x)=|\langle\psi|k_{x}\rangle|^{2}\propto\left|\exp[\frac{ik_{x}d}{2}-k_{x}^{2}\sigma_{d}^{2}]\langle\theta|\psi\rangle\right|^{2}$
(5)

where $|\theta\rangle=|0\rangle+|1\rangle e^{i\theta}$, $\theta=k_{x}d$ and
$k_{x}=\frac{xm}{\hbar t^{x,y}_{\text{meas}}}$ is the transverse wavevector.
Projection of the spatial qubit state along $\sigma_{x+}$, $\sigma_{x-}$
($\sigma_{y+}$, $\sigma_{y-}$) can then be implemented by placing spatial
detectors corresponds to $\theta=0,\pi(-\pi/2,\pi/2)$ respectively. Thus the
Pauli-x,y and Pauli-z measurements would normally have to be performed at
different times $t^{z}_{meas}$ (before spreading) and $t^{x/y}_{meas}$ (after
spreading).

### Squeezing requirement

The masses have to entangle sufficiently first. This inevitably requires a
time of $\tau$ over which the gravitational interaction should act between
them. This is the earliest time any measurement can occur as we want to
measure the two masses after they are entangled. Thus we set
$t^{z}_{meas}=\tau$, being careful that hardly any spreading of the
wavepackets $|L\rangle$ and $|R\rangle$ happens in time $\tau$. However, we
want to measure Pauli-x,y on the same state, i.e., the state evolved up to
time $\tau$ as the gravitational interaction cannot be switched off. This
demands that we must also measure Pauli-x,y at (or very nearly) the same time.
So we need to induce some process at $\tau$ so that the wavepackets
immediately spread to $\sim d$ and overlap to interfere. This requires the
application of an additional squeezing operator to the masses immediately
after time $\tau$. This localizes the wavefunctions so much (each of
$|L\rangle$ and $|R\rangle$ are now highly squeezed) that in subsequent free
evolution for $t^{x,y}_{meas}-t^{z}_{meas}$ they spread rapidly to overlap and
interfere. Thus the extra squeezing at $\tau$ enables us to meet the
criterion: $t^{x,y}_{meas}\sim t^{z}_{meas}=\tau$, so that both the Pauli-z
and the Pauli-x,y measurements are essentially performed on the same entangled
state of the two masses.

The spread of wavepackets scales approximately inversely proportional to its
initial width $\sigma_{d}(0)$: $\sigma(t)\sim\frac{\hbar t}{m\sigma_{d}(0)}$.
Therefore, application of squeezing operator at $\tau$, squeezes width of the
wavepacket and speeds up subsequent spreading. This technique can be
accomplished by passing the test masses through local Harmonic potentials with
jumps between frequencies. More precisely, $n$ sudden switches between
frequencies $\omega_{1}$ and $\omega_{2}$, with a quarter period of harmonic
evolution in each frequency, squeezes the wavepackets spatially by a factor
$(\omega_{1}/\omega_{2})^{n}$ [36, 37]. As, for the squeezing, we have to
expend a quarter period of time $\pi/2\omega_{j}$ in each frequency
$\omega_{j}$, this inevitably requires a time

$t_{squeeze}=n(\pi/2\omega_{1}+\pi/2\omega_{2}).$ (6)

On the other hand, the amount of squeezing depends on satisfying
$t^{x,y}_{meas}\approx t^{z}_{meas}=\tau$. We want the $t_{squeeze}$ to be a
negligible time-scale in comparison to the other times of the problem as the
gravitational entanglement is always happening (we cannot stop it). Thus
$n,\omega_{1},\omega_{2}$ must be so chosen that

$t_{squeeze}<<t^{x,y}_{meas}\approx t^{z}_{meas}=\tau$ (7)

Squeezing thus places significant demands on this protocol.

### Casimir screening imposed constraints

To shield the system from unwanted electromagnetic interaction, one places a
conducting plate between the test masses to act as a Faraday cage. For
simpicity we assume that the placement of a perfect conductor completely
blocks the Casimir interaction between test masses. The Casimir screening
scheme introduces additional Casimir force between the plate and test masses.
The plate-mass Casimir interaction is now the dominant force at very small
scale and places constraints on the system. The screen is closest to one of
the components of the superposition ($|R\rangle$ for the mass to the left of
the screen and $|L\rangle$ for the mass to the right of the screen). We do not
want this component to be pulled so close to the screen that it prevents the
overlap between the two components of each mass in order to enable the
Pauli-x,y measurements. Thus we require that the spread of wavepackets (which
we exploit in our scheme) dominates over the displacement due to Casimir
force. This places a new constraint on minimum separation of a mass and the
conducting plate. The Casimir force between a plate and sphere is given by[38]

$F_{ca}=-\frac{3\hbar
c}{2\pi}(\frac{\epsilon-1}{\epsilon+2})\frac{R^{3}}{s^{5}}$ (8)

where $\epsilon$ is the dielectric constant of test masses, $R$ is the radius
of test masses and $s=\frac{D-d}{2}$ is the mass-plate separation.

We require that the spread of wavepacket dominates over the displacement due
to Casimir force by at least one order of magnitude:

$0.1\sigma_{d}(t_{1})\geq D_{ca}$ (9)

where $\sigma_{d}(t)$ is the width of the propagating wave packets, $t_{1}$ is
the total time before the wavepackets overlap. Thus, in terms of the notation
of the previous part of the paper, $t_{1}\sim
t^{x,y}_{meas}-t^{z}_{meas}-t_{squeeze}$, as that is the time over which the
wavepacket expands after squeezing (We know that this spread is required to be
of the order of the slit separation $d$ for the overlap to happen; moreover,
actually $t_{1}<<t^{x,y}_{meas},t^{z}_{meas},t_{squeeze}$).

Classical treatment of Casimir force gives an estimate of its resulting
displacement on test masses given by:

$D_{ca}=-\frac{9\hbar c}{16\pi^{2}}(\frac{\epsilon-1}{\epsilon+2})\frac{1}{\mu
s^{5}}$ (10)

where $\mu$ is the density of test mass.

On the other hand, the spread of Gaussian wave packet $\sigma_{d}(t)$ scales
linearly with time by a factor of $\frac{\hbar}{\sigma_{d}m}$. Since wave
overlaps at a time $t\sim 2m\sigma_{d}d/\hbar$, the width of wavepackets
spread is of same order as the slit separation $\sigma_{d}(t_{1})\sim d$. In
the original setup, $d$ and $\frac{D-d}{2}$ are on the same scale. We may then
take $\sigma_{d}(t_{1})\sim d\sim s$. The above parameter domain is satisfied
for mass density $\mu\sim 3\times 10^{3}kg/m^{3}$, $\epsilon=5.7$, and the
minimum plate-mass separation $s\sim 12\mu m$.

### Induced phase

With the same mechanism as in [9], we can prepare initial state given by eq3.
Taking $D\sim 40\mu m,d\sim 10\mu m$, for mass $m\sim 10^{-15}kg$,
gravitational interaction, if quantum, would induce relative phase
$\Delta\phi\sim 1\times 10^{-2}rad$ after $\tau\sim 3s$ of entangling time. To
certify the induced relative phase, we estimate the entanglement witness [39]

$W=I\otimes
I-\sigma_{x}\otimes\sigma_{x}-\sigma_{z}\otimes\sigma_{y}-\sigma_{y}\otimes\sigma_{z}$
(11)

The expectation value $\langle W\rangle=\mathrm{Tr}(W\rho)$ would be negative,
if the two masses are entangled. Using Eq.(4) for the state, and phases
$\Delta\phi_{LR}=\Delta\phi_{\uparrow\downarrow},\Delta\phi_{RL}=\Delta\phi_{\downarrow\uparrow}$
from below Eq.(2), the expected witness measure at $\tau\sim 3s$ is
$\sim-0.0065$.

### Initial state preparation

Witnessing the quantum nature of gravity through qubit correlations requires
preparation of superposition state of large spatial splitting $d\sim 10-100\mu
m$ of a micron scale mass , $m\sim 10^{-15}-10^{-14}$kg[9, 34]. This is
considerably beyond what has already been realized (e.g., $10^{-25}$kg mass
over $0.5m$[40], or $10^{-22}$kg mass over $0.25\mu$m [41],[42]). However,
there are several proposed schemes to achieve the required superpositions,
with criteria in terms of temperature, pressure, acceleration/vibration noise
well identified.

The protocol starts from preparing a mass in a pure quantum state in a
harmonic trap, typically the ground state. This mass may or may not have a
spin embedded in it (the protocol of creating the initial state
$\frac{1}{\sqrt{2}}(|L\rangle+|R\rangle)$ will depend on that). We imagine
these objects to be levitated in low frequency static magnetic traps[43, 44,
45, 46]. Feedback cooling can be achieved by first shining light towards the
particle. Scattered photons, which carries information of the particle’s
position, are collected by photo-diode detector. For example, from detecting
$n$ scattered photons, the position of a particle gets determined to the
accuracy of $\lambda/\sqrt{n}$, where $\lambda$ is the wavelength of the
scattered light [47]. This information is then used as feedback to cool the
motion of the micro-sphere via an external damping force. Essentially, the
information gathering rate from detecting scattered photons must overtake the
entropy increase rate of the object from undetected scatterings (photons,
atoms) and noises from the environment. Using the feedback cooling principle,
a mechanical oscillator as massive as 10kg has been prepared close to its
ground state, the center-of-mass motion of which is cooled down to tens of
nano-Kelvin[48], which involved to measuring position to the uncertainty of
$\sim 10^{-20}\text{m}/\sqrt{\text{Hz}}$. Cryogenic diamagnetic levitated
micro-mechanical oscillator has also been realized with very low ($\mu$ Hz)
dissipation rate[43, 44]. For test mass $m\sim 10^{-15}$kg in a trap with
frequency $\sim\mathcal{O}(10-100)$Hz, its ground state spread is
$\sim\mathcal{O}(0.1-1)$nm. As this uncertainty being much larger than the
precision to which position has been localized recently in feedback cooling
[48], it is reasonable to suppose that feedback cooling to nearly the ground
state is also imminent for the above systems (for example, will require $n\sim
10^{8}$ scattered photons to be detected, with undetected photons being much
lower, in a time scale over which no collisions with air molecules take
place). Following the above stage either spinless or spinful methods can be
used to create the superposition.

If we have a spin embedded in the mass, a popular avenue to create a small
superposition ($\leq 100$nm) is to use the Stern-Gerlach effect [49, 50, 51,
9, 29]. The test mass in an initially pure localized state $|C\rangle$ (say,
the ground state) is prepared with its spin in a state
$\frac{1}{\sqrt{2}}(|\uparrow\rangle+|\downarrow\rangle)$ (by microwave
pulses), and the trap is suddenly switched off. The spin state undergoes
spatial splitting due to an inhomogeneous magnetic field gradient so that the
system evolves as

$\frac{1}{\sqrt{2}}(|\uparrow\rangle+|\downarrow\rangle)|C\rangle\rightarrow\frac{1}{\sqrt{2}}(|L\uparrow\rangle+|R\downarrow\rangle).$
(12)

A subsequent measurement of the spin in a different basis, say, the
$|\pm\rangle$ basis, and, for example, getting the $|+\rangle$ outcome, will
prepare the mass in the spatial superposition

$\frac{1}{\sqrt{2}}(|L\rangle+|R\rangle).$ (13)

Care must be taken so that the spin measurement does not reveal the position
of the mass itself to a better precision than the $|L\rangle$ and $|R\rangle$
difference. However, we now require to amplify this superposition as we
require (in the case where there is a screening of electromagnetic
interactions) a spatial splitting of $\sim 10\mu$m, while the diamagnetism
induced by the magnetic field gradient used for the splitting, restricts the
splitting [52, 53]. Interestingly, the superposition could be amplified using
a current carrying wire providing a diamagnetism induced repulsion between
wire and each split component [54]. Alternatively, spins may be subject to
nonlinear gradients to accumulate a velocity difference before catapulting to
a large size [55]. With magnetic field gradient $\sim 10^{3}Tm^{-1}$, the
desired separation of $10\mu m$ can be obtained after $\sim 1$s of flight
time[55]. It is easy to verify that during such intervals of time, the
wavepacket spread remains in the $\sim\mathcal{O}(0.1-1)$nm regime. During
these protocols, spin coherence does not need to be retained during the
amplifications (diamagnetic repulsion/catapulting) the electronic spin does
not play an active role. So, just before the amplification stage, one could
map it to much more coherent nuclear spins, or alternately, simply measured in
a different basis as noted above, to obtain directly the
$\frac{1}{\sqrt{2}}(|L\rangle+|R\rangle)$ state. noted above so as to have
only a spatial superposition. Moreover, for the application herewith (using
spatial qubits) we do not need to complete an interferometer – only create the
large splitting superposition, which removes a significant challenge.

It is possible that even without spins the state
$\frac{1}{\sqrt{2}}(|L\rangle+|R\rangle)$ can be produced as far as $\sim
1\mu$m sized distances between $|L\rangle$ and $|R\rangle$ are concerned. For
example, the first few stages of the on-chip interferometer of Ref.[56] based
on coherent inflation and a $\hat{x}^{2}$ (position2) measurement can be used
(till the superposition is generated). One can then, in principle, combine
with an diamagnetic repulsion aided further spatial splitting of the
$|L\rangle$ and $|R\rangle$ terms [54] so as to reach $\sim 10\mu$m size.

### Decoherence

During propagation, background gas collision and black body radiation
unavoidably decoheres quantum coherence of the superposition state. Adopting
the model in [57], air molecule collision decoheres the $10^{-15}$kg test mass
at a rate of $\sim 0.776s^{-1}$ under pressure $\sim 10^{-15}$Torr. Blackbody
radiation arises from emission, absorption and scattering of thermal photons.
The emission localization parameter typically dominates because the internal
temperature is usually larger than the external temperature. If internal
temperature can be cooled down to $T_{i}=4$K, black body radiation decoheres
the test mass at a rate of $\sim 0.72s^{-1}$, which corresponds to $\sim 10$s
of coherence time. The required cooling to keep coherence of the superposition
state is challenging with state of art technology.

### Squeezing challenges

Take initial width $\sim 0.4$nm, the wavepacket expands to $\sim 0.85$nm after
3 seconds of flight. We will then have to squeeze the state by about 7 orders
of magnitude to $1.5\times 10^{-16}$m, so that it expands to $\sim 20\mu m$ in
the next $0.03s$, where interference occurs.

Optical squeezing is the most common technique for microscopic objects.
However, momentum recoil due to photon scattering induce decoherence and
heating on the levitated objects, the rate of which scales with the object
size[58, 59, 60]. For large mass $\sim 10^{-15}$kg, this approach becomes
infeasible, the coherence time is only $\sim 10^{-6}$s. Alternatively, one may
adopt diamagnetic trapping for squeezing[43, 44, 45, 46]. Diamagnetic trapping
typically operates at much lower frequencies compare to its optical
counterpart, therefore, making it advantageous to hold large masses. Potential
energy per unit volume of a diamagnetic mass in trapped in magnetic field
reads:

$U\approx-\frac{\chi_{m}}{2\mu_{0}}B^{2}+\rho gr$ (14)

where $\chi_{m}$ is the mass magnetic susceptibility, $\mu_{0}$ is the
magnetic permeability in vacuum, $g$ is the gravitational acceleration, $B$ is
induction of the magnetic field when the particle is absent and $r$ is the
vertical displacement.

The mechanical frequency is therefore:

$\omega_{m}=\sqrt{\frac{\chi_{m}}{\mu_{0}}}\frac{\partial B}{\partial r}$ (15)

where $\frac{\partial B}{\partial r}$ is the field gradient.

To control the motional superposition state during the squeezing procedure,
the thermal decoherence rate $\gamma_{th}=\bar{n}\gamma$ needs to be
suppressed, where $\gamma$ is the mechanical dissipation rate and
$\bar{n}=\frac{k_{B}T}{\hbar\omega_{m}}$ is the average phonon number. To keep
coherence during $n$ periods of quarter oscillations, the dissipation rate
must, therefore, satisfy the following relation:

$\gamma<\frac{\hbar\omega_{m}^{2}}{nk_{B}T}$ (16)

In order to achieve the required squeezing, 7 times of successive changes
between two periods of harmonic potentials of 100Hz and 1000Hz would suffice.
The environmental temperature can be kept at $\sim 10mK$ with commercial
dilution refrigerator. The dissipation rate then must be kept below micro-
hertz $\gamma<1\mu$ Hz.

For solid state systems, the main contribution to dissipation is the direct
coupling between the system and the substrate. Ultra-low dissipation of micro
hertz has been reported with diamagnetic levitated objects, where permanent
magnets are used for trapping. In [44], where the levitated mass is similar to
our scheme, damping rate of $\mu$Hz is achieved at pressure $\sim 3\times
10^{-7}Torr$ and room temperature. The major contribution of damping came from
background gas collision, which scales linearly with pressure. Dissipation due
to gas collision could be significantly reduced by lowering temperature and
pressure. Comparing with the condition to keep coherence during the
propagation stage, one finds that the required level of vacuum is much less
demanding during the squeezing stage.

However, permanent magnet based scheme maybe unfeasible for squeezing since
successive changes of trapping potential are required. Alternatively, one may
use magnetic field generated by current-carrying coil, which introduces an
additional type of dissipation due to field fluctuation. One has to consider
specific current sources for that and we do not go to that technicality
herewith. Moreover, there could be other sources of random forces at given
times $\delta F(t)$. Thus we herewith estimate the constraint those forces
have to satisfy. Although the magnetic field noise will be switched off after
attaining the required squeezing, and free spreading of wavepacket under
propagation will ensue, we assume some random force noise being always present
and constrain it (this will surely be an overestimate as far as forces from
magnetic field noise are concerned). Random forces will give a decoherence
rate of

$\Gamma\sim\frac{S_{FF}(\Omega)d^{2}}{\hbar^{2}},$ (17)

where $S_{FF}(\Omega)=\int\delta F(0)\delta F(t)e^{i\Omega t}dt$ is the force
noise spectrum at the frequency $\Omega\sim 1/\tau$ of our experiment., and
$d$ is the spatial splitting of each superposition. Keeping $\Gamma<1$Hz gives
us the constraint that random force noise should be kept below
$\sqrt{S_{FF}}\sim 10^{-29}\text{N}/\sqrt{\text{Hz}}$. Note that although this
may sound challenging, faster frequency noise does not affect the experiment
particularly, while noise at this Hz frequency should be determinable by
precision measurements over a long duration of 1 s to be taken into account in
the experiment. Note that this requirement is not unique to the spatial qubit
method. Rather in the spatial qubit method we are using free propagation for a
large fraction of the time when, at least randomness in superposition
creating/delocalizing forces will be inactive.

##  IV  Conclusions:

We have analyzed the feasibility of applying the massive spatial qubit
methodology to witness the quantum nature of gravity. This will enable a
spinless witnessing of the gravitational entanglement growth between two
masses ($\sim 10^{-15}$ kg each), essentially through position measurements on
masses, but still using qubit-qubit correlations to measure the entanglement.
The core property used here is the spreading of a free quantum wavepacket
which brings two initially localized states $|L\rangle$ and $|R\rangle$ to
interfere so that observables such as $\sigma_{x}=|L\rangle\langle
R|+|R\rangle\langle L|$ can be measured from the interference pattern. It is
expedient work within the remit of a Faraday shielding scheme so as to block
unwanted electromagnetic interactions between two masses, so that the masses
can be brought closer and still interact only gravitationally. But here the
necessity for the wavepacket spreading to dominate over the Casimir force of
the Faraday screen to its nearest component arises. This, in turn, implies a
minimum distance to the screen, which, in turn, dictates a minimum separation
$d\sim 10\mu$m between $|L\rangle$ and $|R\rangle$ to have a significant
entanglement growth rate. This minimum $d$ then also necessitates a
wavefunction spreading from a very localized width $\sigma_{d}$ of each of
$|L\rangle$ and $|R\rangle$ rapidly to $\sim d$ in an extremely short time-
scale, so that the spatial qubit $\sigma_{z}$ and $\sigma_{x}$ measurements
are accomplished at nearly the same time (on the same entangled state of the
two masses). This is turn, necessitates squeezing the width immediately after
significant gravitationally generated entanglement is attained between the
masses (at a time $\tau$). The requirements and a method for achieving this
squeezing is described along with the challenges (it requires position
squeezing of the wavepacket by 7 orders of magnitude). As long as the above
can be met, and spatial superpositions can be generated (for which one may use
spins or other spinless methods), the evidencing through a qubit-qubit
entanglement witness can take place purely through position measurements.

###### Acknowledgements.

DH and US would like to acknowledge partial support from the DST-ITPAR grant
IMT/Italy/ITPAR-IV/QP/2018/G. DH also acknowledges support from the NSAI
Senior Scientist fellowship. AM’s research is funded by the Netherlands
Organization for Science and Research (NWO) grant number 680-91-119. SB would
like to acknowledge Grant Nos. ST/W006227/1.

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