A strong Bell correlation witness between spatially separated pairs of

D. K. Shin1, B. M. Henson1, S. S. Hodgman1, T. Wasak2, J. Chwedenczuk´ 3, and A. G. Truscott1∗ 1Research School of Physics and Engineering, Australian National University, Canberra 0200, Australia 2Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany and 3Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL02093 Warszawa, Poland (Dated: November 26, 2018) The violation of a Bell inequality is a striking demonstration of how contradicts local 1 realism [1]. Although the original argument was presented with a pair of - 2 particles, so far Bell inequalities have been shown to be violated using entangled pairs of photons [2,3], with recent measurements closing all possible loopholes in such a scheme [4–6]. Equivalent demonstrations using massive particles have proven to be much more challenging, generally relying on post-selection of data [7,8] or measuring an entanglement witness that relies on quantum mechanics [9]. Here, we use a collision between two Bose-Einstein condensates to generate the momentum-spin entangled pairs of ultracold helium atoms. We show that a maximally entangled Bell triplet state results and report a direct observation of a strong Bell correlation witness. Based on the high degree of entanglement and the controllability of ultracold atomic systems, extensions to this scheme would allow a demonstration of nonlocality with massive entangled pairs, following Bell’s original idea. Other applications include the demonstration of the Einstein-Podolsky-Rosen paradox [10], quantum metrology and tests of phenomena from exotic theories sensitive to such systems including gravitational decoherence [11] and quantum gravity.

The basic scheme of a Bell test, originally proposed by John ence [11]. Bell [1], involves measuring correlated detector events be- One promising experimental system for demonstrating such tween a pair of spin- 1 particles arriving at spatially separated 2 entanglement is ultracold atoms [13, 14]. Indeed, many char- detectors labelled A and B. Bell envisioned using an entan-√ acteristic quantum effects have been realised recently in ul- |Ψ−i = (|↑i ⊗ |↓i − |↓i ⊗ |↑i ) / 2 gled A B A B , tracold atomic systems ranging from the generation of non- where the states |↑i and |↓i are the spin eigenstates for each classical atomic pairs [15], Hong-Ou-Mandel interference particle. The particles are subject to independent rotations and [16], and the observation of spatially separated entanglement then the spins of the pairs are measured. If the correlation [17] and Einstein-Podolsky-Rosen (EPR) steering [18, 19] in value violates the so-called Bell inequality, then any descrip- an ensemble. A quantum mechanical (QM) witness of many- tion of the system will be incompatible with local realism, body Bell correlations has also been observed in the collec- which requires the particles to be in a defined state at all times tive spin of a Bose-Einstein condensate (BEC) [9]. However, and not to communicate faster than the speed of light. In quan- the spins were not spatially separated, which makes it chal- tum mechanics, the particles form an entangled state and only lenging to extend such a scheme to demonstrate nonlocality. occupy a definite state once the measurement occurs. Entan- Promising progress on a test for momentum entanglement in glement forms the basis for a range of emerging and future an atomic pair has also been reported [20]. quantum technologies, such as quantum computing or metrol- ogy [12]. In this paper, we report on the generation and detection of a strong Bell correlation witness between the atomic spins In Bell tests with photons, the non-classical state comes across a spatially separated pair of metastable helium (He*) from parametric down-conversion. The photons are entan- atoms. The experiment consists of the three essential compo- gled in polarisation (analogous to spin), which is rotated us- nents necessary to realise a Bell test: a correlated atomic pair ing waveplates and then measured. While the earliest exper- source, a rotation of the spins of both atoms corresponding iments with photons [2,3] contained several loopholes that to an independently configurable measurement basis and the still allowed for interpretations consistent with local realism, momentum and spin resolved single-particle detection neces- more recent experiments have succeeded in closing all loop- sary for evaluating pair correlations. Each stage is described holes [4–6]. The first experiments using massive particles in detail below and shown schematically in Figure1. Briefly, were performed by measuring spin correlations between high the pair source is the binary scattering product from a col- arXiv:1811.05681v2 [quant-ph] 23 Nov 2018 energy particles [7,8], while a more recent loophole-free ex- lision between two oppositely spin-polarised BECs (Fig.1a) periment looked at entanglement between the spins in solid which naturally separate in time. The spins of both atoms state systems [4]. However, such schemes rely on entangle- in each pair are then rotated by the same angle (Fig.1b) fol- ment between internal degrees of freedom, and are thus un- lowed by a direct measurement of their momentum and spin able to be extended to entanglement between motional de- (Fig.1c). Over many experimental runs we extract the correla- grees of freedom. This is a major motivation for looking at tions between the spins, which show excellent agreement with entanglement between atoms, as spatially separated, motional the predictions of quantum mechanics, and witness the Bell entanglement could provide insights into proposed theories correlations in our system. We experimentally demonstrate of quantum gravity, by measuring their gravitational decoher- this Bell correlation witness with a significant spatial sepa- 2

3 ration of ≈ 0.1 mm across the entangled pairs of atoms, and 2 S1 (see Methods for details). The atomic sublevels determine the form of quantum entanglement using a quantum |J = 1, mJ = 1i = |↑i and |J = 1, mJ = 0i = |↓i form the state tomography technique operationally akin to the Bell test. qubit subspace (see level diagram in Fig.1(b)), with the atoms By experimentally observing a strong Bell correlation witness initially fully spin-polarised in the |↑i state. Following trap for a range of rotation angles, the quantum state of the pairs switch-off, a π/2-pulse from a two-photon stimulated Raman 3 3 is demonstrated to be a Bell triplet state, suitable for showing process via the λ = 1083 nm 2 S1 − 2 P0 transition (see quantum nonlocality in a more general Bell test. Fig.1(b)) then simultaneously flips half of the atoms’ spin to |↓i and imparts a velocity of ∼ 120 mm/s along the z- axis, opposite to gravity (see Fig.1(a)). In the centre of mass frame, the two condensates split apart at vr ≈ ±60 mm/s and spontaneously scatter atoms into correlated pairs of opposite momenta and spin via binary elastic s-wave scattering, form- ing a uniformly distributed√ spherical halo in momentum with radius kr = 2π/ 2λ [21, 22]. The opposite spin-states of the colliding BECs entangle the pairs in spin as well as mo- mentum (see inset of Fig.1(a)). With the momenta of each pair given by (k, −k = A, B), the state is symmetric un- der exchange of labelling by momentum and exhibits com- plete anti-correlation in spin. Bogoliubov scattering theory predicts that the state of the pair is the archetypal√ Bell triplet + |Ψ i = (|↑iA ⊗ |↓iB + |↓iA ⊗ |↑iB) / 2 (see Methods for details). Such a state is maximally entangled, useful in var- ious quantum information tasks [12], and, more importantly to this work, a viable candidate for demonstrating nonlocality [23]. Following the collision pulse, the scattering halo evolves freely in a uniform magnetic field of ∼ 0.5 G for tsep = 0.8 ms (see Fig.1(b)). The halo expands spherically such that each entangled pair, located at diametrically opposite re- gions of the halo, is spatially separated by dsep ≈ 0.1 mm. A pair of co-propagating Raman beams (see Fig.1(b)) that are wider than the size of the halo by over an order of mag- nitude provide a uniform rotation corresponding to Rˆy(θ) =  (A)  (B) (A) (B) exp −iθσˆy ⊗ exp −iθσˆy , where σˆy and σˆy rep- resent the y-component of Pauli matrices for spins at A and B, while imparting no net momentum change to the atoms. The rotation is independent of the ’s momentum and position, is applied to the whole atomic ensemble, with the rotation an- gle θ controlled by the optical pulse duration. A key feature FIG. 1. Experimental schematic for the generation and detection of + entangled pairs of atoms. (a) Binary collisions entangle pairs of parti- of the |Ψ i state is that it is not rotationally invariant under a cles in quantum mechanics due to the symmetry in certain wavefunc- uniform rotation of both atoms in the pair by a single angle θ, tions (inset). In this work, such entanglement is realised in pairs of which enables us to measure the entanglement of the state. atoms from the collision products of oppositely spin-polarised BECs. Immediately after the rotation pulse, a magnetic field gradi- (b) The scattered pairs form a spherical shell as antipodal points in ent is applied in the z-direction (see Fig.1(c)). This projects momentum, where the BECs lie on the two poles along the collision ˆ axis. Pairs spatially separate in time at which point individual atom’s the atoms into the Sz eigenstates {|↑i , |↓i} via the Stern- spin is rotated by an angle θ. (c) An applied magnetic field gradient Gerlach effect, separating the two spin states in the vertical spatially separates the atoms by spin, before individual atoms are de- direction. Since only the mJ = 1 state has a non-zero mag- tected with full 3D momentum and spin resolution. The images on netic moment, only |↑i feels a magnetic force, causing the the right show the atom count density in the zx-plane for two values state to spatially separate from |↓i atoms at the detector and of θ. Spin correlations between the scattered pairs (diametrically op- allowing state-resolved detection. posite regions on the purple rings) will exhibit quantum nonlocality. The bright yellow ellipses correspond to the BECs, which saturate The atoms then freely fall under gravity onto the detector the detector. located 0.848 m below, from which we obtain the momentum and spin information for individual He* atoms. Figure1(c) Our experiment starts with a magnetically trapped BEC shows a typical image from an average of many experimental of helium-4 atoms in the long-lived metastable state shots, displaying two completely separated halos for each of 3 the experimental configurations in which there was no rotation Fig.2(c), confirming that our pair source behaves as expected. pulse and a π/2-rotation that evenly mixes spins. Atoms in the Furthermore, the inverse proportionality is consistent with the |↑i state form the lower halo, which is slightly non-spherical predictions of Bogoliubov theory, which describes the pair- due to inhomogeneity in the magnetic field gradient causing a scattering process in the low-gain regime. Importantly, we are spatially dependent force around the halo (see Fig.1(c)). Such able to reach correlation amplitudes of ∼ 60, although due to distortion corresponds to a misalignment of the ideal back-to- signal to noise considerations we actually operate in a regime back pairing in momentum and is removed in the data anal- (2) of g↑↓ (0) ≈ 30, an amplitude sufficient to demonstrate a vi- ysis (see Methods for details). Since the mJ = 0 states are olation of a Bell inequality [14]. The corresponding average unaffected by magnetic fields, the |↓i-halo maintains the s- mode occupation in the scattering halo of ∼ 0.03 means we wave spherical shell shape at the detector (see the upper halo are operating in the low-gain regime, where the dominant con- in Fig.1(c)). tribution to the halo comes from scattering of single pairs. To characterise the spin rotation a |↑i-polarised scattering halo was initially prepared from a Raman sequence similar to our previous work to generate mJ = 0 halos (see [22, 24] and Methods). The atoms are then rotated by Rˆy(θ) and the Rabi oscillations are observed (see Fig.3) with negligible coupling to the mJ = −1 state.

FIG. 2. Tunable momentum-spin anti-correlated atomic pair source. (a) Schematic of atoms with momentum and spin degrees of free- dom in the scattering sphere. A 2D planar slice in momentum space is taken for simplicity. (b) Two-body cross-correlation function in (2) momentum-spin g↑↓ from the un-rotated pair source with an aver- age mode occupation of n = 0.058(2), averaged over 2,100 experi- mental runs. (c)The dependence of the degree of anti-correlation on the average halo mode occupation. Two different angles between the Raman beams (30◦ and 90◦) were used to produce the experimental FIG. 3. Coherent control of atomic spin in a metastable helium data. The dashed line depicts the theoretical prediction. Demonstra- scattering halo. Rabi oscillation in the population fraction from a tion of nonlocality strictly requires a minimum correlation strength (2) √ single rotation pulse with an amplitude of 0.85(4) and effective Rabi g > 3 + 2 2 0 in the pair source of ↑↓ . All error bars indicate the frequency Ω = 2π ·50.3(3) kHz. The Larmor precession frequency standard deviation in the mean (see Methods for details). of the atomic spins is ΩL ≈ 2π·1.4 MHz for a field of ∼ 0.5 G. The shaded region around the fitted model (solid line) indicates variations To characterise the two-state scattering halo we look at two- in the Rabi oscillation characteristics at different momentum zones particle correlations between atoms on opposite sides of the of the scattering sphere. halo with either parallel or anti-parallel spin-pairing, given by By rotating the dual spin halo and measuring the resulting P correlations between atoms in each state (see Methods for de- (2) k∈V hnˆk,inˆ−k+∆k,ji gij (∆k) = P , (1) tails), a correlator k∈V hnˆk,ii hnˆ−k+∆k,ji g(2) + g(2) − g(2) − g(2) where i, j ∈ {↑, ↓} denote spin states, nˆq,m the number of D (A) (B)E ↑↑ ↓↓ ↑↓ ↓↑ B(θ) = σˆz σˆz = (2) atoms with momentum q and spin m, and V the volume in θ (2) (2) (2) (2) g↑↑ + g↓↓ + g↑↓ +↓↑ momentum space occupied by the s-wave scattering halo [24] (shown schematically in Fig.2(a)). Figure2(b) shows an ex- is obtained (the θ subscript denotes the average in the rotated perimentally measured correlation function for scattering halo state). This correlator is displayed in Fig.4(a), with the ex- with no rotation, which displays a large peak indicative of perimental results showing excellent agreement with the the- strong correlations between atoms in different spin states on oretical prediction B(θ) = − cos 2θ for the Bell triplet state opposite sides of the halo. The pair correlation amplitude is |Ψ+i. This is the first strong indication that the two atoms are inversely proportional to the mode occupancy n for a sponta- strongly entangled. neous pair source [24], which we tune by varying the number To prove the non-classical properties of our two-atom sys- of atoms in the colliding BECs. This relationship is verified in tem, we first show that the pairs are entangled. Note that for 4 all non-entangled states, the maximum range of the correlator The authors would like to thank Alain Aspect, Michael Bar- (2) is bounded by unity son, Danny Cocks, Matteo Fadel, Karen Kheruntsyan, Wolf-

0 0 gang Mittig, Margaret Reid, Jacob Ross and Tilman Zibold S(θ, θ ) = |B(θ) − B(θ )| 6 1 (3) for insightful discussions. This work was supported through (see Methods for details). We detect a clear violation of this Australian Research Council (ARC) Discovery Project grants bound in Fig.4(a), which proves the system is entangled — a DP120101390, DP140101763 and DP160102337. DKS is necessary ingredient for the violation of a Bell inequality for supported by an Australian Government Research Training any quantum system. Program Scholarship. SSH is supported by ARC Discovery Since we have shown that the atomic spins are vector quan- Early Career Researcher Award DE150100315. JC is sup- tities under rotation (see Figure3 and Methods), we can now ported by Project no. 2017/25/Z/ST2/03039, funded by the exclude a wide class of local hidden variable (LHV) theories. National Science Centre, Poland, under the QuantERA pro- Violation of the inequality gramme.  π   π  √ S θ, θ + = B(θ) − B θ + 2, (4) 2 2 6 from two complementary measurements certifies the exclu- ∗ sion of situations in which one subsystem gives binary out- [email protected] [1] J. S. Bell. On the Einstein Podolsky Rosen paradox. Physics, comes, whereas the second consists of a vector quantity (see 1:195–200, Nov 1964. Methods for details). This is violated in our system, as shown [2] Stuart J. Freedman and John F. Clauser. Experimental test of in Fig.4(b), since for θ = 0 we observe S(0, π/2) = 1.77(6). local hidden-variable theories. Phys. Rev. Lett., 28:938–941, Finally, we compare our results to the predictions of Bo- Apr 1972. goliubov theory applied to the scattering process (see Meth- [3] Alain Aspect, Jean Dalibard, and Gerard´ Roger. Experimental ods for details), which has been tested for a wide range of pair test of bell’s inequalities using time-varying analyzers. Phys. 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√ FIG. 4. Non-classical correlations in scattering halos. (a) A strong correlation B, exceeding 1/ 2 (shaded region), signals the potential to observe the violation of a Bell inequality based on a quantum mechanical model. The QM prediction of the violation was observed by B(π/2) = 0.90(1) with a significance of 18 standard deviations from 18,000 experimental runs when the spins were rotated by π/2. The + dashed line is the theoretical prediction for the Bell triplet Ψ and the solid line is a guide to the eye. (b) Direct observation of non- classicality in the pair correlations based on the QM witness of Bell correlations S. Correlations lying in the shaded and hatched regions indicate the presence of quantum entanglement and Bell nonlocality, respectively. 6-sigma violation of the Bell witness inequality (4) is observed by S(0, π/2) = 1.77(6) All error bars correspond to the standard error evaluated from bootstrapping (see Methods for details).

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METHODS

Experimental apparatus and procedure

∗ The He BEC is initially prepared in the mJ = 1 state in a bi-planar quadrupole Ioffe configuration magnetic trap as explained in our previous papers [22, 26], with harmonic fre- quencies of (ωx, ωy, ωz)/2π ≈ (15, 25, 25) Hz. The mag- netic trap is switched-off abruptly, here denoted as time t = 0, from which it takes ∼ 2 ms for the magnetic√ field to stabilise to a uniform field of B0 ≈ 0.5 (ex + ez) / 2 G, which splits the degeneracy in spin by fl = gµ0B ≈ 1.4 MHz and is maintained throughout until the Stern-Gerlach sequence. At t = 3 ms the π/2 collision Raman pulse for creating + ◦ |Ψ i-pairs, lasting ∼ 10 µs, is applied from two 90 -crossed√ laser beams L1/L2, propagating along the (ex ± ez)/ 2 di- rections, and σ−/π- polarised with respect to the quantisation axis defined by B0, respectively. Each beam’s optical fre- 3 3 quencies were far-detuned from the 2 S1 − 2 P0 transition by ∆ ≈ 3 GHz such that ∆/Γ ≈ 2000  1, making the spontaneous single photon absorption rate negligible. A sin- gle photon recoil is ~k0 = 2π~/λ, where λ = 1083.20 nm. The atoms evolve freely in the stabilised magnetic field B0 for 0.8 ms, at which point the scattering halo, uniformly ex- ˙ panding at a rate dsep ≈ 120 mm/s (given by the recoil mo- menta from the two photons absorbed by the He* atoms), FIG. 5. Characterisation of the rotation pulse. (a) Atom count reaching a diameter of dsep ≈ 96 µm. The spin rotation pulse density at the detector in the zx-plane, integrated over −12 mm < is then applied at t = 3.8 ms, from a second stimulated Raman y < 15 mm, for different spin rotation pulses applied to |↑i-polarised transition coupled to the same transition and detuned as above, scattering halo. (b) Atom counts in each scattering halo. Signifi- cant loss in total number is observed (black diamond), which is most using a single beam L3 that propagates along the x-axis. An RF-pulsed acousto-optic modulator produces a two-tones op- likely due to Penning ionisation based on the strong correlation with population in mJ = 0. For longer pulse durations the atom number tical pulse in L3, with the two frequencies fulfilling the co- in the scattering volume increases due to spontaneous absorption of propagating resonance condition for the two photon Raman the Raman beams by the BECs. Solid lines are sine curve fits to data. process. Furthermore, the beam was elliptically polarised with propagation along x-axis such that σ+ polarisation was extin- guished along B0, which would otherwise couple the qubit Characterisation of spin rotation subspace to the mJ = −1 substate. The beam waist at the trap was σ3 ≈ 1.1 mm, an order of magnitude larger than spatial extent of the atomic ensemble at the time of rotation This section provides the experimental details used to ob- sequence, which provided a uniform rotation operation for all tain the Rabi oscillation in Fig.3, from which we concluded scattered atoms in the halo. that the experimental operation in Fig.1(b) corresponds to a After the rotation pulse, the Stern-Gerlach sequence is im- coherent rotation of the qubit. The characterisation follows plemented by pulsing current through a large coil concentric a sequence almost identical to the main experiment (Fig.1), to the z-axis, that selectively pushes the mJ = +1 atoms except that only mJ = +1 states are prepared initially (see along the −z direction. Subsequently, the atoms are de- circuit diagram in Fig.3(a)). Here the collision pulse is ap- tected by an 80 mm diameter microchannel plate and delay plied at t ≈ 0 so that a Bragg transition (no change in internal line detector located 848 mm below the trap. The free-fall state) can be driven with the same beam geometry used to duration gives the time at detection t ≈ 416 ms, while the produce the entangled pairs (see Fig.1(a)), producing a |↑i- detector has a spatio-temporal resolution of approximately polarised scattering halo (see left Fig.5(a)). At the time and 120 µm × 120 µm × 3 µm [27] and a quantum efficiency location of the Bragg collision pulse, the magnetic field points of ∼ 10%. along the x-axis similar to [22], after which it is stabilised to 7

in both spin-states, each atom from the |Ψ+i pair is almost equally likely to be lost by Penning ionisation. The loss of an atom from any pair results in the detection of a single-hit event, i.e. a single-hit event will be detected in A with ↑ or ↓, with no correlated hit in B. Such events are thus naturally (2) treated by the correlation functions gij as an uncorrelated background event (see Eq. (1)). The presence of Penning ioni- sation will therefore strictly can only reduce the observed cor- relator to the asymptotic uncorrelated state B = 0, such that (2) in the extreme case where half of every pair is lost, gij = 1 for all back-to-back correlations will be observed. 0 The increase in total number (Nloss < 0) for longer pulse durations (τ > 17 µs) is due to the constant spontaneous ab- sorption of the Raman beams and subsequent decay by the BECs (2 3S → 2 3P → 2 3S where each process is accom- FIG. 6. Ramsey-type fringe from two separated π/2-pulses with a 1 J 1 phase delay in the second rotation pulse. The large fringe visibility panied with a single photon recoil), into a scattering volume of 0.95(2) demonstrates the ability to independently configure the in momentum space intersecting the s-wave scattering halo. axis of rotation (n) around the xy-plane of the Bloch sphere. The entangled atoms of interest originally occupying the halos are indeed equally subject to such spontaneous processes, and contribute to an additional loss term, negligible in comparison to that of BECs due to the relative atom numbers involved. In B0 for the rotation pulse as described in the previous section. The rotation pulse is applied at t = 3.8 ms so that the halo is our experiment, a large detuning of the Raman beams from resonance minimised the rate of spontaneous absorption, as dsep ≈ 0.46 mm in diameter, approximately five times larger than the |Ψ+i-halo at the point of rotation, but still signifi- demonstrated by the nearly absent spontaneous effects even cantly smaller than the beam waist. from the BECs. The resulting atom number density at the detector is shown in Fig.5(a) for various durations of rotation pulse τ. The to- Coherent Rotation tal number of atoms detected in the truncated scattering vol- ume (details are given in the next section) N 0 is shown in α As shown in Fig.3, a Rabi oscillation through 2π-rotation Fig.5(b) for various rotation pulse durations. By defining can be induced between the |↑i and |↓i states, with negli- the population fraction of the internal state α = ±1, 0, by gible coupling to the m = −1 state. To ensure that the P (α) = N 0 /N 0, where N 0 = P N 0 is the sum over the J α α α Raman pulse is producing the desired rotation of the atomic triplet, we obtain the Rabi oscillation shown in Fig.3(a). Al- spin on the Bloch sphere, we implement a Ramsey-type in- though negligible transfer to m = −1 was achieved, we no- J terferometry where a secondary π/2-pulse with a phase de- ticed a loss in the total number of atoms detected in the scat- lay φ follows the first π/2-pulse Rˆ (π/2). Following an tering volume, N 0 (τ) = N 0(0)−N 0(τ) (see Fig.5(b)). Two y loss identical initialisation scheme to the Rabi oscillation exper- key features evident in the behaviour of the total atom number iment, the fixed delay between the two π/2-pulses was set during the Raman transition are the loss strongly correlated at T ≈ 10 · T , and the phase delay between the two Ra- with the presence of m = 0 states and the steady, but weak L J man optical fields scanned over all range between 0 and 2π, increase of the total population, which we explain below. relative to the Rˆy reference pulse. We observe a Ramsey- The significant decrease in the total atom number, observed type fringe with almost perfect visibility of 0.95(2) in Fig.6 0 to be up to 25% and correlated with the N0 population (see which along with the Rabi oscillation (Fig.3) provides a clear Fig.5(b)), is due to Penning ionisation between pairs of He* demonstration of the desired unitary rotations on the atomic atoms [28], which is enhanced by 4 orders of magnitude for spin Rˆφ(θ) = exp(−iθ(cos φ σˆx + sin φ σˆy)). spin unpolarised pairs, compared to pairs of mJ = +1 atoms. In our experiment, Penning ionisation would most strongly affect the entangled pairs at the earliest time following their Transformation of the scattering halo production, since the local density of atoms is the highest dur- ing the spatial overlap of the pairs with the BECs, allowing for This section provides details on the data analysis used in more frequent Penning ionising collisions. After the BEC and preprocessing the raw data from detector coordinates (position entangled pair wavefunctions have spatially separated, there and time-of-flight) to the velocity/momentum coordinates rel- should be a negligible fraction of atoms lost, since the |Ψ+i evant for the physical system. Since the atoms are in free-fall pair source was prepared with very low numbers - an order for the majority of the time from the magnetic trap switch-off of in magnitude lower than the scattering halo used to char- (see previous section on experimental procedure), the posi- acterise the Raman pulse in Fig.5. In the presence of BECs tions of atoms at the detector essentially correspond to veloc- 8 ities (interchangeable with momentum) [24]. As seen from Fig.5(a), the spatial distributions of scattered atoms at the de- tector is aspherical for mJ = +1 (bottom) and spherical for mJ = 0 (middle). The deformation in the spatial distribution of the mJ = +1 scattering halo from the ideal spherical shell arises due to inhomogeneous forces from the stray magnetic field present in the vacuum chamber during free fall. Since the accurate determination of the atomic momenta is crucial to identifying the scattered atomic pairs, a distortion correct- ing shape transform is applied to the raw spatial distribution of atoms (r) to retrieve the momentum distribution (k). First a spatial distribution (detector coordinate) of atoms in the BECs and the scattering halo are distinguished by the internal state and the new coordinate origin defined at approx- imately the centre of the corresponding halo. Background atoms from the BECs, thermal fraction and miscellaneous sources other than the scattered pairs are then removed by only keeping counts lying inside the truncated spherical shell defined by 0.6 < r/rtof < 1.2 and |rz/rtof| < 0.8, where rtof ≈ 25 mm is the radius of the scattering halo at the detec- tor. The resulting r-distribution is then fitted with an ellipsoid which defines the desired smooth shape transform consisting of three orthogonal linear scalings about the centre to reduce each principal axes to unity, which can be suitably identi- FIG. 7. Evaluated g(2) correlation functions of the scattering ha- fied as the normalised momentum coordinate in the centre of los after various rotations, for pairs with near opposite momenta and (2) mass reference frame k (see Fig.2(a)). A final filter to re- various spin-pairing configurations. (a) ∆ky = 0 slices of g (∆k) move the remaining background restricts the k-distribution to for various spin rotation pulses (separated by rows) and spin-pairing configurations (separated by columns). (b) Normalised correlations 0.9 < k < 1.1 and |kz| < 0.75, which corresponds to the truncated momentum space V investigated in this work. An for the back-to-back condition (∆k = 0). The asymmetry in corre- ~ lation between ↑↑ (blue circles) and ↓↓ (red squares) arises from the additional relocation of the coordinate origin by K was how- shape distortion of |↑i atoms’ trajectories by stray magnetic fields (2) ever necessary to centre the Gaussian profiles of gij (∆k) at during time-of-flight. Error bars correspond to the standard error es- ∆k ≈ 0, which is crucial in the implementation of Bell test timated from bootstrapping. Solid lines are damped sine curve fits to as it effectively defines the two detection ports in each arm: data. (k, ↑), (k, ↓), (−k, ↑), (−k, ↓) corresponding to the conven- tional A+, A−, B+, B− detection events, respectively. amount of acquired data. The relative strengths in spin correlations with various ro- Correlation functions tation pulse durations are summarised in Fig.7(b). Here, the normalised second-order correlation is defined as This section provides details on the two-particle correlation functions in the scattering halos which were ultimately used to 2g(2) G = ij , construct the B correlator. Figure7(a) shows the effect of the ij P (2) rotation sequence on the correlation properties of the initially α,β∈{↑,↓} gαβ spin and momentum anti-correlated pairs by second-order cor- relation functions g(2) as defined in Eq. (1). As expected for where the second-order correlation without any argument de- the |Ψ+i scattered pairs (τ = 0 µs) no correlation is seen be- notes the value at the exact back-to-back (BB) condition in (2) (2) tween oppositely scattered pairs with the same spin, however momentum (gij = gij (∆k ≈ 0)). The BB correlation with a common π/2 rotation (τ = 5 µs) the spins are always strengths were determined from a cubic bin of width 0.0138 parallel, and finally return to anti-parallel by the nearly spin- in the normalised momentum unit. As demonstrated in the flipping π pulse (τ = 10 µs). The asymmetry in the Gaussian theory section, the general bipartite correlator can be deter- correlation profiles between different spin types are due to the mined directly from the normalised correlations by Eq. (36). effects from stray magnetic field during the atoms’ free-fall The observed asymmetry in the correlation strengths between before detection (see previous section for details) which dis- ↑↑ and ↓↓ combinations can be explained by the asymmetry torts the atomic velocities and thus the correlation profile. The in the correlation volumes due to the shape distortion of the difference in signal-to-noise ratios of the correlation functions mJ = 1 scattering halo, such that the correlation volume in- for different rotation sequences are due to the variations in tegrated strengths recover the expected symmetry. 9

Error analysis Note that the components of the Bloch vector lying in the zx- plane can be parameterised as follows All statistical uncertainties in the correlation functions, and (λ) ! (λ) ! similarly for other variables, were determined from boot- cos φa ~ cos φb ~a(λ) = α(λ) (λ) , b(λ) = β(λ) (λ) , strapping described here. From the complete dataset from sin φa sin φb which the representative correlation function g is determined (12) (see Eq. (1)), subsets (fractional size η) are sampled with re- placement and analysed identically to produce a distribution where α(λ) 6 1 and β(λ) 6 1. With this parameterisation, of outcomes G. A robust estimate of the standard error of the amplitude A is equal to the mean (correlation function in this case) is then given by p 2 2 2 σg¯ = η · Var(G), independent of the sampling size. A = (Czz − Cxx) + (Czx + Cxz) 2 = hαβ(sin φa sin φb − cos φa cos φb)i 2 Amplitude criterion for entanglement + hαβ(cos φa sin φb + sin φa cos φb)i 2 2 = hαβ cos(φa + φb)i + hαβ sin(φa + φb)i . (13) The correlator defined in Eq. (2), assuming local rotations around the y-axis by a common angle, can be written as fol- Using the Cauchy-Schwarz inequality lows 2 2 2 1 hαβ cos(φa + φb)i 6 (αβ) cos (φa + φb) (14a) B(θ) = [Cxx + Czz + (Cxx − Czz) cos 2θ 2 2 2 2 hαβ sin(φa + φb)i 6 (αβ) sin (φa + φb) (14b) +(Cxz + Czx) sin 2θ], (5) we obtain where 2 2 2 2  D (A) (B)E A 6 (αβ) cos (φa + φb) + sin (φa + φb) Cij = σˆi σˆj (6) 2 2 2  = (αβ) cos (φa + φb) + sin (φa + φb) 2 and i, j = x, z. The minimum and the maximum of B(θ) with = (αβ) 6 1. (15) respect to θ yield A — the amplitude of oscillations, namely Therefore A > 1 implies entanglement between the qubits, 1 1 [C + C − A] B(θ) [C + C + A] , (7) which was used in Eq.3. 2 xx zz 6 6 2 xx zz where Non-locality q 2 2 A = (Cxx − Czz) + (Cxz + Czx) . (8) In our setup, the spin-rotation beams are much larger than If the two qubits form a separable (i.e., non-entangled) state, the halo size at this point, thus all atoms in the halo are rotated their composite density matrix reads by the same angle θ and we have access only to the diagonal Z part of E, as B(θ) = E(θ, θ). An extension to implement (A) (B) %ˆ = dλ p(λ)ˆ%λ ⊗ %ˆλ , (9) independent rotations in each atom of the pairs would be ex- perimentally possible, but is beyond the scope of this current where p(λ) is some probability distribution of a variable λ work. and %ˆ(i)(λ) are the single-qubit density matrices (i = A, B). Still, using Eq. (2) we can test a wide range of LHV theo- These single-particle matrices are given by ries. These also take binary outcomes in A and B (i.e., ↑ / ↓), but assume that, on average, the results in one of the sub- 1 (A) 1ˆ(A) ˆ(A) %ˆλ = ( + ~a(λ)~σ ), (10) systems (either A or B) behave like components of a vector, 2 while making no assumptions about the other part. To restrict 1 (B) 1ˆ(B) ~ ˆ(B) %ˆλ = ( + b(λ)~σ ). our system to such an LHV theory, we analyse the properties 2 of rotations in A and B in a dedicated series of experiments, Here ~σˆ(A/B) is a vector of Pauli matrices for the qubit with the results shown in Figs3 and6 demonstrating that they A/B and ~a(λ) and ~b(λ) are the corresponding Bloch vectors do indeed rotate as vectors. (|~a(λ)| = 1 for pure and |~a(λ)| < 1 for mixed states for A Consider two subsystems, where quantities A and B are and analogously for B). measured. The joint probability for observing A and B fulfils Using the separable state (9) this takes the form the postulates of local realism if Z X n (A) (B)o P (A, B) = p(λ)P (A|λ)P (B|λ), (16) Cij = tr %ˆσˆi σˆj = dλ p(λ)ai(λ)bj(λ). (11) λ 10

P B B where P (A|λ) or P (A|λ) are the conditional probabilities for while A P (A)JA = J 6 1. Thus observing A or B given some value of a hidden variable λ, √ A B A B governed by the probability distribution λ. The conditional J1 J2 − J2 J2 6 2. (25) probability for observing B given some result A is When the system is composed of two qubits, J i is replaced P (A, B) P (B|A) = with a corresponding Pauli operator and the inequality be- P (A) comes X p(λ)P (A|λ) = P (B|λ)  π  D (A) (B)E D (A) (B)E √ P (A) S θ, θ + = | σˆ1 σˆ1 − σˆ⊥ σˆ⊥ | 6 2 (26) λ 2 X = P (λ|A)P (B|λ). (17) for all systems compatible with the LHV model outlined λ above. To test it, one can analyse the combination of the cor- Now, imagine that given the quantity measured in A, labelled relator B in Eq.4 and plotted in Fig.4. A as Ji , can have binary outcomes for two local settings i = A A B x, z, i.e., Jx = ±1 and Jz = ±1. The two quantities Ji measured in B are, on average, assumed to be components of CHSH inequality for pair-scattering systems a vector of length 1, which in particular implies The most general LHV theory that can be tested with two −1 J B 1. (18) 6 i 6 qubits assumes binary outcomes of the measurements in A Note that we do not specify how this average is calculated, and B. Both the original Bell inequality [1] or its modification and the outcomes in B can be binary as well. proposed by Clauser, Horne, Shimony and Holt (CHSH) [25] The average outcome in B, say in the x direction, given the use the above assumption. The CHSH inequality A result in is 0 0 0 0 B X B BCHSH = E(θ, φ) + E(θ , φ ) + E(θ , φ) − E(θ, φ ) 6 2, Jx A = P (B|A)Jx B (27) X X requires independent rotations in A and B with Rˆ(A)(θ) = = P (λ|A) P (B|λ)J B, (19) y x  (A) ˆ(B)  (B) λ B exp −iθσˆy and Ry (φ) = exp −iφσˆy and the mea- where surement of the corresponding correlator B X B B D E −Jλ PQ(B|λ)Jx Jλ , (20) (A) (B) 6 6 E(θ, φ) = σˆz σˆz . (28) B θ,φ B and Jλ is the length of the vector in B given the value of λ. To derive the CHSH inequality for the system where atoms Thus the upper bound reads scatter in a pair, we use the bosonic field operators Ψˆ α(r) for X X each spin component α = ±1, 0. The Hamiltonian (the sum- J B = P (B|A)J B P (λ|A)J B = J B. (21) x A x 6 λ A mation convention is used) for a J = 1 BEC reads B λ

√1 B Z  2 We take two orthogonal directions in the zx-plane, (J1 − 3 ~ † † 2 Hˆ = d r ∇Ψˆ (r) · ∇Ψˆ α(r) + V (r)Ψˆ (r)Ψˆ α(r) B 2m α α J2 ), and according to the above argument we obtain c0 † B B ˆ † ˆ ˆ ˆ J − J + Ψα(r)Ψ (r)Ψβ(r)Ψα(r) B 1 A 2 A B 2 β −JA 6 √ 6 JA . (22) 2 # c1 † † + Ψˆ (r)Ψˆ (r)F 0 · F 0 Ψˆ 0 (r)Ψˆ 0 (r) . (29) Now, we modify the inequality (22) by multiplying the two 2 α β αα ββ β α averages by the corresponding results in A. Using that out- comes in A are binary, we obtain The coefficients c0/1 are related to the scattering lengths a0/2 A B A B in total interaction channels J = 0, 2 J1 J1 − J2 J2 2 2 B A A B 4π a0+2a2 4π a2−a0 −JA 6 √ 6 JA . (23) by c0 = ~ and c1 = ~ , while F = 2 m 3 m 3 (Fx,Fy,Fz) is a vector of spin-1 matrices. Finally, we average this inequality by the outcomes in A. The Though the scattering of mJ = 0, 1 pairs from BECs is correlators are equal to governed by terms proportional to c0 and c1, in the low- X A B X A X B density limit spin changing collisions are less probable, and P (A)Ji Ji A = P (A)Ji P (B|A)Ji the term proportional to c1 can be safely neglected. Also A A B in this regime, the dynamics of Ψˆ can be found using the X A B A B α = P (A, B)Ji Ji = Ji Ji , Bogoliubov approximation. Each component is decomposed A,B into the dominant c-number term and a quantum correction, (24) ˆ Ψˆ α(r) = φα(r) + δα(r). The coherent fields φα are governed 11 by time-dependent Gross-Pitaevskii-type equations, whereas ory [14]: E(θ, φ) = −E cos(θ + φ), showing a dependence ˆ the fields δα are subject to dynamical Bogoliubov equations only in the sum of the angles and oscillations with the ampli- which contain both δˆ and δˆ† terms. (2) (2) α β tude E = (g↑↓ − 1)/(g↑↓ + 1), which is expressed in terms Since, in the process of the halo formation, the mean-field of the two-particle correlation function: energy due to the BECs is small compared to the kinetic en- D E ergy of the scattered atoms, the only relevant term propor- RR 0 ˆ† ˆ† 0 ˆ 0 ˆ dkdk δ↑(k)δ↓(k )δ↓(k )δ↑(k) tional to c0 is the production term, and the equation of motion (2) AB 2 2 g↑↓ = D ED E. (32) ˆ ~ ∇ ˆ ˆ† RR 0 ˆ† ˆ ˆ† 0 ˆ 0 is i~∂tδα = − 2m δα + c0φβφαδβ. The presence of the field dkdk δ↑(k)δ↑(k) δ↓(k )δ↓(k ) ˆ† AB δβ describes the scattering of atoms from BECs and the for- 2 ˆ† 2 ˆ† mation of the halo. The terms proportional to φ0δ0 and φ1δ1 Here, the average is calculated in the state prior to rotations. govern the quantum depletion of the BECs, irrelevant for the The expression for BCHSH√optimised over angle settings yields the condition |E| > 1/ 2, which in turn is equivalent to scattering process. As a result, the Bogoliubov equations are √ δˆ (2) symmetric for 0/1. Therefore, we can consider the same Bell g↑↓ > 2 2 + 3 for the Bell inequality (27) to be violated. sequence as in Ref. [14]. The key quantities in the Bell inequality (27) are probed by The Bell test starts with the mixing of the two spin com- setting equal angles in the correlator,√ i.e., E(θ, θ) = B(θ). ponents mJ = 0, 1 (from now on denoted as ↓ / ↑) inde- The observation of |B(θ)| > 1/ 2 signals the detection of pendently in two opposite regions of the halo, A and B, by the Bell correlations, and the potential for the Bell inequality the angles φ and θ over the y-axis. The many-body angular violation in an actual Bell test experiment with independent momentum operators (and the atom-number operators) are settings of the angles in separated regions A and B. Finally, we point out that in the low-gain regime, when only a single Z   ˆα 1 dk ˆ† ˆ ˆ† ˆ pair of qubits is scattered, the many-body angular momentum Sx = δ↑(k)δ↓(k) + δ↓(k)δ↑(k) , (30a) 2 α2π operators are replaced by the Pauli matrices in A and B, and Z   the correlator (31) takes the form of Eq. (28). ˆα 1 dk ˆ† ˆ ˆ† ˆ Sy = δ↑(k)δ↓(k) − δ↓(k)δ↑(k) , (30b) 2i α2π Z   ˆα 1 dk ˆ† ˆ ˆ† ˆ B correlator Sz = δ↑(k)δ↑(k) − δ↓(k)δ↓(k) , (30c) 2 α2π Z   ˆ 1 dk ˆ† ˆ ˆ† ˆ The correlation coefficient E(θ, φ) given above can be Nα = δ↑(k)δ↑(k) + δ↓(k)δ↓(k) , (30d) 2 α2π readily evaluated from the single-particle detection resolved in momentum and spin, since with α = A, B. Finally, a normalised correlator is constructed P (k) (−k) hSˆz Sˆz iθ,φ ˆ(A) ˆ(B) E(θ, φ) = k∈V hSz Sz iθ,φ P (k) (−k) E(θ, φ) = , (31) hNˆ Nˆ iθ,φ (A) (B) k∈V hNˆ Nˆ iθ,φ D ˆ (A) ˆ (A)  ˆ (B) ˆ (B)E N↑ − N↓ N↑ − N↓ where the subscript θ, φ denotes averaging over the rotated = θ,φ . (33) D ˆ (A) ˆ (A)  ˆ (B) ˆ (B)E state. N↑ + N↓ N↑ + N↓ θ,φ This correlator E(θ, φ) from Eq. (31) satisfies the Bell in- equality of Eq. (27) for all LHV theories [29]. The corre- Therefore, it can be expanded in terms of products of number lator E can be evaluated analytically in the Bogoliubov the- operators across the regions A/B

D ˆ (A) ˆ (B)E D ˆ (A) ˆ (B)E D ˆ (A) ˆ (B)E D ˆ (A) ˆ (B)E N↑ N↑ + N↓ N↓ − N↑ N↓ − N↓ N↑ E = , (34) D ˆ (A) ˆ (B)E D ˆ (A) ˆ (B)E D ˆ (A) ˆ (B)E D ˆ (A) ˆ (B)E N↑ N↑ + N↓ N↓ + N↑ N↓ + N↓ N↑

where for convenience the subscript (θ, φ) for labelling the a second-order correlation function general rotated state is assumed for all correlators. Since the density of the scattering halo is symmetric in momentum (s- D (k)E wave scattering) and spin, N¯ = Nˆm for all m ∈ {↑, ↓} and k ∈ V , observe that each term in Eq. (34) corresponds to D ˆ (A) ˆ (B)E ¯ 2 (2) Ni Nj = N gij (35) 12

Therefore, the general correlation coefficient E(θ, φ) in Eq. (34) can be written in terms of g(2) as

g(2) + g(2) − g(2) − g(2) E(θ, φ) = ↑↑ ↓↓ ↑↓ ↓↑ . (2) (2) (2) (2) (36) g↑↑ + g↓↓ + g↑↓ +↓↑