## Splitting the entanglement

When particles in a quantum mechanical system are entangled, a measurement performed on one part of the system can affect the results of the same type of measurement performed on another part—even if these subsystems are physically separated. Kunkel *et al.*, Fadel *et al.*, and Lange *et al.* achieved this so-called distributed entanglement in a particularly challenging setting: an ensemble of many cold atoms (see the Perspective by Cavalcanti). In all three studies, the entanglement was first created within an atomic cloud, which was then allowed to expand. Local measurements on the different, spatially separated parts of the cloud confirmed that the entanglement survived the expansion.

## Abstract

Modern quantum technologies in the fields of quantum computing, quantum simulation, and quantum metrology require the creation and control of large ensembles of entangled particles. In ultracold ensembles of neutral atoms, nonclassical states have been generated with mutual entanglement among thousands of particles. The entanglement generation relies on the fundamental particle-exchange symmetry in ensembles of identical particles, which lacks the standard notion of entanglement between clearly definable subsystems. Here, we present the generation of entanglement between two spatially separated clouds by splitting an ensemble of ultracold identical particles prepared in a twin Fock state. Because the clouds can be addressed individually, our experiments open a path to exploit the available entangled states of indistinguishable particles for quantum information applications.

The progress toward large ensembles of entangled particles is pursued along two different paths. In a bottom-up approach, the precise control and characterization of small systems of ions, atoms in optical lattices, and photons is pushed toward increasingly large system sizes, reaching entangled states of 14 ions (*1*), 4 atomic lattice sites (*2*), or 10 photons (*3*). Complementary, large numbers of up to 3000 mutually entangled ultracold atoms (*4*–*6*) can be generated, for which the state characterization is advanced top-down toward resolving correlations on the single-particle level. Because the atoms cannot be addressed individually, ultracold atomic ensembles are controlled by global ensemble parameters, such as the total spin. Ideally, the atoms are indistinguishable, either with respect to the observable, such as the spin in hot vapor cells (*7*), or in all quantum numbers in Bose-Einstein condensates (BECs) (*8*–*13*). Is it possible to make these particles distinguishable—and addressable—again, while keeping the high level of entanglement?

The generation of entanglement in these systems is deeply connected with the fundamental indistinguishability of the particles (*14*). For example, two indistinguishable bosons 1 and 2, which are prepared in two independent modes a and b, are described by an entangled triplet state , owing to bosonic symmetrization. Although this type of entanglement seems to be artificial, the state presents a resource for violating a Bell inequality (*15*). Equivalently, an ensemble containing the same number of distinguishable spin-up and spin-down atoms is not necessarily entangled, whereas a twin Fock state, the corresponding ensemble with indistinguishable bosons, exhibits full many-particle entanglement (*16*, *17*). This form of entanglement is directly applicable for atom interferometry beyond the standard quantum limit (*11*). However, most quantum information tasks require an individual addressing of subsystems. Despite the experimental progress in entanglement creation in BECs—including the demonstration of Einstein-Podolsky-Rosen (EPR) correlations (*18*) and Bell correlations (*19*–*21*), as well as the demonstration of strongly correlated momentum states (*22*–*24*)—a proof of entanglement between spatially separated and individually addressable subsystems has not been realized so far. The possible applications of such a resource range from spatially resolved quantum metrology to tests for fundamental sources of decoherence or Bell tests of quantum nonlocality.

Here, we report the creation of particle entanglement in an ensemble of up to 5000 indistinguishable atoms and prove entanglement between two spatially separated clouds.

Our experiments started with the preparation of a ^{87}Rb BEC in a crossed-beam optical dipole trap. The ensemble of 20,000 particles was transferred to the hyperfine level *F* = 1, *m _{F}* = 0. Spin-changing collisions created up to 5000 entangled atoms in the Zeeman levels

*m*= ±1, which resided in a spatially excited mode of the dipole trap (

_{F}*25*,

*26*) (Fig. 1). The output state consisted of a superposition of twin Fock states with varying total number of atoms

*N*=

*N*

_{+1}+

*N*

_{–1}. Each twin Fock state is characterized by an equal number of atoms

*N*

_{+1}=

*N*

_{–1}in the two Zeeman levels

*m*= ±1 (

_{F}*11*). Because the total number of particles

*N*was measured during detection, the system was well described by a single twin Fock state with a definite particle number. Self-similar expansion (

*26*) allowed for an imaging of the undistorted but magnified density profiles. An inhomogeneous magnetic field separated the atoms in order to record the atomic densities for each Zeeman level.

The spatially excited mode of the ensembles in *m _{F}* = ±1 provides a natural splitting into a left and right cloud along a line of zero density. Hence, we divided the initial twin Fock state into two spatially separated parts, (left side) and (right side). This process can be described as a beam splitter of the initially populated antisymmetric input mode . The splitting introduces additional quantum noise caused by a coupling with the (empty) symmetric input mode . In principle, an ideal twin Fock state shows a maximal entanglement depth (

*16*); all atoms that make up a twin Fock state are entangled with one another. Therefore, any splitting results in the appearance of quantum correlations between the clouds. Such a creation of entanglement is analogous to so-called vacuum-class entanglement (

*27*) in optics, in which a single-mode squeezed laser beam is coupled to a vacuum state on a beam splitter, creating entanglement between the two output modes (

*28*).

The resulting quantum correlations can be visualized on the multiparticle Bloch sphere (Fig. 2A). Here, the atoms in the levels *m _{F}* = ±1 are represented by spin-½ particles, whose spins

**j**

^{(}

^{k}^{)}sum up to a total spin

**J**. On the Bloch sphere, the lines of latitude represent the number imbalance between the two levels, and the lines of longitude represent the phase difference. An ensemble in a twin Fock state can be depicted as a ring on the Bloch sphere, characterized by a vanishing imbalance,

*J*

_{z}= (

*N*

_{+1}–

*N*

_{–1})/2 = 0, and an undetermined phase difference.

If we divide the cloud, the collective spins **J**^{(a)}, **J**^{(b)} of the two parts have to sum up to the original collective spin of the full ensemble **J** = **J**^{(a)} + **J**^{(b)}. Therefore, the *z* components of the collective spins are perfectly anticorrelated . Furthermore, because the spin length is maximal, the collective spins of the two parts have to point in a similar direction in the *x*-*y* plane and thus have similar azimuthal angles ϕ^{(a)} ≈ ϕ^{(b)}.

Hence, if the particle number difference of cloud b is measured to yield , the conditioned state of cloud a satisfies . If the value is measured on cloud b, the state of cloud a has to fulfill ; if the value is measured on cloud b, the state of cloud a has to fulfill . The different possible measurements on cloud b yield precise predictions for the measurement results of cloud a, which cannot be explained by a single quantum state that is independent of the chosen type of measurement. In this sense, the described system is analogous to the thought experiment by Einstein, Podolsky, and Rosen (*29*), in which entanglement is witnessed by the variances of the predictions (*30*, *31*). Therefore, it should be possible to detect entanglement between the spatially separated parts of a twin Fock state.

To this end, we derived an entanglement criterion that optimally exploits the described spin correlations (*32*). The spin correlations are represented by prediction operators and for *m* = *x*, *y*. Here, the *x* and *y* components are normalized so that the optimal value is 1, according to , with and the spin length *j _{n}* =

*N*/2 for

_{n}*n*= a, b. We arrive at a simple separability criterion(1)where . Any separable state, including mixtures of product states of the form with a fluctuating number of particles, fulfills this inequality. A violation of this criterion indicates that the state is inseparable and therefore entangled. For perfectly symmetric states, as we would expect in the ideal case, the right-hand side (RHS) of Eq. 1 is equal to 1. Any deterioration from perfect symmetry is quantified by and . Containing the characteristic product of the prediction uncertainties, our inequality for three-dimensional spins has similarities to the famous two-dimensional continuous-variable entanglement criteria (

*31*,

*33*,

*34*), specifically to the EPR criterion (

*33*). It presents a general entanglement criterion, which is particularly sensitive for a spatially separated twin Fock state.

An application of criterion in Eq. 1 requires an evaluation of the spin correlations between the two clouds a and b. The measurement results for and are readily obtained from the absorption images. The measurement of the orthogonal direction is performed with a sequence of resonant microwave pulses before the particle number detection (*32*). The pulses lead to an effective rotation of the spins by π/2. Because the microwave phase is independent of the atomic phases, the rotation yields a measurement of the spin component *J*_{⊥} along an arbitrary angle in the *x*-*y* plane. Because our quantum state is symmetric under rotations around the *z* axis, both owing to the initial symmetry and the influence of magnetic field noise, the measured distributions of *J*_{⊥} can be identified with both *J _{x}* and

*J*.

_{y}The histograms of *J _{z}* and

*J*

_{⊥}/

*j*are shown in Fig. 2, B and C, for a mean total number of 3460 particles in both clouds together. The

*J*data show the expected anticorrelation, whereas the

_{z}*J*

_{⊥}measurements are strongly correlated. The strength of these correlations can be quantified by evaluating the prediction uncertainties—the width of the distributions in the diagonal directions in the histograms: and .

The prediction variance is presented in Fig. 3A as a function of the total number of atoms. The shown fluctuations, obtained by subtracting independent detection noise (*32*), remain well below the shot-noise limit and are equivalent to a number squeezing of –11.0(5) dB. The orthogonal quantities (Fig. 3B) are slightly influenced by small position fluctuations of the clouds, increasing the standard deviation by a factor of 1.8 above shot noise. The quantities and are shown in Fig. 3C. We obtained a value of up to 0.94, close to the ideal value of 1, indicating a sufficiently clean preparation of an almost symmetric state.

From these results, we could test a violation of the separability criterion. In Fig. 4B, the orange diamonds correspond to an evaluation of the RHS of the criterion (Eq. 1), which would ideally be 1 (black line). For most values of *N*, the left-hand side (LHS), represented in Fig. 4B by the green circles, is well below the RHS, signaling entanglement in the system. At the best value of the total number of 3460 atoms, the experimental data violate the separability criterion by 2.8 standard deviations. Therefore, our measurements cannot result from classical correlations and prove the generation of entanglement between spatially separated clouds from particle-entangled, indistinguishable atoms.

The presented creation of spatial entanglement opens a path to create highly entangled quantum states in spatially separated modes for a broad range of applications. This includes the field of quantum metrology, where our results can be applied in order to obtain an improved spatial resolution—for example, in the sensing of electromagnetic fields. Owing to the possibility of addressing the separated modes—for example, with Raman beams or spatially modulated light shifts—the created states are a resource for quantum information protocols. For example, it presents a resource for the synthesis of any pure symmetric state with only single-particle projective measurements (*35*, *36*). Moreover, the spatially separated twin Fock state carries nonlocal EPR and Bell correlations. The presented experimental scheme can be extended to a multiparticle Bell test of quantum mechanics with the addition of local addressing of the two modes and a single-particle–resolving atom counting (*37*). Last, if the created twin Fock state is fully separated into single atoms—for example, in an optical lattice—all constituent particles become individually addressable.

Complementary to our work, in (*38*), spatially distributed multipartite entanglement has been observed, whereas spatial entanglement patterns were observed in (*39*).

## Supplementary Materials

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**C.K. thanks M. Cramer for the discussion at the 589. Heraeus seminar that led to the initial idea for the experiments. C.K. thanks A. Smerzi, L. Santos, and W. Ertmer for regular inspiring discussions and a review of the manuscript.

**Funding:**This work was supported by the European Union (European Research Council Starting Grant 258647/GEDENTQOPT, CHIST-ERA QUASAR, COST Action CA15220, European Research Council Consolidator Grant 683107/TempoQ and QuantERA CEBBEC); the Spanish Ministry of Economy, Industry and Competitiveness and the European Regional Development Fund FEDER through grant FIS2015-67161-P (MINECO/FEDER); the Basque government (project IT986-16); the National Research, Development, and Innovation Oﬃce (NKFIH) (grant K124351); the Deutsche Forschungsgemeinschaft (DFG) (Forschungsstipendium KL 2726/2-1 and project KL2421/2-1); the FQXi (grant FQXi-RFP-1608); and the Austrian Science Fund (FWF) through the START project Y879-N27. We also acknowledge support from DFG through RTG 1729 and CRC 1227 (DQ-mat), project A02.

**Author contributions:**K.L., J.P., B.L., I.K., and C.K. performed the experiments. G.V., I.A., M.K., B.L., C.K., and G.T. contributed to the development of the entanglement criterion. All authors discussed the results and contributed to the data analysis and the writing of the manuscript.

**Competing interests:**The authors declare no competing financial interests.

**Data and materials availability:**The experimental data presented in the figures are available at https://doi.org/10.5281/zenodo.1186798. Other materials may be requested from the corresponding author.