Article
Reference
Macroscopic Optomechanics from Displaced Single-Photon
Entanglement
SEKATSKI, Pavel, ASPELMEYER, Markus, SANGOUARD, Nicolas Bruno
Abstract
Displaced single-photon entanglement is a simple form of optical entanglement, obtained by
sending a photon on a beam splitter and subsequently applying a displacement operation. We
show that it can generate, through a momentum transfer in the pulsed regime, an
optomechanical entangled state involving macroscopically distinct mechanical components,
even if the optomechanical system operates in the single-photon weak coupling regime. We
discuss the experimental feasibility of this approach and show that it might open up a way for
testing unconventional decoherence models.
SEKATSKI, Pavel, ASPELMEYER, Markus, SANGOUARD, Nicolas Bruno. Macroscopic
Optomechanics from Displaced Single-Photon Entanglement. Physical review letters, 2014,
vol. 112, no. 8
DOI : 10.1103/PhysRevLett.112.080502
Available at:
http://archive-ouverte.unige.ch/unige:36534
Disclaimer: layout of this document may differ from the published version.
1 / 1
Macroscopic Optomechanics from Displaced Single-Photon Entanglement
Pavel Sekatski,
1,2
Markus Aspelmeyer,
3
and Nicolas Sangouard
1
1
Group of Applied Physics, University of Geneva, CH-1211 Geneva 4, Swi tzerland
2
Institut for Theoretische Physik, Universitat of Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria
3
Vienna Center for Quantum Science and Technology, Faculty of Physics, Univ ersity of Vienna,
Boltzmanngasse 5, 1-1090 Vienna, Austria
(Received 30 August 2013; published 27 February 2014)
Displaced single-photon entanglement is a simple form of optical entanglement, obtained by sending a
photon on a beam splitter and subsequently applying a displacement operation. We show that it can
generate, through a momentum transfer in the pulsed regime, an optomechanical entangled state involving
macroscopically distinct mechanical components, even if the optomechanical system operates in the single-
photon weak coupling regime. We discuss the experimental feasibility of this approach and show that it
might open up a way for testing unconventional decoherence models.
DOI: 10.1103/PhysRevLett.112.080502 PACS numbers: 03.67.Bg, 03.65.Ta, 03.65.Yz, 42.50.Wk
Introduction.—Can a macroscopic massive object be in a
superposition of two well distinguishable positions? It has
been argued that such superpositions undergo intrinsic
decoherence, e.g., due to a nonlinear stochastic classical
field [1–3] or caused by the superposition’s perturbation of
spacetime [4,5]. These decoherence mechanisms are differ-
ent from conventional decoherence that occurs through
entanglement with the environment [6] and that has been
nicely demonstrated in Refs. [7–11]. In contrast, testing for
unconventional decoherence models requires a combina-
tion of large masses and superpositions of states corre-
sponding to well separated positions. Matter-wave
interferometry with large clusters [12] or with submicron
particles [13] is one possible route. Another approach is to
manipulate states of motion of massive mechanical reso-
nators, a fast moving field of research that has now
succeeded in entering the quantum regime [14–17].In
the framework of optically controlled mechanical devices
[18], the proposals [19,20] have the potential to create a
superposition of mechanical states with a distance of the
order of the mechanical zero-point fluctuation where the
effects of unconventional decoherences might be observ-
able [21,22]. However, this requires (i) one to work in the
single-photon strong coupling regime, (ii) a coupling rate
at least of the order of the mechanical frequency so that the
displacement induced by a single photon is larger than
the mechanical zero-point spread, and (iii) one to work in
the resolved sideband regime where the mechanical fre-
quency is larger than the cavity decay rate to allow ground
state cooling. While (i) and (ii) can be relaxed, e.g., using
nested interferometry [23] and (iii) can be circumvented by
cooling, e.g., via pulsed optomechanical interactions [24],
the distance between the superposed states remains small, of
the order of the mechanical ground state extension.
Here, we show how to create macroscopic optomechan-
ical entanglement with relatively simple ingredients. Our
proposal starts with an optical entangled state of the type
j
¯
þi
A
j−i
B
− j¯−i
A
jþi
B
, involving two spatial modes A and
B. Concretely, this state is obtained by sending a single
photon into a beam splitter (with output modes A and B)
and by subsequently applying a phase-space displacement
on A. The displaced photons in A then interact with a
mechanical system M through radiation pressure. If the
interaction between A and M falls within the pulsed regime
[24–26] where the pulse duration is much smaller than the
mechanical period, the optical and mechanical modes
entangle, j
¯
þi
AM
j−i
B
− j¯−i
AM
jþi
B
. Because j
¯
þi
A
and
j¯−i
A
are well distinguishable in photon number, the
mechanical components ρ
ðÞ
M
¼ tr
A
fj
¯
ih
¯
j
AM
g are well dis-
tinct in the phase space even in the weak coupling regime
and if the coupling rate is smaller than the mechanical
frequency. This relaxes the constraints on the initial cooling
of the mechanical oscillator and makes our proposal well
suited to test unconventional decoherence processes, as we
show below.
Optomechanical entanglement.—Consider an optome-
chanical cavity described by H ¼ ℏω
m
m
†
m−
ℏg
0
a
†
aðm
†
þ mÞ, where ω
m
is the angular frequency of
the center of mass motion of the mechanical system, m, m
†
(a, a
†
) are the bosonic operators for the phononic (pho-
tonic) modes, and g
0
¼ðω
c
=LÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ℏ=2Mω
m
p
for a Fabry-
Perot cavity with a mechanically moving end mirror (ω
c
is
the optical angular frequency, L is the cavity length, and M
is the effective mass of the mechanical mode). The form of
the optomechanical interaction, proportional to a
†
a
¯
x
m
[where
¯
x
m
¼ x
0
ðm þm
†
Þ is the position operator, x
0
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ℏ=2Mω
m
p
being the mechanical zero-point fluctuation
amplitude] tells us that starting with a superposition of
photonic components which are well distinguishable in
photon number space, we can create a superposition of
mechanical states corresponding to well distinct momenta.
Displaced single-photon entanglement exhibits such a
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0031-9007=14=112(8)=080502(5) 080502-1 © 2014 American Physical Society
property [27,28] and has the advantage of being easily
prepared, see Fig. 1. It can be written as
1
ffiffiffi
2
p
ðDðβÞjþi
A
j−i
B
− DðβÞj−i
A
jþi
B
Þ; (1)
where ji ¼ 2
−ð1=2Þ
ðj0ij1iÞ, j0i being the vacuum, and
j1i the single photon Fock state [29]. DðβÞ¼e
βa
†
−β
⋆
a
stands for the displacement operator and can be imple-
mented using an unbalanced beam splitter and a coherent
state [30]. Although the photon number distributions for
DðβÞjþi
A
and DðβÞj−i
A
partially overlap (their variance is
given by β
2
þð1=4Þ), their mean photon numbers β
2
β þð1=2Þ are separated by 2β [27]. (Here β is considered
real, as throughout Letter). In other words, their distance in
the photon number space is of the order of the square root
of their size. This makes the state (1) macroscopic in the
sense that its components can be distinguished without a
microscopic resolution [28].
Consider first the case where jni
A
photons interact
with the mechanical mode initially prepared in its
motional ground state j0i
M
. According to Ref. [19],
they induce a coherent displacement of the mechanical
state whose amplitude varies periodically in time
e
iðg
2
0
n
2
=ω
2
m
Þðω
m
t−sinðω
m
tÞÞ
jðg
0
n=ω
m
Þð1 − e
−iω
m
t
Þi
M
jni
A
. The
first exponential term corresponds to the variation of the
cavity length and is quadratic in the photon number because
the mean position of the mechanical oscillator depends on
the photon number. To avoid this nonlinear behavior, we
consider the pulsed regime where the interaction time τ is
much smaller than the mechanical period [sinðω
m
τÞ ∼ ω
m
τ,
cf. below for the detailed conditions]. Right after this
interaction, the propagator has the simple form
e
ig
0
τa
†
aðmþm
†
Þ
and after a free evolution of duration t, the
overall propagator can be written as UðtÞ¼
e
ig
0
τa
†
aðe
iω
m
t
mþe
−iω
m
t
m
†
Þ
e
−iω
m
tm
†
m
. An initial state j0i
M
jni
A
now evolves towards jnαðtÞi
M
jni
A
where jnαðtÞi
M
is a
coherent state with a fixed amplitude and a periodic phase
nαðtÞ¼−ig
0
nτe
−iω
m
t
. In other words, the n photons kick
the mechanical mode that gets an additional momentum
2g
0
nτp
0
at time t ¼ 0 (p
0
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ℏMω
m
=2
p
is the initial
mechanical momentum spread). The mechanical state then
starts to rotate in phase space. It reaches a minimal position
−2g
0
nτx
0
after π=2ω
m
, then gets a momentum −2g
0
nτp
0
after π=ω
m
and so on.
Let us now come back to the initial state (1). The pulse in
A enters the optomechanical cavity, the mechanical mode
being in j0i
M
, as before. A time t after the interaction, the
state of the system is
1
ffiffiffi
2
p
X
k
a
ðþÞ
β
ðkÞjki
A
jkαðtÞi
M
j−i
B
−
X
k
a
ð−Þ
β
ðkÞjki
A
jkαðtÞi
M
jþi
B
; (2)
where a
ðÞ
β
ðkÞ¼ð1=
ffiffiffi
2
p
Þe
−ðβ
2
=2Þ
ðβ
k
=
ffiffiffiffi
k!
p
Þð1 ððk=βÞ− βÞÞ
are the probability amplitudes for having k photons in
DðβÞji
A
. Since
P
k
ða
ðþÞ
β
ðkÞÞ
⋆
a
ð−Þ
β
ðkÞ¼0, the mechani-
cal mode entangles with the optical modes. Specifically,
after π=2ω
m
, the state (2) involves two mechanical states
ρ
ðÞ
M
¼
P
k
ja
ðÞ
β
ðkÞj
2
j−g
0
τki
M
h−g
0
τkj, each having a vari-
ance ð1 þ g
2
0
τ
2
ð1 þ 4β
2
ÞÞx
2
0
in space and for which the
mean position is separated by 4g
0
τβx
0
(see Fig. 2). These
two mechanical states can thus be distinguished with a
detector having a resolution δx ∼ 2g
0
τβx
0
, see below. For
g
0
τβ ≥ 1, such a detector cannot resolve two phononic
Fock states with n and n þ 1 excitations (no microscopic
resolution) and the entangled state (2) can fairly be defined
as being macroscopic.
Macroscopic correlations.—We now show how to dem-
onstrate that the mechanical mode involves macroscopi-
cally distinct states ρ
ðÞ
M
. More precisely, we show that B
and M are correlated, i.e., when the state of B is projected
into j−i (jþi), the mechanical mode is found in ρ
ðþÞ
M
[ρ
ð−Þ
M
]
a quarter of a mechanical period after the interaction, (cf.
Fig. 3) and that these correlations can be revealed without
the need for a microscopic resolution. This is done by
tracing out A, and by measuring the
¯
X ¼ 2
−1=2
ðb
†
þ bÞ
quadrature of B and the mirror position. The latter can be
realized following Ref. [24], by observing through a
FIG. 1. A single photon is sent through a beam splitter and
creates an entangled state between the two output modes A and B:
A then undergoes a displacement and couples to a mechanical
system by momentum transfer in the pulsed regime.
FIG. 2 (color online). Trajectory of the mechanical state in the
phase space. (I) The mirror first gets a momentum proportional to
the mean photon number. The superposition of two mechanical
states (corresponding to the two ovals) result from the interaction
with a superposition of DðβÞj−i
A
and DðβÞjþi
A
. (II) After a
quarter of a period, the positions of the two superposed states are
maximally distinct and are correlated with the
¯
X quadrature of the
mode B. (III) By measuring the position after a multiple of a half
period, the information about the number of photons in A
(contained in the mirror) is erased, which enables us to observe
the entanglement between A and B.
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quadrature measurement the phase acquired by a strong,
short light pulse reflected by the mechanical oscillator. We
attribute the value þ1 (−1) to a positive (negative) result of
the quadrature measurement on B and þ1 (−1) if the mirror
is found to be shifted more to the left (right) with respect
to its mean position −g
0
τx
0
ð1 þ2β
2
Þ. For an uncertainty
δx on the measurement of the mirror position, the
probability P
E
for having the same results { 1, 1}
is given by ð1=4Þþðg
0
τβ=2π
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þg
2
0
τ
2
β
2
þ δx
2
=ð4x
2
0
Þ
p
Þ
(for β ≫ 1) while the probability for having different
results P
E
∓
¼ð1=2Þ − P
E
. Therefore, the correlations
between the outcomes (the probability for having
correlated results minus the probability for having
anticorrelated results) are given by ð2=πÞðg
0
τβ=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ g
2
0
τ
2
β
2
þ δx
2
=ð4x
2
0
Þ
p
Þ. In the regime of interest
g
0
τβ ≥ 1, even a coarse grained measurement with the
resolution δx ¼ 2g
0
τβx
0
leads to substantial correlations
∼0.45. This is a consequence of the macroscopic character-
istic of the optomechanical state (2).
Testing unconventional decoherence models.—Figure 4
shows how to probe the effect of mirror decoherence. First,
the mechanical position is measured at any time that is a
multiple of half a mechanical period where no information
is obtained about the state of A. Finding the mirror at the
position y projects the overall state into
1
ffiffiffi
2
p
X
k
a
ðþÞ
β
ðkÞe
i
ffiffi
2
p
g
0
τky
jki
A
j−i
B
−
X
k
a
ð−Þ
β
ðkÞe
i
ffiffi
2
p
g
0
τky
jki
A
jþi
B
jyi
M
: (3)
Actively controlling the relative length of paths A and B
to get rid of the undesired phase term e
i
ffiffi
2
p
g
0
τky
and
subsequently applying Dð−β Þ leaves the optomechanical
state in ð1=
ffiffiffi
2
p
Þðj1i
A
j0i
B
− j0i
A
j1i
B
Þjyi
M
. The modes A
and B can then be combined on a beam splitter and varying
their relative phase leads to interference fringes, ideally
with a unit visibility (V). (Note here that from the values
of the probabilities p
mn
of detecting m ∈ f0; 1g photons in
A and n ∈ f0; 1g in B, a lower bound on the negativity
between A and B can be obtained N
AB
≥
ð1=2Þð
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðp
00
− p
11
Þ
2
þðVðp
01
þ p
10
ÞÞ
2
p
− ðp
00
þ p
11
ÞÞ
through the approach presented in Ref. [31].) Decoherence
of the mirror operates as a weak measurement of the photon
number on A (see the Supplemental Material [32]).
Therefore, if the measurement of the mechanical position
is delayed, more and more “which path” information is
revealed, which decreases the visibility as the delay time
increases. In particular, we compare conventional (envi-
ronmentally induced) decoherence with unconventional
decoherence proposed by gravitationally induced collapse
[4,5] and by quantum gravity [33] (see the Supplemental
Material [32]). For sufficiently large β, i.e., macroscopic
entanglement, and small thermal dissipation we find an
experimentally feasible parameter regime, in which the
unconventional decoherence rates surpass the conventional
ones, hence opening up the possibility for experimental
tests (see below). Finally, note that the observed visibility is
degraded if the mirror position is not accurately measured.
A small imprecision δx would indeed introduce an addi-
tional phase on A that prevents its redisplacement to the
single photon level and degrades the quality of the
interference between A and B [27,34]. Quantitatively,
V ≈ 1 −
3
2
δϕ
4
β
4
þ oðδϕ
4
β
4
Þ; (4)
where δϕ ¼ðδx=x
0
Þð2ðδx=x
0
Þ
2
þ 1Þ
−ð1=2Þ
ffiffiffi
2
p
g
0
τ. A high
accuracy δx ≲ ðx
0
=ðg
0
τβÞ
2
Þ is thus required to observe
high visibility and to see the effect of mirror decoherence.
Witnessing optomechanical entanglement.—We can
prove that the mirror is entangled with the optical modes
from an entanglement witness that uses the values of {P
E
,
P
E
∓
} and N
AB
only (see the Supplemental Material [32]).
The witness is based on the following intuitive argument:
since B is a qubit, the only way for M to be correlated to B and
for B to be entangled with the joint system AM is that M is
entangled with AB: Concretely, we can conclude about
optomechanical entanglement if N
AB
>
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
þE
þ
P
−E
þ
p
þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
þE
−
P
−E
−
p
. We emphasize that in contrast to the correla-
tion measurement, the detection of entanglement N
AB
requires a measurement of the mirror position with a very
high accuracy (through V). We are retrieving what seems to
be the essence of macroentangled states: although they
involve components that can easily be distinguished without
microscopic resolution, one needs detectors with a very high
precision to reveal their quantum nature [28,35].
Experimental feasibility.—We now address the question
of the experimental feasibility in detail. First, we require
FIG. 3. Setup for checking that the result of the homodyne
measurement on B is correlated with the position of M even if the
position measurement does not have a microscopic resolution.
FIG. 4. Setup for probing the effect of decoherence on the
interference between A and B obtained after erasing the which
path information gained by the mirror. A feedback loop is needed
to control the path length of A (relative to B) depending on the
result of the measurement of the mirror position.
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4g
0
τβ ≳ 1, which allows one to observe significant corre-
lations between M and B.
To further guarantee a high visibility of the interference
between A and B, the system needs to operate in the linear
regime. For a pulsed optomechanical interaction (τ ≪ ω
−1
m
),
the nonlinear response of the optomechanical system
degrades the visibility of the interference pattern according
to V → Vð1 − ϵÞ where ε ¼ðg
0
τβÞ
6
ðω
2
m
=g
2
0
β
2
Þ [36].This
undesired effect is thus negligible if g
0
β ≫ ω
m
.Therequire-
ment of observing a high interference visibility also imposes
the mirror position to be accurately estimated, cf. Eq. (4).It
has been established in Ref. [24] that the maximum accuracy
is obtained by choosing an input drive with a duration
∼ðln 2=κÞ. The achievable precision then depends on its
number of photons N
p
via ðδx=x
0
Þ¼ðκ=
ffiffiffi
5
p
g
0
ffiffiffiffiffiffi
N
p
p
Þand is
thus high if g
0
ffiffiffiffiffiffi
N
p
p
> κ. The primary limitation for N
p
is
the power that can be homodyned before photodetection
begins to saturate. Assuming a saturation power of 10 mW
results in N
p
∼ 5 × 10
16
=κ. To build up a proposal as simple
as possible, we consider thecasewhereasinglelocaloscillator
with a controllable amplitude is used both for implementing
the displacement and for measuring the mirror position
(τ ∼ ðln 2=κÞ). Using Eq. (4), this results in the reduced
visibility V → Vð1−
¯
ϵÞ where
¯
ϵ ∼ 2 × 10
−35
κ
2
β
4
.
The mechanical device also needs to be prepared in
its ground state. More precisely, if the mechanical oscillator
is initially in a thermal state with a mean occupation nth,
the interference visibility is unchanged but the observed
correlations decrease according to ð2=πÞðg
0
τβ=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n
th
þ 1 þ g
2
0
τ
2
β
2
þ δx
2
=ð4x
2
0
Þ
p
Þ. High correlations can
thus be observed if
ffiffiffiffiffiffi
n
th
p
≪ g
0
τβ; i.e., the constraint on
the initial cooling is relaxed for macroscopically distinct
mechanical states. Cooling in the pulsed regime can be
obtained through various schemes [24,25]. For example,
Refs. [24,37] show that two subsequent pulses (identical to
the pulses used for the measurement of the mechanical
position) that are separated by π=2ω
m
allow one to cool
the mechanical mode to an effective thermal occupation
of n
eff
¼ð1=2Þð
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þðκ
4
=g
4
0
N
2
p
Þ
q
− 1Þ.Forg
0
ffiffiffiffiffiffi
N
p
p
> κ,
this results in n
eff
≪ 1, i.e., ground state cooling.
For concreteness, we consider a mechanical mirror
with resonance frequency ω
m
¼ 2π × 20 × 10
3
s
−1
(ð2π=ω
m
Þ¼50 μs) and an effective mass M ¼ 60 ng in
a 0.5 cm long cavity (g
0
=ω
m
¼ 5 × 10
−3
). We require
correlations larger than 0.5 (4g
0
τβ ¼ 6) and an error on
the overall visibility of ∼1%. This imposes a cavity finesse
of ∼8000 (β ∼ 40000, N
p
∼ 4 × 10
9
, κ ∼ 2π × 2 × 10
6
s
−1
,
τ ∼ 60 ns). For comparison, the highest reported finesse
in an optical Perot-Fabry cavity with micromirrors is
1.5 × 10
5
[38].
The photons in A need to be stored on the time scale of
the decoherence being probed. A simple fiber loop allows
one to reach delay times up to 100 μs without significant
loss at telecom wavelength. Much longer delays can be
obtained with such a technique if one is willing to use
postselections [39].
The surrounding temperature T must also be low enough
so that the effect of conventional (environmentally induced)
decoherence [6] is negligible on the time scale of the
decoherence being probed. This requires T ≪
ðℏω
m
Q
m
=k
B
Þð1=2ðg
0
τÞ
2
β
2
Þð1=2πnÞ for n mechanical peri-
ods. In other words, for a base temperature of T ¼ 800 mK
and a mechanical quality factor of Q
m
¼ 10
6
, conventional
decoherence operates on a time scale of 1 μs, which is long
enough to observe optomechanical entanglement. Lower
temperatures and/or higher Q
m
are required for testing
unconventional decoherence models. For example, for
quantum gravity induced collapse [33], we find a time
scale ∼415 μs following Ref. [40], which would be testable
with the proposed device with Q
m
∼ 1.5 × 10
7
and
T ∼ 20 mK where conventional decoherence operates on
∼630 μs. Gravitationally induced decoherence [4,5] pro-
vides another example, despite the known ambiguity with
respect to the mass distributions. Under the assumption
where the mass is distributed over spheres corresponding to
the size of atomic nuclei, we find a time scale of ∼10 μs
following Ref. [22]. This is testable with the proposed
device for T ∼ 300 mK and Q
m
∼ 10
7
where conventional
decoherence operates on ∼30 μs. Note that in addition to
absolute decoherence rates, the scaling behavior with
respect to mechanical parameters, e.g., the mass, provides
an independent assessment of the nature of the observed
decoherence (see the Supplemental Material [32]).
Conclusion.—We have proposed a way for creating and
detecting macroscopic optomechanical entanglement that
combines displaced single-photon entanglement and pulsed
optomechanical interaction. Our proposal can be imple-
mented in a wide variety of systems. The optomechanical
photonic crystal cavity device introduced in Ref. [41] could
exhibit correlations of 0.6 and an interference visibility of
0.95 at a temperature of a few kelvins, while more massive
systems, like the one proposed before, open up a way to
measure unconventional decoherence models.
We thank N. Gisin, K. Hammerer, S. Hofer, N. Timoney,
and P. Treutlein for discussions. This work was supported
by the Swiss NCCR QSIT, the Austrian Science Fund FWF
(SFB FoQuS, P24273-N16, SFB F40-FoQus F4012-N16),
the Vienna Science and Thechnology Fund WWTF, the
European Research Council ERC (StG QOM), and the
European Commission (IP SIQS, ITN cQOM).
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PRL 112, 080502 (2014)
PHYSICAL REVIEW LETTERS
week ending
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