# Einstein-Podolsky-Rosen experiment with two Bose-Einstein condensates

Paolo Colciaghi  These authors contributed equally to this work  Department of
Physics, University of Basel, 4056 Basel, Switzerland  Yifan Li  These authors
contributed equally to this work  Department of Physics, University of Basel,
4056 Basel, Switzerland  Philipp Treutlein  Department of Physics, University
of Basel, 4056 Basel, Switzerland  Tilman Zibold<EMAIL_ADDRESS>Department of Physics, University of Basel, 4056 Basel, Switzerland

###### Abstract

In 1935, Einstein, Podolsky and Rosen (EPR) conceived a Gedankenexperiment
which became a cornerstone of quantum technology and still challenges our
understanding of reality and locality today. While the experiment has been
realized with small quantum systems, a demonstration of the EPR paradox with
spatially separated, massive many-particle systems has so far remained
elusive. We observe the EPR paradox in an experiment with two spatially
separated Bose-Einstein condensates containing about 700 Rubidium atoms each.
EPR entanglement in conjunction with individual manipulation of the two
condensates on the quantum level, as demonstrated here, constitutes an
important resource for quantum metrology and information processing with many-
particle systems. Our results show that the conflict between quantum mechanics
and local realism does not disappear as the system size is increased to over a
thousand massive particles.

##  I Introduction

According to quantum mechanics, complementary properties of a physical system
such as its position and momentum or two orthogonal components of its spin
cannot be simultaneously known with arbitrary precision. This is expressed by
the Heisenberg uncertainty principle Heisenberg (1927, 1930), which imposes a
lower bound on the product of the two uncertainties, so that better knowledge
of one property necessitates greater uncertainty of the other and vice versa.
This is in stark contrast to classical physics, where all properties of a
system can be simultaneously known, in principle with arbitrary precision.

In their seminal work Einstein  _et al._ (1935), EPR considered this aspect
for a bipartite quantum system where the parts $A$ and $B$ are entangled
through interactions. After spatially splitting the system, measurements on
the two parts yield strongly correlated outcomes, which allow one to use
measurements on $A$ to predict properties of $B$ (and vice versa). The
uncertainty product of such predictions for two complementary properties of B
falls below the Heisenberg uncertainty bound of $B$ Reid (1989). The fact that
the distant system A can be used for better predictions than what is locally
possible at B revealed a conflict between quantum mechanics and the classical
principle of local realism, which became known as the “EPR paradox”
Schrödinger (1935a). In a local realist world, each system possesses its
properties independent of observation and independent of actions performed on
spatially separated systems. In a quantum world, on the other hand, the
measurements on $A$ change the quantum state of the distant system $B$, a
scenario Schrödinger called “steering” Schrödinger (1935a). It was later found
that not all entangled states are able to show such strong correlations Werner
(1989); Wiseman  _et al._ (2007). Only a strict subset, the so-called EPR
entangled states, are able to demonstrate an EPR paradox Reid  _et al._
(2009). Furthermore, EPR entanglement was identified as a resource for quantum
technologies such as quantum metrology, quantum teleportation, entanglement
swapping, or randomness certification Cavalcanti and Skrzypczyk (2017); Uola
_et al._ (2020); Yadin  _et al._ (2021).

The EPR paradox has been observed with small systems of few photons or atoms,
in its original form Ou  _et al._ (1992); Hagley  _et al._ (1997); Bowen  _et
al._ (2003); Reid  _et al._ (2009); Kheruntsyan  _et al._ (2012) or in the
form of Bell testsFreedman and Clauser (1972); Lamehi-Rachti and Mittig
(1976); Aspect  _et al._ (1982); Hensen  _et al._ (2015); Giustina  _et al._
(2015); Shalm  _et al._ (2015); Rosenfeld  _et al._ (2017). How far quantum
behavior extends into the macroscopic world is an open question Leggett
(2002), which can be addressed by performing EPR experiments with increasingly
macroscopic, massive systems Leggett (2007); Cavalcanti and Reid (2007); Reid
_et al._ (2009). Generating and verifying sufficiently strong entanglement
between massive many-particle systems is a challenging task, requiring
excellent isolation from the environment, high fidelity coherent manipulations
and low noise detection. Previous experiments demonstrated entanglement (non-
separability) between spatially separated atomic ensembles Julsgaard  _et al._
(2001); Chou  _et al._ (2005), mechanical oscillators Kotler  _et al._ (2021);
de Lépinay  _et al._ (2021), and hybrid systems Lettner  _et al._ (2011);
Thomas  _et al._ (2021). However, the observed correlations were not strong
enough to demonstrate an EPR paradox.

Atomic Bose-Einstein condensates (BECs) are many-particle systems that are
particularly well suited to investigate non-classical phenomena at the
quantum-to-classical boundary Pezzé  _et al._ (2018). Being composed of
neutral atoms in ultra-high vacuum, they couple very weakly to their
environment and thus show excellent coherence. Furthermore, they are reliably
initialized in pure states and can be manipulated and detected with high
fidelity by means of radio-frequency, microwave and optical fields. Multi-
particle entanglement in BECs has been demonstrated in the form of squeezed
spin states Pezzé  _et al._ (2018). These states are EPR entangled, which has
been verified by measuring spin correlations within a single cloud of atoms
Peise  _et al._ (2015); Fadel  _et al._ (2018); Kunkel  _et al._ (2018); Lange
_et al._ (2018), making them a promising starting point for an EPR experiment.

Figure 1:  Schematic of an EPR experiment with two particles (left) and with
two many-particle systems (right), where the spin degree of freedom is
considered. In both cases, the particles are entangled by interactions and
subsequently split into two different locations. In the case of the many-
particle system, the interactions produce multipartite entanglement, which is
inherited by the split systems in form of bipartite entanglement between their
collective spins.

In this work we perform an EPR experiment with two BECs, in close analogy with
the original Gedankenexperiment, see Fig. 1: We first prepare a BEC in a
squeezed spin state, in which all atoms are entangled with each other. We then
physically split it into two distinct condensates, which can be individually
manipulated and detected on the quantum level. The two condensates inherit the
entanglement from the initial state, resulting in correlated measurement
outcomes and allowing us to demonstrate the EPR paradox between two massive
many-particle systems.

##  II Experimental sequence

Figure 2:  Experimental sequence for spatial splitting and independent
coherent control of the two condensates. (a) Space-time schematic. Orange and
green color refer to condensate A and B, respectively, whereas magenta
represents the splitting pulse. Bottom left: main control parameters for
splitting and state manipulation. The timing is aligned with the sketch above,
but not to scale. On the right, typical absorption images are shown, from
which the atom numbers in all four states are determined. This corresponds to
a measurement of the selected components of the collective spins
$\hat{\mathbf{S}}^{A}$ and $\hat{\mathbf{S}}^{B}$. (b) Hyperfine levels of the
87Rb ground state with transitions for coherent splitting and manipulation of
$\hat{\mathbf{S}}^{A}$ and $\hat{\mathbf{S}}^{B}$. (c) Individual Rabi
oscillations of the two collective spins after spatial splitting. Shown are
the normalized spin components
${n_{A,B}=(N_{1}^{A,B}-N_{2}^{A,B})/(N_{1}^{A,B}+N_{2}^{A,B})}$. The
discontinuity in the horizontal axis denotes a change in time scale.

Our experiment starts by preparing a two-component 87Rb BEC of approximately
1400 atoms in a static magnetic trap on an atom chip Böhi  _et al._ (2009).
The two components refer to the internal degree of freedom: The atoms occupy
the hyperfine levels
${\left|1A\right\rangle\equiv\left|F=1,m_{F}=-1\right\rangle}$ and
${\left|2A\right\rangle\equiv\left|F=2,m_{F}=1\right\rangle}$ of the
electronic ground state, see Fig. 2(b), forming a collection of pseudo-spin
$1/2$ particles. Due to the nearly identical collisional interaction strength
and magnetic moments of these states, the atoms occupy a single spatial mode.
Hence, we can describe the BEC as a collective spin $\hat{\mathbf{S}}$, the
sum of the individual atomic spins Pezzé  _et al._ (2018). For instance, the
$z$ component of the collective spin is given by half the population
difference between the two states,
${\hat{S}_{z}=(\hat{N}_{1}-\hat{N}_{2})/2}$, which we can directly measure. By
coherently driving the two-photon transition between $\left|1A\right\rangle$
and $\left|2A\right\rangle$ with radio frequency and microwave fields (orange
in Fig. 2(b)), we are able to perform arbitrary spin rotations with high
fidelity.

We create entanglement between the atomic spins in the initial BEC through
interactions, by inducing one-axis twisting dynamics Kitagawa and Ueda (1993)
with a state-dependent potential on the atom chip Riedel  _et al._ (2010);
Ockeloen  _et al._ (2013); Schmied  _et al._ (2016); Fadel  _et al._ (2018).
After an interaction time of $\approx 40$ ms and a subsequent spin rotation,
the BEC is prepared in a squeezed spin state polarized along $\hat{S}_{x}$
with squeezed spin component $\hat{S}_{z}$ (see Fig. 2(a)), whose variance is
reduced by ${-7~{}\mathrm{dB}}$ compared to a coherent spin state Pezzé  _et
al._ (2018).

We then coherently split the spin-squeezed BEC into two spatially separated,
individually addressable condensates (see Fig. 2(a)). While coherent splitting
has been demonstrated in early experiments with BECs Andrews  _et al._ (1997),
the particular challenge we face here is to spatially split a two-component
condensate, while maintaining nearly perfect overlap and coherence between the
components. This is necessary so that high-fidelity coherent spin rotations
can be carried out after the splitting on each condensate separately.

To split the BEC, we first release it from the trap and accelerate it with a
magnetic field gradient to reduce the overall expansion time. We then
coherently transfer a fraction of the atoms to states with zero magnetic
moment by simultaneously driving the transitions
${\left|1A\right\rangle\rightarrow\left|1B\right\rangle\equiv\left|F=2,m_{F}=0\right\rangle}$
and
${\left|2A\right\rangle\rightarrow\left|2B\right\rangle\equiv\left|F=1,m_{F}=0\right\rangle}$
with a two-tone microwave pulse (magenta in Fig. 2). We choose to transfer
half of the populations (pulse duration ${t_{\pi/2}\approx 70~{}\mathrm{\mu
s}}$ for both transitions), which allows us to realize a nearly ideal 50:50
beam splitter for the atoms. A subsequent pulse of the magnetic field gradient
selectively accelerates the atoms in states $\left|1A\right\rangle$ and
$\left|2A\right\rangle$, spatially separating the system into two distinct
two-component BECs, which we call system $A$ (composed of states
$\left|1A\right\rangle$ and $\left|2A\right\rangle$) and system $B$ (composed
of states $\left|1B\right\rangle$ and $\left|2B\right\rangle$). Since the
overlap of the states in each system is preserved by the splitting mechanism,
we can describe them as two collective spins $\hat{\mathbf{S}}^{A}$ and
$\hat{\mathbf{S}}^{B}$. Both transition frequencies are insensitive to
magnetic field fluctuations, ensuring long coherence times Li  _et al._
(2020).

Once the two condensates are split, we can coherently drive the transitions
${\left|1A\right\rangle\leftrightarrow\left|2A\right\rangle}$ and
${\left|1B\right\rangle\leftrightarrow\left|2B\right\rangle}$ with distinct
radio-frequency and microwave signals, which allows us to perform arbitrary
spin rotations on $\hat{\mathbf{S}}^{A}$ and $\hat{\mathbf{S}}^{B}$
independently – as demonstrated by the individual Rabi oscillations shown in
Fig. 2(c). Subsequently, projective measurements of both collective spins are
carried out by resonant absorption imaging: The atoms in states with $F=2$
($\left|2A\right\rangle$ and $\left|1B\right\rangle$) are detected on a first
image by a resonant laser pulse. On a second image we detect atoms with $F=1$
($\left|1A\right\rangle$ and $\left|2B\right\rangle$), after they have been
optically pumped to $F=2$. Due to the large separation between the two BECs
($\approx 80~{}\mathrm{\mu m}$ at the time of the first image and $\approx
100~{}\mathrm{\mu m}$ at the time of the second), we can count the atoms
present in all four states separately and thus obtain a measurement of the
spin components ${\hat{S}_{z}^{A}=(\hat{N}_{1}^{A}-\hat{N}_{2}^{A})/2}$ and
${\hat{S}_{z}^{B}=(\hat{N}_{1}^{B}-\hat{N}_{2}^{B})/2}$ (see Fig. 2(a)). Other
spin components can be measured by coherently rotating the collective spins
before detection.

##  III EPR criteria

According to quantum mechanics, $\hat{\mathbf{S}}^{B}$ satisfies the
Heisenberg uncertainty relation ${\mathcal{E}_{\mathrm{Hei}}^{B}\equiv
4~{}\mathrm{Var}(\hat{S}_{z}^{B})\mathrm{Var}(\hat{S}_{y}^{B})/|\langle\hat{S}_{x}^{B}\rangle|^{2}\geq
1}$, which places a lower bound on the uncertainty product of
$\hat{S}_{z}^{B}$ and $\hat{S}_{y}^{B}$. It follows from the spin commutation
relations and applies to repeated measurements on identically prepared
systems. A similar relation holds for $\hat{\mathbf{S}}^{A}$. However, if the
two spins are entangled, their measurement outcomes are correlated, allowing
us to use measurement outcomes obtained on $\hat{\mathbf{S}}^{A}$ to predict
those obtained on $\hat{\mathbf{S}}^{B}$. The accuracy of this prediction is
quantified by the inferred variances
$\mathrm{Var}_{\mathrm{inf}}\left(\hat{S}_{z,y}^{B}\right)\equiv\mathrm{Var}\left(\hat{S}_{z,y}^{B}-\hat{S}_{z,y}^{B,\mathrm{inf}}\right)$,
which involve linear estimates $\hat{S}_{z,y}^{B,\mathrm{inf}}\equiv-
g_{z,y}\hat{S}_{z,y}^{A}+c_{z,y}$ of $\hat{S}_{z,y}^{B}$ using
$\hat{S}_{z,y}^{A}$. Here, $g_{z,y}$ and $c_{z,y}$ are real numbers that can
be chosen to optimize the prediction, i.e. to ensure that
$\langle\hat{S}_{z,y}^{B,\mathrm{inf}}\rangle=\langle\hat{S}_{z,y}^{B}\rangle$
and to minimize the inferred variances.

The EPR paradox is demonstrated if the product of the inferred variances of
two non-commuting spin components is lower than the associated Heisenberg
uncertainty bound Reid  _et al._ (2009), that is if the inequality

$\mathcal{E}_{\mathrm{EPR}}^{A\rightarrow
B}\equiv\frac{4~{}\mathrm{Var}_{\mathrm{inf}}\left(\hat{S}_{z}^{B}\right)\mathrm{Var}_{\mathrm{inf}}\left(\hat{S}_{y}^{B}\right)}{\left|\left\langle\hat{S}_{x}^{B}\right\rangle\right|^{2}}\geq
1$ (1)

is violated. Observing a violation of Eq. (1) challenges our classical notions
of locality and reality: it implies that an experimenter with access to a
spatially separated system $\hat{\mathbf{S}}^{A}$ can make better predictions
about $\hat{\mathbf{S}}^{B}$ than what is possible if one has full local
control over $\hat{\mathbf{S}}^{B}$ alone.

A related but weaker criterion exists for entanglement (non-separability)
Giovannetti  _et al._ (2003): Two spins $\hat{\mathbf{S}}^{A}$ and
$\hat{\mathbf{S}}^{B}$ are in an entangled state if the inequality

$\mathcal{E}\mathrm{{}_{Ent}}\equiv\frac{4~{}\mathrm{Var}\left(g_{z}\hat{S}_{z}^{A}+\hat{S}_{z}^{B}\right)\mathrm{Var}\left(g_{y}\hat{S}_{y}^{A}+\hat{S}_{y}^{B}\right)}{\left(\left|g_{z}g_{y}\right|\left|\left\langle\hat{S}_{x}^{A}\right\rangle\right|+\left|\left\langle\hat{S}_{x}^{B}\right\rangle\right|\right)^{2}}\geq
1$ (2)

is violated. $\mathcal{E}_{\mathrm{EPR}}^{A\rightarrow B}$ is always larger or
equal than $\mathcal{E}_{\mathrm{Ent}}$, reflecting the fact that an
observation of the EPR paradox requires stronger correlations than a
demonstration of entanglement Wiseman  _et al._ (2007).

##  IV Observation of the EPR paradox

Figure 3:  Spin correlations between the two BECs and illustration of the
inferring mechanism. The grey dots are individual data points of simultaneous
measurements of spin components $S_{z}^{A}$ and $S_{z}^{B}$ (left plot) and
$S_{y}^{A}$ and $S_{y}^{B}$ (right plot) of the two systems. Data points for
$S_{y}^{B}$ are corrected for the phase shift due to the measured trigger
jitter of the microwave generator (Appendix D). The blue histograms are their
marginal distributions, $2\sigma$ intervals are indicated by blue dotted
lines. The correlations of measurement results ($2\sigma$ covariance ellipses
in blue) allow one to infer measurement results of one system from the other.
This reduces the variance of the prediction as shown by the histograms and the
reduced $2\sigma$ intervals in red. For comparison, the $2\sigma$ variance
ellipses of ideal non-entangled states with the same number of atoms are shown
in yellow.

To evaluate these criteria, we perform measurements of either the $x$, $y$, or
$z$ spin components simultaneously on both systems, repeating the experiment
many times with identical preparation. The measurement basis is selected by
rotating the two collective spins individually when they are completely
separated, at a distance of more than $14~{}\mathrm{\mu m}$, see Fig. 2(a).
The outcome of every measurement of $\hat{S}_{y}^{A,B}$ or $\hat{S}_{z}^{A,B}$
(approx. 1600 repetitions each) is represented by a point in the correlation
plots of Fig. 3. Strong correlations are visible between $A$ and $B$ in both
spin components. The inference corresponds to an affine transformation (e.g.
$S_{z}^{B}\mapsto S_{z}^{B}+g_{z}{S}_{z}^{A}-c_{z}$), which reduces the
variance, as can be seen from the marginal histograms and $2\sigma$ intervals
of the raw data (blue) and the transformed quantities (red). Splitting a
squeezed spin state with a beam-splitter-like process retains part of the
reduced (increased) fluctuations in the squeezed (anti-squeezed) component.
This can be seen from the comparison of the variance ellipse of a separable
coherent spin state with the same atom number (yellow lines in Fig. 3) to the
covariance ellipse of the data (blue).

The measurements of $\langle\hat{S}_{x}^{A,B}\rangle$ quantify the lengths of
the collective spins $\hat{\mathbf{S}}^{A,B}$ and thus determine the
Heisenberg uncertainty bounds. Their values normalized to the atom numbers
correspond to the interferometric contrast, which is a measure of the overall
coherence of the process. We obtain ${96\%}$ contrast for both
$\hat{\mathbf{S}}^{A}$ and $\hat{\mathbf{S}}^{B}$, comparable to the contrast
we measure without splitting the condensate, indicating that the reduction is
mostly due to spin squeezing Wineland  _et al._ (1994) and imperfections in
the initial state preparation, not due to the splitting process.

Combining all measurements we obtain

$\displaystyle\mathcal{E}_{\mathrm{EPR}}^{A\rightarrow B}$
$\displaystyle=0.81\pm 0.03~{},$
$\displaystyle\mathcal{E}_{\mathrm{EPR}}^{B\rightarrow A}$
$\displaystyle=0.77\pm 0.03~{},$ $\displaystyle\mathcal{E}_{\mathrm{Ent}}$
$\displaystyle=0.35\pm 0.02~{},$ $\displaystyle\mathcal{E}_{\mathrm{Hei}}^{B}$
$\displaystyle=10.2\pm 0.4~{},$ $\displaystyle\mathcal{E}_{\mathrm{Hei}}^{A}$
$\displaystyle=9.2\pm 0.5~{},$

demonstrating both entanglement and the EPR paradox between the condensates
$A$ and $B$. The Heisenberg uncertainty products
$\mathcal{E}_{\mathrm{Hei}}^{A}$ and $\mathcal{E}_{\mathrm{Hei}}^{B}$ are
larger than unity due to technical noise. The EPR criteria are much smaller
due to the variance reduction by inferring. We note that all criteria are
determined without subtraction of any technical noise and that we observe the
paradox both ways, inferring from $A\rightarrow B$ and from $B\rightarrow A$.

##  V Individual control of the BECs

Figure 4:  Individual manipulation of the two entangled BECs on the quantum
level. $\hat{\mathbf{S}}_{B}$ is rotated by an angle $\theta$ around the $x$
axis with respect to $\hat{\mathbf{S}}_{A}$, as sketched in the inset on the
top left. The red filled circles represent $\mathcal{E}_{\mathrm{Ent}}$, red
empty circles $\mathcal{E}_{\mathrm{EPR}}^{B\rightarrow A}$, and blue squares
$\mathcal{E}_{\mathrm{Hei}}^{A}$. The EPR paradox (entanglement) is observed
if $\mathcal{E}_{\mathrm{EPR}}^{B\rightarrow A}$
($\mathcal{E}_{\mathrm{Ent}}$) falls below the dashed line at unity. The
insets at the bottom show the spin correlations similar to Fig. 3, with
$\hat{S}_{z}$ on top and $\hat{S}_{y}$ at the bottom, for $\theta=0,\;\pi/2$
and $\pi$.

Although we rotate $\hat{\mathbf{S}}_{A}$ and $\hat{\mathbf{S}}_{B}$
individually, the results presented so far were obtained by measuring the two
spins in the same bases. Since many applications of EPR entanglement require
performing different measurements on the two systems, we demonstrate that our
experiment is able to maintain the entanglement between $\hat{\mathbf{S}}_{A}$
and $\hat{\mathbf{S}}_{B}$ in this process. Figure 4 shows the results of
measurements in which $\hat{\mathbf{S}}_{B}$ is rotated by an angle $\theta$
around the $x$ axis with respect to $\hat{\mathbf{S}}_{A}$, as sketched in the
inset on the top left. The case $\theta=0$ (leftmost points in Fig. 4)
corresponds to the same configuration as above, i.e. simultaneous measurements
of the same spin component. As $\theta$ increases, the correlations decrease
and almost completely vanish for $\theta=\pi/2$, where orthogonal spin
components are measured on the two condensates. This can be directly seen in
the correlation plots at the bottom of Fig. 4. For $\theta=\pi$ the
correlations reappear, but with opposite sign, since the spin components are
now anti-aligned. The parameters $\mathcal{E}_{\mathrm{Ent}}$ and
$\mathcal{E}_{\mathrm{EPR}}^{B\rightarrow A}$ fall again below unity,
demonstrating that the manipulation preserves the quantum correlations.

In the setting $\theta=\pi/2$ our experiment realizes a situation discussed by
Schrödinger Schrödinger (1935b), where the values of two complementary
properties of system $A$ are apparently obtained in a single experimental run:
one ($\hat{S}_{z}^{A}$) by direct measurement on $A$ and the other
($\hat{S}_{y}^{A}$) by exploiting the strong correlations to infer its value
from the simultaneous measurement on $B$. Under the local realist assumptions
that measurements reveal pre-existing properties of a system and that
simultaneous measurements on spatially separated systems do not disturb each
other, the restrictions imposed by the Heisenberg uncertainty relation could
thus be overcome Schrödinger (1935b). Today, however, we know that local
realism is inconsistent with the results of increasingly rigorous experimental
tests of Bell inequalities Freedman and Clauser (1972); Lamehi-Rachti and
Mittig (1976); Aspect  _et al._ (1982); Hensen  _et al._ (2015); Giustina  _et
al._ (2015); Shalm  _et al._ (2015); Rosenfeld  _et al._ (2017). In the spirit
of Peres’ statement that “unperformed experiments have no results” Peres
(1978), we should thus refrain from inferring a value for $\hat{S}_{y}^{A}$ if
it is not actually measured on system $A$.

##  VI Conclusions

Our experiment demonstrates the EPR paradox, a cornerstone of quantum physics,
between two systems with a large number of massive particles. This shows that
the conflict between quantum mechanics and the classical principles of
locality and realism does not disappear in systems of increasing size and
complexity, at least up to the level demonstrated here. The key to this result
is the high degree of coherence of our splitting technique together with the
ability to perform high-fidelity coherent rotations of the individual systems
after splitting. The ability to measure the two collective spins in different
bases is also an essential prerequisite for a future Bell test, which in
addition requires non-Gaussian measurements or state preparation, e.g. single-
atom resolving detection Oudot  _et al._ (2019).

EPR entanglement is a valuable resource for both quantum metrology and quantum
information processing. The noise reduction gained from the inference in Eq.
(1), quantified by the difference between the Heisenberg products and the EPR
criteria, translates to a metrological enhancement that can be exploited in
quantum sensing Yadin  _et al._ (2021); Fadel  _et al._ (2022). One can, e.g.,
use one system as a small sensor to probe fields and forces with high spatial
resolution and the other one as a reference to reduce the quantum noise of the
first system. Furthermore, EPR entanglement is the resource that guarantees
the efficacy of certain quantum information protocols such as quantum
teleportation, entanglement swapping, one-side device-independent quantum key
distribution, or randomness certification Reid  _et al._ (2009); Cavalcanti
and Skrzypczyk (2017); Uola  _et al._ (2020). For this resource to be useful,
however, the systems need to be spatially separated and individually
addressable, which we realize here for the first time in massive many-particle
systems. Therefore, in addition to its foundational significance, our work
demonstrates the necessary ingredients to exploit EPR entanglement in many-
particle systems as a resource.

###### Acknowledgements.

The authors acknowledge contributions to improving the experimental apparatus
by Simon Josephy and Clara Piekarsky. This work was supported by the Swiss
National Science Foundation (Grant No. 197230).

##  Appendix A Addressability and strength of the transitions

In our experiment, multiple transitions in the hyperfine-split ground state of
87Rb are coupled with radio-frequency and microwave driving fields in order to
coherently split and manipulate the two-component BEC. To preserve the
entanglement in this process, it is essential to prevent loss from the system
by spurious driving of undesired transitions. This requires a careful choice
of parameters since several of the transitions involved are nearly degenerate
and differ only by quadratic Zeeman shifts, which are of the same order of
magnitude as the Rabi frequencies (1-10 kHz).

In the experimental sequence up until the splitting, the populated states and
applied driving frequencies are such that they do not lead to any undesired
couplings. However, the driving fields used for the splitting pulse and for
the subsequent rotations of $\hat{\mathbf{S}}^{A}$ are close to resonance with
other relevant transitions, see Fig. 5. It is possible in principle to
suppress the undesired transitions by polarization selection rules, using only
driving fields with the correct circular polarization. However, since the
condensate is close to the metallic atom chip surface, this is challenging to
attain in our experiment and we observe that the driving fields contain all
polarization components. We therefore have to ensure the selectivity by
careful choice of driving field frequencies and strength.

In the case of the splitting pulse (Fig. 5(a)), the desired transition
${\left|1A\right\rangle\leftrightarrow\left|1B\right\rangle}$ is close to
resonance with
${\left|2B\right\rangle\leftrightarrow\left|F=2,m_{F}=-1\right\rangle}$
($9~{}\mathrm{kHz}$ detuning) and the desired
${\left|2A\right\rangle\leftrightarrow\left|2B\right\rangle}$ with
${\left|1B\right\rangle\leftrightarrow\left|F=1,m_{F}=1\right\rangle}$
($9~{}\mathrm{kHz}$ detuning). These additional close-to-resonant transitions
can cause atoms to be transferred from system $B$ further to states where they
are lost during the application of the magnetic gradient pulse due to the
different magnetic moments. This loss mechanism is mitigated by the fact that
states $\left|1B\right\rangle$ and $\left|2B\right\rangle$ are initially
unpopulated and become populated only during the splitting $\pi/2$ pulse. In
addition to this, since we only need to transfer half of the population with
the splitting pulse, we can choose a detuning to make the undesired
transitions further off resonant. We carefully choose a combination of
detuning and small enough Rabi frequency for this pulse for which we observe
no losses to the undesired states, corresponding to a detuning of
$2.2~{}\mathrm{kHz}$ and a pulse duration ${t_{\pi/2}^{\mathrm{split}}\approx
70~{}\mathrm{\mu s}}$.

To rotate $\hat{\mathbf{S}}^{A}$ after the splitting, the states
$\left|1A\right\rangle$ and $\left|2A\right\rangle$ are coupled by a two-
photon transition via the intermediate state $\left|F=2,m_{F}=0\right\rangle$
(see Fig. 5(b)). The same driving fields are near resonance with the
transition ${\left|1B\right\rangle\leftrightarrow\left|2B\right\rangle}$,
driven as a two-photon transition with intermediate states
$\left|F=2,m_{F}=-1\right\rangle$ and $\left|F=1,m_{F}=-1\right\rangle$
($10~{}\mathrm{kHz}$ detuning). Since all transitions involved are two-photon
transitions with correspondingly weak effective two-photon Rabi frequencies,
this two-photon detuning can be exploited to ensure good selectivity of the
pulse. The desired transition
$\left|1A\right\rangle\leftrightarrow\left|2A\right\rangle$ is driven on two-
photon resonance with $\pi/2$ time of ${t_{\pi/2}^{A}\approx 960~{}\mathrm{\mu
s}}$, ensuring that the two-photon detuning of the undesired transitions is
sufficient to render spurious rotations of $\hat{\mathbf{S}}^{B}$ negligible.

Figure 5:  Level structure of the 87Rb hyperfine split ground state in the
linear Zeeman regime (not to scale). The color coding is consistent with Fig.
2: Orange indicates system $A$, green system $B$, and magenta the splitting
transitions. The desired transitions are indicated by solid lines, the
undesired close-to-resonance transitions are represented by dashed lines. (a)
Transitions involved in the splitting of the condensate into systems $A$ and
$B$. (b) Transitions involved in driving spin rotations of
$\hat{\mathbf{S}}^{A}$ after splitting.

##  Appendix B Imaging

We detect the atoms by resonant absorption imaging Ketterle  _et al._ (1999),
which is based on comparing the shadow cast by the atoms in a resonant laser
beam (absorption image) with a reference image taken without atoms. Since the
criteria Eq. (1) and Eq. (2) depend on the atom number, the accurate
calibration of the imaging system is crucial. This is ensured with three types
of calibration measurements: First of all, the conversion from absorbed light
to atom number is made independent of the laser intensity following the method
described in Ref.Reinaudi  _et al._ (2007). Second, the detectivity of the
four states is calibrated by ensuring that the detected total atom number is
independent of the relative population in the four states when driving Rabi
oscillations between them. Last, the total atom number is calibrated by
observing the projection noise of a coherent spin state Muessel  _et al._
(2013) prepared in an equal superposition of $\left|1A\right\rangle$ and
$\left|2A\right\rangle$.

Since the calibration relies on projection noise measurements on equal
superpositions of the involved states, it is most accurate for the
measurements of $\hat{S}_{y}$ and $\hat{S}_{z}$ that enter the inferred
variances in the EPR and entanglement criteria. In the measurements of
$\hat{S}_{x}$, nearly all atoms occupy one state, and we observe a slight
decrease on the order of $3\%$ in the detected total atom number. Since we can
exclude atom loss, we attribute this effect to an effective decrease in
detectivity due to the increased density of the cloud. To correct for this
effect, we determine $\langle\hat{S}_{x}^{A,B}\rangle=\langle
n_{x}^{A,B}\rangle\langle N_{y,z}^{A,B}\rangle/2$, i.e. we measure the
relative atom number
${n_{x}^{A,B}=(N_{1,x}^{A,B}-N_{2,x}^{A,B})/(N_{x}^{A,B})}$ corresponding to
the interferometric contrast, and multiply by $\langle
N_{y,z}^{(A,B)}\rangle$, the total atom number detected in the corresponding
measurements along $y$ and $z$.

In order to minimize detection noise, we choose the region of interest for
counting atoms to be as small as possible, while still including nearly all of
the atomic signal. We do this by selecting elliptical regions that include
$\approx 97\%$ of the total detected atom number. To avoid any artificial
noise reduction due to the small discarded signal (see the supplementary
materials of Ref. Fadel  _et al._ (2018) for a discussion of this effect), we
perform the atom number calibration after the region of interest has been
determined.

Since detection noise is mainly composed of photon shot noise, we can reduce
it by choosing longer imaging pulses, as long as atomic diffusion during the
imaging pulse doesn’t increase the cloud size on the image too much. We find
the lowest imaging noise for an imaging pulse duration of $70~{}\mathrm{\mu
s}$. During this time, the radiation pressure accelerates the atoms, inducing
a relevant Doppler shift. To maintain the resonance throughout the process, we
chirp the frequency of the imaging pulses.

We use a fringe removal algorithm for optimized reference images based on
linear combination of actual reference images Ockeloen  _et al._ (2010).
Besides minimizing the effect of interference fringes on the reference images,
this procedure further reduces the contribution of photon shot noise of the
reference images.

With these optimizations the resulting photon shot noise corresponds to about
$\pm 3$ atoms in every state.

##  Appendix C Data analysis

To determine the EPR and entanglement criteria, we take 4500 measurements
overall within 42 hours of measurement time. We subdivide the data into blocks
for which we evaluate the criteria separately. Each block consists of hundred
$\hat{S}_{z}$, hundred $\hat{S}_{y}$, and twenty $\hat{S}_{x}$ measurements.
The measurements along $x$ are performed ten times along the positive and
negative spin directions, respectively, to reduce bias from a potential
difference in the detectivity. This procedure renders the analysis robust
against the effect of small, slow drifts of the experimental conditions during
the many hour long measurement run, while the quantum noise of the atoms,
which occurs in each shot of the experiment, is unchanged. The values of the
EPR and entanglement criteria listed in the main text are the averages of the
values obtained for the individual blocks. We verified that analyzing the
whole data set in one block does not change the conclusions of our paper, in
this case we obtain $\mathcal{E}_{\mathrm{EPR}}^{A\rightarrow B}=0.87\pm
0.04~{}$ and $\mathcal{E}_{\mathrm{EPR}}^{B\rightarrow A}=0.82\pm 0.04$.

##  Appendix D Radio-frequency and microwave signal generation and local
oscillator phase

The microwave and radio-frequency driving fields are generated with several
commercial function generators. To allow phase control and simultaneous
generation of the two frequencies needed for the splitting of the two-
component condensate, we use an IQ -modulated microwave signal (microwave
generator: Rohde & Schwarz SGS100A). The modulation signal is generated by a
two-channel arbitrary waveform generator (Keysight 33522B). The radio-
frequency signal needed for the two-photon transition of system $A$ is
generated with a separate radio-frequency source (Photonics Technologies VFG
150).

The sequence of pulses is generated by programming lists of frequencies,
amplitudes, and phases that are executed upon trigger and by gating the
respective function generator signals. The digital signals used for triggering
and gating are derived from a common sampling clock. All function generators
are referenced to a GPS-disciplined 10 MHz crystal oscillator (Stanford
Research Systems FS752), providing long term frequency stability. However,
absolute phase stability is not ensured since not all devices share a common
sampling clock.

In a standard Ramsey experiment, a constant phase offset of the local
oscillator does not matter, since the initial coupling of the two states and
the final readout of the phase is done with the same local oscillator and an
offset of its phase drops out. This scenario applies in our case to the spin
$A$, composed of the states $1A$ and $2A$. The two-photon transition between
these two states is driven by the microwave generator, the IQ modulator and
the RF source. A constant phase offset in any of these devices will not affect
the readout of the phase of spin $A$ and thereby $\hat{S}^{A}_{y}$.

For spin $B$ the situation is more complicated. The phase of system $B$, and
thereby $\hat{S}^{B}_{y}$, is determined by three processes. Initially, spin
$B$ inherits the phase of spin $A$, which as above is determined by the phase
of the microwave generator, the IQ modulator and the RF source. The second
process is the actual splitting in spin space by the two tone microwave
signal, which is generated by the microwave generator and the IQ modulator.
The same devices are also generating the coupling for the last process, which
is the final rotation of spin $B$. In contrast to spin $A$, the RF source only
appears in the first process and therefore a phase offset of this source with
respect to the other devices does not drop out in the phase sensitive
measurement of $\hat{S}^{B}_{y}$.

Since the RF signal and the IQ modulator signal are not derived from the same
sampling clock, fluctuations in the relative triggering delay of the devices
by $\delta t$ will lead to phase changes of $\omega_{\mathrm{RF}}\delta t$ in
system $B$, where $\omega_{\mathrm{RF}}\approx 2\pi\times 1.79$ MHz. We
measure typical timing fluctuations of $\delta t$ on the order of 4 ns. The
fluctuations are actually small when compared to the large variance observed
in the measurement of $\hat{S}_{y}^{B}$, since this is the anti-squeezed
direction. However, they are relevant for the value of the much smaller
inferred variance.

These fluctuations are a technical limitation that can be resolved in the
future with a different setup of generators for the driving fields. In the
present experiment, we correct them by directly measuring $\delta t$. We
determine $\delta t$ in each shot of the experiment from a measurement of the
phase of the radio-frequency generator with respect to the starting time of
the IQ modulator sequence with an oscilloscope. This classical information can
then be used for a better estimation of
$\hat{S}^{B}_{y}=\hat{S}^{B}_{y,\mathrm{measured}}+g_{\delta t}\delta t$ with
an optimized gain parameter $g_{\delta t}$. In Fig. 3, the data points of
$\hat{S}^{B}_{y}$ are corrected in this way. Furthermore, we use this
correction in the evaluation of the EPR criterion Eq. (1). We stress that
while this correction based on additional classical information can reduce
classical noise, it cannot lead to a violation of the EPR inequality, which
can only be achieved by sufficiently strong entanglement between the two
systems Reid  _et al._ (2009). The correction based on this classical
information is not applied in the evaluation of the entanglement criterion.

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