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PHYSICAL REVIEW RESEARCH 5, 033202 (2023)
Massive spatial qubits: Testing macroscopic nonclassicality and Casimir entanglement
Bin Yi ,1Urbasi Sinha,2Dipankar Home,3Anupam Mazumdar,4and Sougato Bose 1
1Department of Physics and Astronomy, University College London, Gower Street, WC1E 6BT London, United Kingdom
2Raman Research Institute, C. V. Raman Avenue, Sadashivanagar, Bengaluru, Karnataka 560080, India
3Center for Astroparticle Physics and Space Science (CAPSS), Bose Institute, Kolkata 700 091, India
4Van Swinderen Institute, University of Groningen, 9747 AG Groningen, Netherlands
(Received 6 September 2021; revised 25 May 2022; accepted 23 May 2023; published 22 September 2023)
An open challenge in physics is to expand the frontiers of the validity of quantum mechanics by evidencing
nonclassicality of the center of mass state of a macroscopic object. Yet another equally important task is to
evidence the essential nonclassicality of the interactions which act between macroscopic objects. Here we
introduce a new tool to meet these challenges: massive spatial qubits. In particular, we show that if two distinct
localized states of a mass are used as the |0and |1states of a qubit, then we can measure this encoded spatial
qubit with a high fidelity in the σx,σ
y,andσzbases simply by measuring its position after different duration of
free evolution. This technique can be used reveal the irreducible nonclassicality of the spin and center of mass
entangled state of a nanocrystal implying macrocontextuality. Further, in the context of Casimir interaction, this
offers a powerful method to create and certify non-Gaussian entanglement between two neutral nano-objects.
The entanglement such produced provides an empirical demonstration of the Casimir interaction being inherently
quantum.
DOI: 10.1103/PhysRevResearch.5.033202
I. INTRODUCTION
It is an open challenge to witness a nonclassicality in the
behavior of the center of mass of a massive object [1,2].
While there are ideas to observe nonclassicalities of ever more
massive objects [3–13], the state-of-the-art demonstrations
have only reached up to macromolecules of 104amu mass
[14,15]. Such demonstrations would test the limits of quantum
mechanics [16–21], would be a stepping stone to witness
the quantum character of gravity [22–26], and would open
up unprecedented sensing opportunities [27]. Identifying new
tools to probe macroscopic nonclassicality (by which here we
mean in terms of large mass) is thus particularly important.
Here we propose and examine the efficacy of precisely such
a tool: a mechanism to read out a qubit encoded in the spatial
degree of freedom of a free (untrapped) mass (a purely spatial
qubit). A principal merit of this scheme is that measuring
the spatial qubit operators σx,σ
y, and σzexploits solely the
free time evolution of the mass (Hamiltonian H=ˆp2/2m),
followed by the detection of its position. As the mass is not
controlled/trapped by any fields during its free evolution,
decoherence is minimized.
As a first application, we show that our tool enables the
verification of an irreducible nonclassicality of a particular
joint state of a spin (a well established quantum degree spin)
and the center of mass of a macroscopic object, whose quan-
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tum nature is yet to be established. To this end, we use the
state produced in a Stern-Gerlach apparatus which is usually
written down as an entangled state of a spin and the position
[25,28–32]. Such Stern-Gerlach states have been created with
atoms with its spatial coherence verified after selecting a spe-
cific spin state [31,33]. However, there are, as yet, no protocols
to verify the entanglement between the spin of an object in
a Stern-Gerlach experiment and the motion of its center of
mass in a way which can be scaled to macroscopic objects.
We show that this can be accomplished via the violation of
a Bell’s inequality in which the spin and the positions of the
mass are measured. This violation will also prove the non-
classicality of a large mass in terms of quantum contextuality
[34,35].
Next, we propose a second application once the quantum
nature of the center of mass degree of freedom of macro-
scopic objects is assumed (or established in the above, or in
some other way). This application has import in establishing
the quantum nature of the interactions between macroscopic
objects. We show how our spatial qubit methodology can en-
able witnessing the entanglement created between two neutral
nanocrystals through their Casimir interaction. This has two
implications: (a) It will empirically show that the extensively
measured Casimir interaction [36–38] is indeed quantum
(e.g., is mediated by virtual photons similar to [39,40]—if
photons are replaced by classical entities they would not en-
tangle the masses [22–24,41,42]). (b) As the entangled state
is generated by starting from a superposition of localized
states, it is non-Gaussian. While there are ample methods for
generating [43–45] and testing [46] Gaussian entanglement of
nanocrystals, there is hardly any work on their non-Gaussian
counterparts.
2643-1564/2023/5(3)/033202(10) 033202-1 Published by the American Physical Society
YI, SINHA, HOME, MAZUMDAR, AND BOSE PHYSICAL REVIEW RESEARCH 5, 033202 (2023)
FIG. 1. Spatial detection for σx,σ
ymeasurements: a pair of de-
tectors (color: orange) located at phase angle θ=0,θ =πperform
σxmeasurement. The detectors (color: purple) at θ=π/2,θ =
−π/2 perform σymeasurement.
We are achieving our tool by combining ideas from two
different quantum technologies: photonic quantum informa-
tion processing and the trapping and cooling of nanocrystals.
In the former field a qubit can be encoded in the spatial mode
of a single photon by passing it through an effective Young’s
double slit [47]. These qubits, called Young qubits, and their
d-level counterparts [48,49], have been exploited in quantum
information [50,51]. On the other hand, we have had a rapid
development recently in the field of levitated quantum nano-
objects [7,8,52] culminating in their ground-state cooling and
the verification of energy quantization [53,54]. While sev-
eral schemes for verifying quantum superposition of distinct
states of such objects have been proposed to date, in these
schemes, either the x,y, and zmotions are measured as infinite
dimensional systems [10,55,56] rather than being discretized
as an effective qubit, or never measured at all (only ancillary
systems coupled to them are measured [11,12]). Here we
adapt the idea of Young qubits from photonic technologies
to massive systems. Note that a very different encoding of a
qubit in the continuous variables of a harmonic oscillator was
proposed long ago for quantum error correction [57], which is
not suited to an untrapped nanocrystal.
II. QUBIT ENCODING AND ITS MEASUREMENT
IN ALL BASES
Our encoding is intuitive: |0and |1states of a qubit are
represented by two spatially separated (say, in the xdirection)
nonoverlapping wavepackets whose position and momenta
are both centered around zero in the other two commuting (y
and z) directions. Explicitly, these states (writing only the x
part of their wavefunction) are
|0= 1
√σdπ1/4∞
−∞
exp −(x+d/2)2
4σ2
d|xdx,(1)
|1= 1
√σdπ1/4∞
−∞
exp −(x−d/2)2
4σ2
d|xdx,(2)
with dσd. These states are schematically depicted in Fig. 1
in which only the xdirection is depicted along with their
evolution in time. For simplicity, we will omit the acceleration
due to the Earth’s gravity (as if the experiment is taking place
in a freely falling frame), which can easily be incorporated
as its effect commutes with the rest. Thus we only consider
one-dimensional (1D) time evolution in the xdirection. In this
paper, we will only require two states: (a) a state in which a
spin embedded in a mass is entangled with the mass’s spatial
degree of freedom in the state |φ+= 1
√2(|↑,1+|↓,0)for
our first application, and (b) the spatial qubit state |+ =
1
√2(|0+|1) as a resource for our second application. Prepa-
ration of the above adapts previous proposals and will be
discussed with the respective applications.
We now outline our central tool: the method of measuring
the above encoded spatial qubit in various bases. The spatial
detection can be performed by shining laser light onto the test
masses [58,59]. The Rayleigh scattered light field acquires a
position-dependent phase shift. The scheme is limited only
by the standard quantum limit [58] (quantum back action)
of phase measurement when a large number of photons are
scattered from the mass. The resolution scales with the num-
ber nof scattered and detected photons as λ/√n, hence the
power collected at the detector (see Eq. (13) of Ref. [59]),
and the detection time (as long as this is lower than the
dynamical time scale, it is independent of whether the particle
is trapped/untrapped). Thus the detection time should be as
much as one needs for the required resolution, but much less
than the time span of the experiment. For the protocols we
will present in this paper, we will base our calculations on a
∼60 nm diameter diamond nanoparticle (about m∼10−19 kg
mass). By the above methodology, for a 60 nm diameter
particle, the detection resolution can reach 200 ±20 fm/√Hz
with laser power ∼385 µW at the detector, at environmental
pressure ∼0.01 mbar [59]. Thus for an integration time of
δtint ∼4µs, the resolution reaches ∼1 Å, which corresponds
to just ∼108photons. Moreover, we are measuring this mass
as a free particle, with an initial position spread. So the
standard quantum limit of position measurement here (for a
free particle) is [60]∼¯hδtint
m∼2/3 Å. Thus the measure-
ment precision required is of the same order as the standard
quantum limit (one does not need to go beyond it). As the
whole measurement is μs, any noise of frequency lower than
MHz will not affect it (simply remains constant during each
measurement run). Moreover, lower frequency noise causing
variation between, say, groups of runs, could be measured
efficiently by other proximal sensors and taken into account.
Also note that the spatial detection is performed at the very
end of the protocol, so the question of back action on further
position measurements does not arise.
Due to the spreading of the wavepackets along yand z
directions, when we determine whether the object is in a given
position x=x0at some measurement time t, we are essen-
tially integrating the probability of detecting it over a finite
region y(t) and z(t). The operator σz=|00|−|11|
is trivial to measure, as we simply shine a laser centered at
x=d/2 much before the wavepacket states |0and |1have
started to overlap [at a time tz
meas d(2σdm)/¯h; the error
in σzmeasurement as a function of tz
meas is described in the
Appendix A; timing errors δttz
meas have very little effect].
As described in the previous paragraph, if ∼108photons are
033202-2
MASSIVE SPATIAL QUBITS: TESTING MACROSCOPIC … PHYSICAL REVIEW RESEARCH 5, 033202 (2023)
FIG. 2. Detection scheme for the entanglement of spin and center of mass of a Stern-Gerlach state: A spin bearing nano-object is measured
to be in a set of zones of size δx, where the size is set by the strength and duration of lasers scattered from the object, which serves to measure
the spatial qubit. Within each spatial zone a suitable method is used to measure the spin in different bases, for example, by rotating the spin
states by microwave pulses followed by fluorescence of certain states under excitation by a laser of appropriate frequency.
scattered and collected, we can tell apart two states |0and |1
separated by ∼1Å.
To measure the spatial qubit σxand σyoperators, we
need a large enough time tx,y
meas d(2σdm)/¯hso that the
wavepackets of the |0and |1states have spread out enough
to significantly overlap with each other and produce an
interference pattern. Moreover, due to the free propaga-
tion, we would expect the measurement time tx,y
meas, final
position x, and the transverse wave vector kxare related
by x=¯hkxtx,y
meas
m(detecting at a position xafter the inter-
ference effectively measures the initial superposition state
of |0and |1in the |kxbasis). Noting the momentum
representation of the qubit states |n={√2σdexp[ ikxd
2−
k2
xσ2
d]exp[−inkxd]|kx}dkx(n=0,1), the probability to de-
tect the object at a position xfor any initial qubit state |ψis
given by
P(x)=|ψ|kx|2∝
exp ikxd
2−k2
xσ2
dθ|ψ
2
,(3)
where |θ=|0+|1eiθin which the parameter θ=kxd=
xmd
¯htx,y
meas (we will call θthe phase angle). Therefore, finding the
object in various positions xis in one to one correspondence
with positive operator valued measurements (POVM) on the
spatial qubit, with the relevant projection on the state being,
up to a normalization factor, as |θθ|.σxmeasurements can
therefore be implemented by placing a pair of position de-
tectors (which will in practice be lasers scattering from the
object) at positions corresponding to phase angle θ=0,θ =
π; Similarly, σymeasurements can be achieved by placing
detectors at θ=π/2,θ =−π/2 (Schematic shown in Fig. 1).
For minimizing the time of the experiment, we are going
to choose tx,y
meas =d(2σdm)/¯h. The efficacy of the σxand σy
measurements as a function of the finite time tx,y
meas for various
ratios σd:dis discussed in the Appendix.
III. NONCLASSICALITY OF THE
STERN-GERLACH STATE
As a first application of this spatial qubit technology,
we consider an extra spin degree of freedom embedded in
a mesoscopic mass. We now imagine that the mass goes
through a Stern-Gerlach apparatus. The motion of the mass
relative to the source of the inhomogeneous magnetic field
(current/magnets) is affected in a spin-dependent manner due
to the exchange of virtual photons between the source and
the spin (Fig. 2) resulting in an entangled state of the spin
and position of the nano-object as given by |φ+= 1
√2(|↑,1
+|↓,0), as depicted as the output of the preparation stage
in Fig. 2. It could also be regarded as an intraparticle entan-
glement (an entanglement between two degrees of freedom
of the same object), which has been a subject of several
investigations [34,61,62].
To measure the spin-motion entanglement in |φ+,we
have to measure variables of spin and spatial qubit. Here we
specifically want to estimate the action of measuring one of
these qubits on the quantum state of the other. During this
measurement, the inhomogeneous magnetic field causing the
Stern-Gerlach splitting is simply switched off so that spin
coherence can be maintained using any dynamical decoupling
schemes as required [63]. Alternatively, one can also use
a more pristine nanodiamond with less surface defects. As
shown in Fig. 2, after a required period of free evolution
tx,y
meas, measurements of the spatial qubit operators are made;
the light shone on the object should not interact at all with
the embedded spin degree of freedom if it is completely
off-resonant with any relevant spin transition. Immediately
033202-3
YI, SINHA, HOME, MAZUMDAR, AND BOSE PHYSICAL REVIEW RESEARCH 5, 033202 (2023)
after measuring the spatial qubit, the spin degree of freedom
is directly measured in various bases. The latter could be
implemented, for example, with a Nitrogen Vacancy (NV)
center-spin qubit in a nanodiamond crystal, where the spin
state is rotated by a microwave pulse, which corresponds to
basis change, followed by a fluorescence measurement by
shining a laser resonant with an optical transition [64]. The
implementation would require cryogenic temperature of the
diamond [65,66]. So the spin coherence time is much greater
than the experimental time scale [63]. As the spin measure-
ment is very efficient, we only need to consider the resolutions
δx,δtof the spatial qubit measurements so that the effective
spatial Pauli Xand Yoperators are then projections onto a
mixed state with phase angle ranging from θ−δθ
2to θ+δθ
2
with δθ =md
¯htx,y
meas δx−xmd
¯h(tx,y
meas )2δt. For purposes of coherence,
which continuously decreases with time, it is best to choose
time of the order of the minimum allowed time for overlap
of the wavepackets, i.e., choose tx,y
meas =d(2σdm)/¯hso that
δθ =δx
2σd−x¯h
4σ2
dmd δt. The approximate Pauli matrices are then
˜σx=1
δθ ⎛
⎜
⎝
0θ+δθ
2
θ−δθ
2
e−iθdθ|θ=0
θ+δθ
2
θ−δθ
2
eiθdθ|θ=00⎞
⎟
⎠
=1
δθ 0−ieiδθ
2+ie−iδθ
2
−ieiδθ
2+ie−iδθ
20,
and similarly, ˜σy=1
δθ (0−eiδθ
2+e−iδθ
2
eiδθ
2−e−iδθ
20).To ve rify the
entanglement we have to show that the spin-motion entangled
state violates the Bell-CHSH inequality B=|AB+AB+
AB−AB| 2 with variables [67]A=τx+τyand A=
τx−τyoperators of the spin (τxand τyare spin Pauli matri-
ces) and B=˜σxand B=˜σyoperators of the spatial qubit.
The expected correlation can be calculated (see Appendix) to
give B=|2√2f(δθ )|2√2 where f(δθ)=2
δθ Re[ieiδθ/2]=
2
δθ cos( π+δθ
2). To obtain a violation of the CHSH inequality the
upper bound of δθ can be calculated as |f(δθ)|= 1
√2,δθ ≈
2.783. Due to high tolerance in spatial detection, one may
pick δθ =π/2 so that the four detectors consisting of Pauli
Xand Ymeasurements are placed adjacent to each other and
cover the full range θ∈[−3/4π,5/4π]. The probability of
detection is ∼17.7% for each repetition of the experiment.
For realization, consider a m∼10−19 kg (108amu) spin-
bearing mass cooled to a ground state in ω∼1kHztrap
[68,69] so that its ground-state spread is =¯h
2mω∼1nm.
The cooling to ground state essentially requires a measure-
ment to the accuracy of the ground-state spread and sufficient
isolation. Essentially, measuring the position of an object to
the above precision (nm) requires 3 ×107photons, which will
hardly heat up the system in a diamagnetic trap. In fact, feed-
back cooling has been achieved for massive (10 kg) masses
[70].
At time t=0 the embedded spin is placed in a superposi-
tion 1/√2(|↑ + |↓), and the mass is released from the trap.
The wave packet then passes through an inhomogeneous mag-
netic field gradient ∼105Tm−1[25]. Due to the Stern-Gerlach
effect, the mass moves in opposite directions corresponding to
|↑ and |↓ spin states and, in a time-scale of tprep ∼50 μs,
evolves to a |φ+state with a separation of d=25 nm be-
tween the |0and |1spatial qubit states [12,22,25–27](all
lower mand dare also possible as they demand lower tprep
and ∂B/∂x). To keep the spin coherence for tprep, dynamical
decoupling may be needed [63]; it is possible to accommodate
this within our protocols—one just needs to change the direc-
tion of the magnetic field as well in tandem with the dynamical
decoupling pulses which flip the spin direction [71]. During
the above tprep, the spread of wavepackets is negligible so that
σdremains ∼1nm.
According to our results above, in order to obtain a CHSH
inequality violation, one has to measure to within δx∼
2σdδθ ∼1 nm resolution. To achieve this resolution, first we
have to ensure that during the whole duration of our protocol,
the acceleration noise has to be below a certain threshold so
as to not cause random displacement greater than 1 nm. Given
tx,y∼50 ms is the longest duration step, the acceleration
noise needs to be ∼10−6ms−2. Next comes the measurement
step where light is scattered from the object, which also needs
to measure to the required resolution. This is possible as there
are feasible techniques that give resolutions of 0.1 pm/√Hz
[58,59] for position measurements by scattering light contin-
uously from an object. Adopting the scheme in Ref. [59], the
resolution can be achieved by scattering light continuously
from the object for about 1 μs, which is 4 orders of magnitude
smaller than the experimental time span. On the other hand if
the timing accuracy δtof tx,y
meas is kept below ∼0.1ms (also
easy in terms of laser switching on/off times), there is a
negligible inaccuracy in θ.
Note that as shown in the Appendix, dephasing between
the spatial states |0and |1at a rate γsimply suppresses the
CHSH violation by a factor e−γt, which could be a new way to
investigate decoherence of the mass from various postulated
models [16–21] and environment. As during the preparation
of the superposition, spatial states are in one-to-one corre-
spondence with spin states (|0↑,|1↓, the spin decoherence
affects the superposition in the same way, and their effect is
mathematically captured by the same γ.
The decoherence of the spatial degree of freedom re-
sults from background gas collision and black-body radiation.
Adopting the formulas from Ref. [10], for our realization, the
contribution to γfrom background gas reaches ∼167.2s
−1
at pressure ∼10−10 Torr. Black-body radiation induces deco-
herence at a rate of ∼274.9s
−1at internal temperature 50 K.
On the other hand, the spin degree of freedom, which may
be encoded in NV centers of nanodiamond crystal, reaches
a coherence time of ∼0.6 s at liquid nitrogen temperature
77 K [63]. As it stands, the coherence of electron spins in
nanodiamonds is lower than what we require for 10−19 kg
mass [72–74]. However, recently, much larger times of 0.4ms
has been achieved via dynamical decoupling [75]. This is
already much larger than the superposition state preparation
time (∼50 µs). There are ideas to incorporate dynamical de-
coupling in the preparation of the initial superposition in our
experiment [71]. There are also ideas of using bath dynamical
decoupling which should not affect the spatial superposition
generation [76]. Note that, strictly speaking, the electronic
spin is only required during the Stern-Gerlach generation
of superposition after which it can be mapped onto nuclear
033202-4
MASSIVE SPATIAL QUBITS: TESTING MACROSCOPIC … PHYSICAL REVIEW RESEARCH 5, 033202 (2023)
FIG. 3. Application in witnessing Casimir-induced entangle-
ment: Two masses, each prepared in a superposition of two states, act
as two qubits 1
√2(|01+|11)⊗1
√2(|02+|12). The system freely
propagates and undergoes mutual interactions for a time τ.This
interaction induces entanglement which can be witnessed from cor-
relations of spatial qubit Pauli measurements. For example, in the
figure, σx,σymeasurements on test mass 1 and σzmeasurements
on test mass 2 are depicted. Casimir interaction induced by virtual
photons as quantum mediators is shown [79].
spins. Before the free packet expansion (interferometry), the
electronic spin can be mapped onto nuclear spin which has
much longer coherence times. At the end of the protocol, we
will need to measure the nuclear spin, for which we may need
to map back to electronic state. Such mapping can happen in
microseconds given the hyperfine couplings [77]. Therefore,
achievable pressure and temperature make the coherence time
sufficient for the realization of our protocol (same estimates
hold for the protocol of the next section).
As both the spin and the mass are measured, it character-
izes the entanglement of the given state irrespective of the
dynamics from which the state was generated, as opposed
to previous protocols which rely on a reversible nature of
the quantum dynamics [3–6]. As opposed to single object
interferometry [10,55,56], here the CHSH violation explores
decoherence of the mass in multiple bases—not only how the
|01|term of the spatial qubit decays (position basis), but also
whether, and if so how, |+−| decays (where |− = |0−
|1)—a novel type of decoherence of even/odd parity basis.
Moreover, as the total spin-motional system is quantum 4 state
system, the CHSH violation can also be regarded as a viola-
tion of the classical notion of noncontextuality [34,35,78].
IV. CASIMIR-INDUCED ENTANGLEMENT
Neutral unmagnetized untrapped masses, ideal for the
preservation of spatial coherence, can interact with each
other via the Casimir interaction [26] (gravity can cause
observable effects in reasonable times only for masses
>10−15 −10−14 kg [22,26]). Two such masses (mass m,ra-
dius R) indexed 1 and 2 are each prepared in the spatial qubit
state |+ (the superposition size, separation between states
|0and |1, being d) while the distance between the centers
of the superpositions is D(Fig. 3). In a time τ,theCasimir
interaction evolves the system to
eiφ
√2|01
1
√2(|02+eiφ01 |12)+|11
1
√2(eiφ10 |02+|12),
(4)
where φ=kR6
D7τ,φ01 =kR6
(D+d)7τ−φ, φ10 =kR6
(D−d)7τ−
φ, in which k=23c
4π(−1)2/(+2)2[80] (See also
Refs. [81,82]), where is the dielectric constant of the mate-
rial of the masses. This formula is valid when the separation is
much larger than the radius of the sphere, which is the regime
of our subsequent calculations. On top of the above evolution,
we assume a local dephasing |01|i→e−γτ|01|ifor both
particles i=1,2 (this can generically model all dephasing
[26,83]). To verify the induced entanglement, one can make
spatial qubit measurements up to uncertainties parametrized
by δθ as outlined previously and then estimate the entangle-
ment witness [84]W=I⊗I−˜σx⊗˜σx−˜σz⊗˜σy−˜σy⊗˜σz
where ˜σxand ˜σyare as discussed before, and we take ˜σz=
0
−∞ |xx|dx −∞
0|xx|dx.IfW=Tr(Wρ) is negative,
the masses are entangled. We find
˜
W=1−1
2e−2γtg2(δθ)(1 +cos(φ10 −φ01))
−e−γtg(δθ)(sin(φ10 )+sin(φ01 )),(5)
where g(δθ)=2
δθ cos( π−δθ
2).
We are going to consider the Stern-Gerlach mechanism
to first prepare the state |φ+, and use that to prepare |+.
We consider a R∼20 nm, m∼1.17 ×10−19 kg mass, and
consider it to have been trapped and cooled it to its ground
state σd∼1nmina1kHztrap[56]. We then release it, and
subject it to a magnetic field gradient of 5 ×104Tm−1[25]
for t∼100 μs so that a Stern-Gerlach splitting of d≈50 nm
develops while there is insignificant wavepacket spreading (σd
remains ∼1 nm). At this stage, a microwave pulse may be
given to rotate the spin state so that the |φ+state evolves
to |0(|↑ + |↓)+|1(|↑ − |↓). A subselection of the |↑
spin state via deflection through another Stern-Gerlach then
yields the state |+ [31,33]. Alternatively, by performing a
controlled-NOT gate with the spatial qubit as the control and
the spin as the target (again, performed quite accurately by
a microwave pulse [56]), |φ+gets converted to |+|↓ so
that the spatial part is our required state. For D≈2.1µm,
then φ10 =φ10 −φ≈0.17, φ01 =φ01 −φ≈−0.14 af-
ter τ∼0.012 s of entangling time, which gives a negative
witness W∼−0.0064. A value of δθ ∼π/6 is the highest
tolerance of error in spatial detection to keep the entanglement
witness negative. The position detectors that consist of Pauli X
and Ymeasurements with width δθ ∼π/6 has ∼5.9% chance
of detection for each repetition of the experiment.
Note here that the form of witness operator compels one to
measure both the ˜σx⊗˜σxoperator and the other two operators
on the same entangled state.˜σzmeasurement is also done at
tz
meas =τ. This is about 0.1 of the overlapping time ∼d(2σdm)
¯h
so that the fidelity of the σzmeasurement is very high (see
Appendix). We then require tx,y
meas −tz
meas τso that the extra
entanglement generated due to interactions after the ˜σzmea-
surement and before the ˜σx/˜σymeasurements is negligible.
This in turn requires us to speed up the development of spatial
overlap between the qubit states due to wavepacket spreading
033202-5
YI, SINHA, HOME, MAZUMDAR, AND BOSE PHYSICAL REVIEW RESEARCH 5, 033202 (2023)
after the ˜σzmeasurement, which can be accomplished by
squeezing both of the wavepackets in position after the time
tz
meas. After 0.01 s of flight, the wavepacket width σd∼1nm
expands to ∼10 nm. Thus we have to squeeze the state by 2
orders of magnitude to ∼1×10−10 m, so that it expands to
∼100 nm, where overlapping occurs, in the next 0.001 s. The
fidelity of XY measurement here is very high (see Appendix).
The slight delay in σx/ymeasurement (0.001 s later than the
σzmeasurement) would cause only a ∼5% error in the wit-
ness magnitude. Note that in order to achieve the required
squeezing, two appropriate periods of unitary evolution in
harmonic potentials of ω1∼1 MHz and ω2∼0.1 MHz would
suffice (nrepeated changes between ω1and ω2separated by
appropriate periods of harmonic evolution will squeeze by
the factor (ω1/ω2)n[85]); if this potential was applied as
an optical tweezer then it will hardly cause any decoherence
γsqueeze,j∼ωj10−5[7]. We additionally need to ensure, for
reasons described earlier, that the acceleration noise is below
10−6ms−2. The whole procedure described above could be
one of the earliest demonstrations of non-Gaussian entangle-
ment between neutral masses. It would also demonstrate the
nonclassical nature of the Casimir interaction, namely that it
is mediated by quantum agents (virtual photons) as in the inset
of Fig. 3.
Compared with other types of interactions, gravitational
interaction at this scale is negligible compared with Casimir
interaction as long as the separation between the two masses
are less than 200 μm for materials with density of diamonds
[22]. The phase induced by electrostatic interaction can be
estimated by
p2t
4π0R3¯h, where
pis electric dipole moment. For
R∼20 nm, t∼10 ms,
pneeds to be much smaller than
10−30 C.m to be considered negligible compared with Casimir
interaction. In current experiments,
pcan be as small as
∼10−23 C.m for 10 μm radius particles, scaling with the vol-
ume of the particle [86,87]. From the scaling, we can expect
electric dipole ∼10−29 C.m for 10 nm radius particles. An
improvement of the electric dipole by a couple of orders of
magnitude is required to make electrostatic interaction negli-
gible in comparison with Casimir interaction. These electric
dipoles typically appear due to defects in crystals such as dis-
locations. It is possible that given the 10 nm size, one can form
a single crystal [88] so that the intrinsic dipole background can
be reduced.
V. CONCLUSIONS
We have shown how to measure a qubit encoded in a
massive object by position detection. We have shown how this
can be applied to (a) stretch the validity of quantum physics
to the center of mass of nano-objects, demonstrating quantum
contextuality, never before tested for macroscopic objects, (b)
entangle spatial qubits encoded in two such objects, extending
non-Gaussian quantum technology, (c) prove empirically the
quantum coherent nature of the Casimir force. Indeed, in the
same open-minded way that one asks whether quantum me-
chanics continues to hold for macroscopic masses [1,2], one
can question whether those interactions between such masses
which are extensive in nature (grow as volume/area/mass)
continue to be mediated by a quantum natured field so as to
be able to entangle the masses. In comparison with standard
approaches for probing the nonclassicality for smaller masses,
we avoid a Mach-Zehnder interferometer—only requiring the
preparation of an original spatial superposition. This is advan-
tageous because of the difficulty of realizing beam splitters
for nano-objects (tunneling probability ∝e−√2mV
¯hxgetting ex-
tremely small), and also for avoiding interactions with mirrors
and beam splitters which can cause decoherence (we exploit a
two-slit experiment as a beam-splitter [89], see Appendix E).
Our methodology can be quantum mechanically simulated
with cold atoms, where other methods to encode qubits in
motional states have been demonstrated [90], before they are
actually applied to nano-objects.
ACKNOWLEDGMENTS
D.H. and U.S. would like to acknowledge partial sup-
port from the DST-ITPAR, Grant No. IMT/Italy/ITPAR-
IV/QP/2018/G. D.H. also acknowledges support from the
NASI Senior Scientist fellowship. A.M.’s research is funded
by the Netherlands Organization for Science and Research
(NWO) Grant No. 680-91-119. S.B. would like to ac-
knowledge EPSRC Grants No. EP/N031105/1 and No.
EP/S000267/1.
APPENDIX A: EFFICACY OF THE PAULI-Z
MEASUREMENT AS A FUNCTION
OF MEASUREMENT TIME
The system freely propagates for a time t, the final state
may be written for an initial state |+ as
x|(t)= 1
[2πσ2]1/41
1
s−i¯ht
2m⎧
⎨
⎩
exp ⎡
⎣−x−d
2σ2
ds2
41
s−i¯ht
2m⎤
⎦
+exp ⎡
⎣−x+d
2σ2
ds2
41
s−i¯ht
2m⎤
⎦⎫
⎬
⎭
,(A1)
where s≡i4φ
σ2+1
σ2+1
σ2
d
,φis global phase added during the
propagation.
Note that Eq. (A1) consists of two terms tracing which path
the object passes through, effectively the predefined Young
qubit. σzmeasurement requires that the wavepackets are well
separated upon measurement. The condition maybe formu-
lated by demanding the probability distribution P0(P1)of|0
(|1) state alone confined in the x<0(x>0) regime
=1−0
−∞ P0dx
∞
−∞ P0dx 1.(A2)
Substituting Eq. (A1), the fraction term can be evaluated at
the σzmeasurement time t=tz
meas:
0
−∞ P0dx
∞
−∞ P0dx =0
−∞ exp −
(x+d
2σ2
ds)21
s
2
s2+¯h2t2
8m2dx
∞
−∞ exp −
(x+d
2σ2
ds)21
s
2
s2+¯h2t2
8m2dx
,(A3)
033202-6
MASSIVE SPATIAL QUBITS: TESTING MACROSCOPIC … PHYSICAL REVIEW RESEARCH 5, 033202 (2023)
where the normalization factor in the probability distribution,
independent of x, cancels out in the calculation.
Let us take σd:d=1 : 50 and tz
meas ≈one-tenth of the
overlapping time d(2σdm)
¯h. We then get ∼(1 −4.7×10−7).
We may thus claim that for the above choice of parameters
the left and right Gaussian wavepackets are well separated,
and Pauli-Z measurement has a good fidelity.
APPENDIX B: EFFICACY OF THE PAULI-XAND PAULI-Z
MEASUREMENT FOR VARIOUS PARAMETERS
We take t=tx,y
meas ∼2σdmd
¯has the time of σx,σymea-
surement, and evaluate how accurate this measurement is
for various ratios σd:dusing the full-time evolution as in
Eq. (A1). Our target is to check how accurately the interfer-
ence pattern is reproduced at correct positions as given by x=
¯hkxtx,y
meas
m. For an initial state |0+|1,ifσd:d=1:10thefirst
peak of the interference pattern adjacent to the central peak
(corresponding to θ=2π) locates at x∼11.797σd.Ifσd:
d=1 : 100, x∼12.559σd.Ifσd:d=1:50,x∼12.536σd.
While assumption x=¯hkxtx,y
meas
mgives a value of x=4πσd∼
12.566σd, which is >∼99.94% accurate for σd
d1
100 ,
>∼99.76% accurate for σd
d1
50 . Therefore, the σx,σymea-
surements have a good fidelity in the σd
d1
50 setting, which
we shall use.
APPENDIX C: METHODS OF COMPUTATION WITH
EFFECTIVE PAULI OPERATORS INCLUDING
UNCERTAINTIES, AND THE INCORPORATION
OF DECOHERENCE
Let ˜σdenote the measured Pauli operator, parametrized by
uncertainty in phase angle δθ. Then,
˜σx=1
δθ 0−ieiδθ
2+ie−iδθ
2
−ieiδθ
2+ie−iδθ
20
=g(δθ)σx,(C1)
where g(δθ)=−ieiδθ
2+ie−iδθ
2=2
δθ cos(π−δθ
2). g(δθ) goes to
1asδθ →0. Similarly,
˜σy=g(δθ)σy.(C2)
Therefore, for arbitrary density state ρ
Tr(˜σxρ)=g(δθ)Tr(σxρ),
Tr(˜σyρ)=g(δθ)Tr(σyρ).(C3)
In particular, let ρx+(ρy+) denote the positive eigenstate of
σx(σy), and ρx+=1
2(11
11
), ρy+=1
2(1−i
i1). Then,
Tr(˜σxρx+)=Tr(˜σyρy+)=g(δθ).(C4)
Furthermore, if decoherence is considered, let ˜ρ=
(ρ00 ρ01e−γt
ρ10e−γtρ11 ), where γdenotes dephasing rate, then we
have
Tr( ˜σx˜ρ)=g(δθ)Tr(σxρ)e−γt,
Tr( ˜σy˜ρ)=g(δθ)Tr(σyρ)e−γt.(C5)
APPENDIX D: DECOHERENCE IN PROBING THE
ENTANGLEMENT OF THE STERN-GERLACH STATE
Consider the initial state |φ+= 1
√2(|↑,R+|↓,L),
where the spatial qubit undergoes decoherence. The density
operator after decoherence can be written as
˜ρ(φ+)=1
2⎛
⎜
⎜
⎜
⎝
00 00
01e−γt0
0e−γt10
00 00
⎞
⎟
⎟
⎟
⎠
,(D1)
Tr(σx⊗σx˜ρ(φ+)) =Tr(σy⊗σy˜ρ(φ+)) =e−γt,
Tr(σx⊗σy˜ρ(φ+)) =Tr(σy⊗σx˜ρ(φ+)) =0.(D2)
Therefore,
|ab+ab+ab−ab| = |2√2g(δθ)e−γt|2√2.
(D3)
APPENDIX E: YOUNG-TYPE QUBIT AS BEAM SPLITTER
Our experimental setup serves the same purpose as a
Mach-Zehnder interferometer in probing contextuality [34],
in which a particle passes through beam splitters and which
path information defines a spatial qubit. In our approach,
a Young-type double slit acts effectively as a lossy beam
splitter [89]. A cubic beam splitter has two input and two
output. The transformation matrix from the former to the latter
ports is described by a two-by-two unitary matrix. The initial
states |0and |1act as the two input. By placing a pair
of detectors in the interference plane, we project the input
states onto a different basis, parametrized by phase angle θ.
For instance, conducting Pauli-Yoperation requires placing
two detectors at phase angle −π/2 and π/2 respectively.
The effective beam splitter therefore transforms the system
from a superposition of |0and |1to the basis spanned
by |0−i|1and |0+i|1. The transformation matrix is
therefore 1
√2(1i
1−i)=1
√2(1i
i×e−iπ/21×e−iπ/2), equivalently, a
50:50 beam splitter followed by a phase shifter with angle
−π/2.
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