Emission of Spin-correlated Matter-wave Jets from Spinor Bose-Einstein Condensates
Kyungtae Kim, Junhyeok Hur, SeungJung Huh, Soonwon Choi, Jae-yoon Choi
EEmission of Spin-correlated Matter-wave Jets from Spinor Bose-Einstein Condensates
Kyungtae Kim, Junhyeok Hur, SeungJung Huh, Soonwon Choi,
2, 3 and Jae-yoon Choi ∗ Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea Department of Physics, University of California Berkeley, Berkeley, CA 94720, USA Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (Dated: February 16, 2021)We report the observation of matter-wave jet emission in a strongly ferromagnetic spinor Bose-Einstein con-densate of Li atoms. Directional atomic beams with | F = , m F = (cid:105) and | F = , m F = − (cid:105) spin states aregenerated from | F = , m F = (cid:105) state condensates, or vice versa. This results from collective spin-mixing scat-tering events, where spontaneously produced pairs of atoms with opposite momentum facilitates additionalspin-mixing collisions as they pass through the condensates. The matter-wave jets of different spin states( | F = , m F = ± (cid:105) ) can be a macroscopic Einstein-Podolsky-Rosen state with spacelike separation. Its spin-momentum correlations are studied by using the angular correlation function for each spin state. Rotating thespin axis, the inter-spin and intra-spin momentum correlation peaks display a high contrast oscillation, indicatingcollective coherence of the atomic ensembles. We provide numerical calculations that describe the experimentalresults at a quantitative level and can identify its entanglement after 100 ms of a long time-of-flight. The collective scattering processes in many-body systemscan lead to remarkable, counter-intuitive phenomena, due toquantum interference effects. Superradiance in an atomic en-semble is a prominent example, where the spontaneous emis-sion process occurs cooperatively, emitting a directional lightwith an enhanced decay rate [1]. Superradiant scattering isobserved in degenerate Bose [2] and Fermi gases [3], and isoften described as a self-amplified atom-light scattering pro-cess [2, 4, 5]. Atoms scattered by incoming light can interferewith condensates at rest, forming a matter-wave grating thatdiffracts successive laser light, enhancing the amplitude of thedensity modulation. These collective behaviors are not solelyrestricted to the optical domain, but can also extend to matterwaves, where directional atomic beams are generated withoutexternal light fields [6]. Under a periodic driving of scatter-ing length, density modulations are spontaneously developed,stimulating further pair-wise collision processes with a cer-tain direction given by the density modulation [6, 7]. Thisends up as a directional atomic beam, resembling fireworks,contrasting a diffusive spherical shell structure out of uncor-related s -wave collisions [8].Recent studies of matter-wave emissions have offered newopportunities to study complex correlations in the high-harmonic generation process [9], quantum phenomena in anon-inertia frame [10], and dissipative many-body quantumdynamics [11]. Such efforts can provide new directions forproducing non-classical quantum states of atomic spins. Inspinor Bose-Einstein condensates (BECs), for example, corre-lated spin states like the squeezed vacuum state [12–14] havebeen generated via spin-mixing collisions, to explore funda-mental questions of quantum physics [15, 16] and its applica-tion to quantum metrology [17–19]. However, most of suchexperiments have focused on the low kinetic energy regime,where the created spin pairs are localized in the trapping po-tential with their source, challenging the local addressing andmanipulation of the quantum state. One wave to overcomethis hurdle would be to realize directional superradiant colli-sions in a spin manifold, which could generate a macroscopic Einstein-Podolsky-Rosen (EPR) state of atoms [20–22].In this Letter, we report the emission of spin-correlatedmatter-wave jets from spinor BECs of Li atoms. The di-rectional atomic beams of | F = , m F = ± (cid:105) ( |↑(cid:105) , |↓(cid:105) ) spinstates are generated from | F = , m F = (cid:105) state condensates,or vice versa. The kinetic energy of the atomic beam is highenough to escape the trapping potential, where we take ad-vantage of strongly ferromagnetic spin interaction to facilitatethe matter-wave amplification of fast-moving particles. Thematter-waves with opposite spin states can be a macroscopicEPR state, and its spin-momentum correlation is revealed byangular correlation functions between emitted spin states. Toinvestigate non-classical correlations, we coherently rotate thespin states and study the responses of the correlation functionsat various rotation angles. The momentum correlation peaksamong these spin states exhibit a high contrast oscillation as afunction of the rotation angle, suggesting that the ensemblesof atoms with opposite spins still maintain collective coher-ence to exhibit interference patterns. Numerical analysis isprovided indicating the non-classical correlations can be ob-served after a sufficiently long expansion time ( ∼
100 ms).Our experimental sequences start by preparing an F = Li atoms in a quasi-two-dimensional(2D) optical trap [23]. The condensates are initially in a po-lar phase due to a few Gauss of the magnetic field along the z -axis. To observe the matter-wave jets from spinor BECs, weapply an RF pulse to the trapped condensates, making equalpopulation of |↑(cid:105) and |↓(cid:105) states. When the quadratic Zeemanenergy, q = h × − | c | = h ×
160 Hz), the initialstate is unstable, producing pairs of atom in the | (cid:105) state ow-ing to the presence of quantum fluctuations [24]. The spon-taneously created atom pairs obtain kinetic energy from thequadratic Zeeman energy and propagate in opposite direc-tion due to momentum conservation [Fig. 1(b)]. The mov-ing atoms can interfere with stationary condensates in the |↑(cid:105) and |↓(cid:105) states via spin-mixing Hamiltonian, displaying spatialmodulations in the scattering amplitude that stimulates a fur- a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b E ne r g y pe r pa r t i c l e RF spin flip
Time E ne r g y B q > |c| E ne r g y B E ne r g y pe r pa r t i c l e Dressing Time q < 0 O p t i c a l den s i t y Δ B O p t i c a l den s i t y (a) (b)(c) (d) (e)(f) Magnetic field Magnetic field
Dressing
FIG. 1. Emission of matter-wave jets from spinor BECs. (a) Zeeman shift of the F = B . Thehyperfine interaction introduces a quadratic Zeeman shift q > Li atoms. (b) An RF pulse flips atoms in the | (cid:105) state to the ( |↑(cid:105) − |↓(cid:105) ) / √ q > | c | ). Quantum fluctuations produce two atom pairs of | (cid:105) stateafter spin-mixing collisions, and the excess internal energy is released in the kinetic energy of the atoms, flying in opposite directions becauseof momentum conservation. (c) Spin-resolved image of the matter-wave jets in | (cid:105) state with q = 1.7 kHz after a hold time of t h = . {|↓(cid:105) , | (cid:105) , |↑(cid:105)} . The inset represents the condensates remaining in the trap (dashed box). (d) Dressing the | F = , m z = − (cid:105) state with | F = , m z = − (cid:105) state by applying a microwave, we tune the q to have a negative value [25]. (e) After switchingon the dressing field, the polar phase is no longer the ground state of the system, creating spin pairs ( |↑(cid:105) and |↓(cid:105) ). Through the de-excitationprocess, the atom obtains kinetic energy from | q | . (f) After sufficient stimulated collisions, matter-wave jets of opposite spins are also observed. ther pair generation process [5, 21]. This, in turn, leads to self-amplification of the modulation amplitude and the emission ofa directional atomic beam in the horizontal plane constrainedby strong 2D potential [26].The spin-mixing collisional strength is characterized bythe spin-dependent interaction coefficient ( c ). We note thatthe Li atoms are favorable for observing the matter-wavejets because of their strong spin interactions [23]. The spin-dependent interaction coefficient of the atoms is as large as46% of the spin-independent interaction coefficient. There-fore, the grating formed by spin-mixing interaction is weaklydephased more by the source condensates than the other al-kali atoms [27], and the traveling spin pairs can be amplifiedself-consistently.The emission of matter-wave jets in the | (cid:105) state is shownin Fig. 1(c). Stern-Gerlach spin separated absorption imagesare used to resolve the source condensates in the trap and thecreated pairs, so that we are able to study its dynamics un-der various hold times t h after the RF pulse. In the first fewmilliseconds of hold time, the condensates are stable with nopopulations in the | (cid:105) state. Then, radially propagating atomicbeams with narrow angular width, matter-wave jets, suddenlyappear at t h ∼ E k ) of the matter-wave jets isalmost equal to the quadratic Zeeman energy [26], indicating that the atom pairs are created from the source condensates af-ter the spin-changing scattering process. Since the quadraticZeeman energy far exceeds the condensate chemical potential, µ = h ×
300 Hz, the matter-wave jets can escape the BECs andtrapping potential.As a complementary experiment, we also investigate theemission of matter-wave jets with |↑(cid:105) and |↓(cid:105) states from polarcondensates. The polar phase becomes dynamically unstablewhen the quadratic Zeeman energy is negative [24], generat-ing spin pairs of the |↑(cid:105) and |↓(cid:105) states [Fig. 1(e)]. Similar tothe previous experiment, we observe two-dimensional matter-wave jets of the created spin states that have kinetic energyfrom the quadratic Zeeman energy, E k (cid:39) | q | . One remark-able difference is that the matter-wave jets can naturally have aspin-momentum correlation (reminding the EPR state), whichwill be discussed after studying its generation mechanism.The creation process can be well understood by investigat-ing the early time dynamics of matter-wave jets [Fig. 2(a)].For both experiments listed above, initial dynamics arewell described by a simple exponential function, N jet ( t ) = N jet ( ) e γ e t , where the γ e ∼
160 Hz. In the long-time limit, thejet populations saturate because of depletion of the source.The exponential growth dynamics is a characteristic featureof the dynamical instability, in which spontaneously createdatoms pairs are parametrically amplified [28]. The micro-scopic origin of such instability can be found in the exis-tence of imaginary eigenfrequencies in the Bogoliubov quasi-particle (atom pair) spectrums [24]. The instability rate is q = -2.2 kHzq = +2.2 kHz q (kHz) t N t E m i tt ed a t o m nu m be r q = 2.2 kHzq = -2.2 kHz (a) (b) E m i tt ed a t o m nu m be r ( ) Total atom number (10 ) FIG. 2. (a) Dynamical generation of matter-wave jets. Emitted atomnumbers over hold time for q = ± . γ e =
160 Hz [26]. (b) Threshold for jet for-mation. Emitted atom number after t h =
15 ms under various initialatom numbers. To extract the threshold atom number, N t , the datais fitted to bilinear curves (dashed lines). Inset shows q t dependenceof N t with power-law fit, N t ∼ q . t (solid line). The error bars mean95% confidence interval of the bilinear fits. maximized for particles with kinetic energy E k = | q | − c ,which is consistent with the observation within measurementuncertainty. The population growth rate of the quasi-particlesis proportional to the spin-dependent interaction energy, γ b = | c | / ¯ h =
320 Hz, which is two times higher than the obser-vation. In order to ascribe the discrepancy, we consider aloss rate κ in a finite-sized system. When atoms with ve-locity v leave the condensates of radius R TF before sufficientspin-mixing collisions, it leads to an atom loss with a rate κ ∼ v / R TF [5, 6]. In the experiments, we have R TF = µ mand | q | = h × . ( v / R TF ) is evalu-ated to 100 Hz, which may account for the observed differencebetween γ e and γ b .From this competing relation, we can expect that the burstmode only occurs when γ b > κ . In other words, for run-away stimulated collisions certain thresholds of atom num-ber ( N > N t ) and quadratic Zeeman energy ( q < q t ) are re-quired. For both initial conditions ( q / h = . q / h = − . | q | , the threshold atom num-ber N t for negative q is smaller than that of positive q . Weattribute such difference to the immiscible dynamics between Azimuthal angle, (°)
Intra-spin correlationInter-spin correlation (a)(b)
FIG. 3. Angular correlation functions of matter-wave jets at after28 ms of TOF under (a) q = 2.7 kHz and (b) q = -2 kHz. (b) The up-per panel shows the correlation between the same spins (intra-spin)and the lower panel shows the correlation between the opposite spins(inter-spin). Shaded areas mark one standard error based on 100 to120 realizations. Black solid lines are time-dependent the Bogoli-ubov calculations for the experimental parameters. the |↑(cid:105) and |↓(cid:105) states [29], which form magnetic spin domainsafter a hold time [Fig. 1(c) inset]. We suspect phase diffu-sion dynamics may occur during the domain wall formation,which could suppress the collective pair production processfor positive q .To uncover correlations in the matter-wave jets, we studythe angular correlation functions. C i j ( φ ) = (cid:10) n i ( φ (cid:48) ) n j ( φ (cid:48) + φ ) (cid:11) (cid:104) n i ( φ (cid:48) ) (cid:105) (cid:10) n j ( φ (cid:48) + φ ) (cid:11) , (1)where n i ( φ ) is the angular density of the emitted atoms inthe | i (cid:105) state and the brackets mean the angular and ensem-ble average, (cid:104) n i (cid:105) = (cid:10) π (cid:82) n i ( φ (cid:48) ) d φ (cid:48) (cid:11) ens . When the jets arein the | (cid:105) state for q > | c | , we study the correlation function C ( φ ) , and when jets are in |↑(cid:105) and |↓(cid:105) states, we study spinversus angular-moment correlations by investigating the inter-spin C ↑↓ ( φ ) and intra-spin states C ↑↑ ( φ ) & C ↓↓ ( φ ) correlationfunctions. For both experiments, we observe one sharp peaknear φ = ◦ and the other broad peak near φ = ◦ . Bothpeaks are well displayed in C [Fig. 3(a)], and for matter-wave jets of opposite spin states, each peak can be found inthe intra- and inter-spin state correlation functions [Fig. 3(b)],respectively.The peak near φ = ◦ is a result of the Hanbury Brown-Twiss (HBT) effect between different angular modes, whichwould be close to 2 in the ideal limit [30, 31]. The variance of (cid:104) N k (cid:105) (atom number with momentum k ) becomes (cid:104) N k (cid:105) [ (cid:104) N k (cid:105) + ] , since σ = [ C ii ( ) − ] (cid:104) N k (cid:105) + (cid:104) N k (cid:105) and C ii ( ) =
2. This re-sult indicates thermal-like fluctuations of the emitted jet pop-ulations. As indicated in the study [32], the reduced densitymatrix of the quantum state displays a Bose-Einstein distri-bution because of the entropy associated with the correlated
120 180 240
Azimuthal angle, ( o )1.01.1
120 180 240
Azimuthal angle, ( o ) (a) (b)(c) FIG. 4. As the spin rotation angle θ is increased from 0 to π / C i j (mean value of C i j ( φ ) in the interval φ ∈ [ ◦ , ◦ ] ) under spin-axis rotation. Solid and dashed lines arenumerical calculations accounting the field gradient effect [26]. Theerror bars represent one standard deviation of the mean. pairs. Indeed, the angular mode population for both spin statesis well described by the thermal distribution [26]. A similarobservation is also reported from the Ref. [10], where the au-thors connect the emerging thermal distribution to the Unruhradiation, and the pair production Hamiltonian can be consid-ered as a boosting transformation in an accelerating frame.The other peak near φ = ◦ implies momentum conser-vation of the emitted atom pair, and its width is broadenedbecause of the near-field effect [7, 31]. That is, the conden-sates cannot be regarded as a point source with the current ex-pansion time so that different modes are overlapped in the de-tection area and interfere. According to the simple near-fieldmodel [7], about three modes ( N m (cid:39)
3) can be overlapped, re-ducing the correlation peaks to C i j ( φ = ◦ ) = + / N m (cid:39) .
1. This is further supported by our time-dependent Bogoli-ubov calculation, which captures the angular correlation func-tions for all spin states at a quantitative level [Fig. 3(a) and(b)]. As we increase the expansion time in the calculation,these modes can be resolved, and the peak becomes higherand narrower.Matter-wave jets of opposite spins are of particular interestas the momentum correlation peak only appears for inter-spinstate, C ↑↓ ( φ = ◦ ) , indicating strongly correlation betweenspins and momentum like EPR state. In a homogeneous sys-tem (or infinite expansion time limit), the quantum state after time evolution can be written as, (cid:79) ε k ≈| q | ∞ ∑ j = λ j ( t ) (cid:104) ˆ a † k , ↑ ˆ a † − k , ↓ + ˆ a † k , ↓ ˆ a † − k , ↑ (cid:105) j | vac (cid:105) . (2)The tensor product excludes opposite momentum ( k = − k ),and λ j ( t ) = ( − i tanh ( ct )) j j !cosh ( ct ) [26]. Here, the triplet Bell pair state | Ψ T (cid:105) = ( ˆ a k , ↑ ˆ a − k , ↓ + ˆ a k , ↓ ˆ a − k , ↑ ) † | vac (cid:105) constitutes the macro-scopic entangled state, whose non-classical correlation can beshown by interfering the two spin states. For example, undera spin rotation ( |↑(cid:105) , |↓(cid:105) → |↑(cid:105)±|↓(cid:105)√ ), the triplet state becomesanother entangled state, | Ψ + (cid:105) = ( ˆ a k , ↑ ˆ a − k , ↑ + ˆ a k , ↓ ˆ a − k , ↓ ) † | vac (cid:105) .Analyzing the angular correlation function C i j ( φ ) for the en-tangled state | Ψ + (cid:105) , the correlation peak near φ = ◦ wouldappear only for the same spin state ( C ↑↑ & C ↓↓ ). Increasing thespin rotation angle θ from 0 to 2 π , these two quantum statesare transformed into each other, displaying oscillations in theintra- and inter-spin state correlation peaks at φ = ◦ .Taking the | F = , m F = (cid:105) upper hyperfine state as an in-termediate state, the spin axis can be rotated by applying amicrowave pulse [13]. When the matter-wave jets start to es-cape the trapped BECs ( t h = . C ↑↓ ( φ = ◦ ) gradually disappears[Fig. 4(a)], while it simultaneously emerges in the intra-spincorrelation functions C ↑↑ and C ↓↓ [Fig. 4(b)]. Displaying themomentum-correlation peaks in Fig. 4(c) as a function of therotation angle, we observe a clear oscillation with almost fullcontrast after 2 π rotation.The observed oscillations are clear signature that ourspin states are in a coherent superposition state with spin-momentum correlations. However, this does not constitutethe witness of macroscopic entanglement in a single-mode, aseach jet stream contains many particles in distinct modes, andthe global wavefunction may not be pure. Searching for non-classical correlations, we take a bipartite separability criterionfor the collective spins [33]. W = ∑ i = x , y , z Var (cid:0) ˆJ i (cid:1) (cid:10) ˆJ (cid:11) , (3)where W < (cid:10) ˆJ (cid:11) = (cid:42) ∑ m = ↑ , ↓ [ N m ( φ ) + N m ( φ + π )] (cid:43) , ˆJ i = ˆ P i ( φ ) ± ˆ P i ( φ + π ) , ˆ P i ( φ ) = (cid:90) ∆ φ / − ∆ φ / [ ˆ n ↑ i ( φ + φ (cid:48) ) − ˆ n ↓ i ( φ + φ (cid:48) )] d φ (cid:48) . The index i = ( x , y , z ) refers to the spin axis, and the transverse(longitudinal) spin length takes a minus (plus) sign. Angular bin size, (°) -1 ExperimentTheory, TOF 28 ms100 ms150 ms W i t ne ss , W FIG. 5. Entanglement witness at various angular bin sizes. The en-tanglement can be certified when W < Figure 5 displays the experimentally measured and theo-retically calculated entanglement witness as a function of theangular bin size ∆ φ . We assume the transverse spin length tobe the averaged value of the ˆJ x and ˆJ y because the phase of thespin state is independent of the microwave phase. Both resultsshow that the entanglement witness is far above the entangle-ment criterion W <
1. The experiment’s deviation from thecalculation is attributed to the residual field gradient effect,atom loss during the microwave dressing, and atom numbercounting uncertainties [26]. Even without these experimentalartifacts, however, the calculation shows that the interferencebetween different angular momentum modes dims the infor-mation of the corresponding pair, so that the non-separabilitycriterion cannot be satisfied with 28 ms of expansion time.This trap geometry requires a long expansion time ( ∼
100 ms)to resolve individual angular momentum modes, where we ob-serve the two correlation peaks become almost symmetric. Inthis respect, reducing the possible number of emitting modescan be a promising way to generate certifiable macroscopicentanglement, for example, by using quasi-one-dimensionaltrap geometry that exhibits only two outgoing modes [34].In conclusion, we have observed directional matter-wavejets of various spin states via the “superradiant” spin-mixingscattering process in spinor condensates. The matter-wave jetsexhibit strong correlations between spins and momentum andhold the promise of being a macroscopic EPR state. This workopens up new perspectives for the study of quantum atomoptics and quantum simulations using spinor BECs. It canbe applied to precision measurement in an atom interferome-ter [19], to test Bell’s inequality with massive particles [35],and to study non-equilibrium quantum dynamics with fast-moving impurities in a quantum liquid [36].The authors thank Andrey Moskalenko and Young-Sik Rafor discussion. This work was supported by the National Re-search Foundation of Korea Grant No.2019M3E4A1080401and Samsung Science and Technology Foundation BA1702- 06. S.C. acknowledges support from the Miller Institute forBasic Research in Science. ∗ [email protected][1] R. H. Dicke, Phys. Rev. , 99 (1954).[2] S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, J. Stenger,D. E. Pritchard, and W. Ketterle, Science , 571 (1999).[3] P. Wang, L. Deng, E. W. Hagley, Z. Fu, S. Chai, and J. Zhang,Phys. Rev. Lett. , 210401 (2011).[4] M. G. Moore and P. Meystre, Phys. Rev. Lett. , 5202 (1999).[5] A. Vardi and M. G. Moore, Phys. Rev. Lett. , 090403 (2002).[6] L. W. Clark, A. Gaj, L. Feng, and C. Chin, Nature (London) , 356 (2017).[7] H. Fu, L. Feng, B. M. Anderson, L. W. Clark, J. Hu, J. W.Andrade, C. Chin, and K. Levin, Phys. Rev. Lett. , 243001(2018).[8] A. P. Chikkatur, A. G¨orlitz, D. M. Stamper-Kurn, S. Inouye,S. Gupta, and W. Ketterle, Phys. Rev. Lett. , 483 (2000).[9] L. Feng, J. Hu, L. W. Clark, and C. Chin, Science , 521(2019).[10] J. Hu, L. Feng, Z. Zhang, and C. Chin, Nat. Phys. , 785(2019).[11] L. Krinner, M. Stewart, A. Pazmi˜no, J. Kwon, and D. Schneble,Nature (London) , 589 (2018).[12] C. Gross, H. Strobel, E. Nicklas, T. Zibold, N. Bar-Gill, G. Kur-izki, and M. K. Oberthaler, Nature (London) , 219 (2011).[13] B. L¨ucke, M. Scherer, J. Kruse, L. Pezz´e, F. Deuretzbacher,P. Hyllus, O. Topic, J. Peise, W. Ertmer, J. Arlt, L. Santos,A. Smerzi, and C. Klempt, Science , 773 (2011).[14] C. D. Hamley, C. S. Gerving, T. M. Hoang, E. M. Bookjans,and M. S. Chapman, Nat. Phys. , 305 (2012).[15] K. Lange, J. Peise, B. L¨ucke, I. Kruse, G. Vitagliano, I. Apel-laniz, M. Kleinmann, G. T´oth, and C. Klempt, Science ,416 (2018).[16] P. Kunkel, M. Pr¨ufer, H. Strobel, D. Linnemann, A. Fr¨olian,T. Gasenzer, M. G¨arttner, and M. K. Oberthaler, Science ,413 (2018).[17] W. Muessel, H. Strobel, D. Linnemann, D. B. Hume, and M. K.Oberthaler, Phys. Rev. Lett. , 103004 (2014).[18] I. Kruse, K. Lange, J. Peise, B. L¨ucke, L. Pezz`e, J. Arlt, W. Ert-mer, C. Lisdat, L. Santos, A. Smerzi, and C. Klempt, Phys.Rev. Lett. , 143004 (2016).[19] L. Pezz`e, A. Smerzi, M. K. Oberthaler, R. Schmied, andP. Treutlein, Rev. Mod. Phys. , 035005 (2018).[20] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. , 777(1935).[21] H. Pu and P. Meystre, Phys. Rev. Lett. , 3987 (2000).[22] L.-M. Duan, A. Sørensen, J. I. Cirac, and P. Zoller, Phys. Rev.Lett. , 3991 (2000).[23] S. Huh, K. Kim, K. Kwon, and J.-y. Choi, Phys. Rev. Research , 033471 (2020).[24] Y. Kawaguchi and M. Ueda, Physics Reports , 253 (2012).[25] F. Gerbier, A. Widera, S. F¨olling, O. Mandel, and I. Bloch,Phys. Rev. A , 041602 (2006).[26] See Supplement Material.[27] J. M. Vogels, K. Xu, and W. Ketterle, Phys. Rev. Lett. ,020401 (2002).[28] C. Klempt, O. Topic, G. Gebreyesus, M. Scherer, T. Henninger,P. Hyllus, W. Ertmer, L. Santos, and J. J. Arlt, Phys. Rev. Lett. , 195303 (2010). [29] S. De, D. L. Campbell, R. M. Price, A. Putra, B. M. Anderson,and I. B. Spielman, Phys. Rev. A , 033631 (2014).[30] R. Hanbury Brown and R. Q. Twiss, Nature (London) , 27(1956).[31] Z. Wu and H. Zhai, Phys. Rev. A , 063624 (2019).[32] G. I. Mias, N. R. Cooper, and S. M. Girvin, Phys. Rev. A ,023616 (2008).[33] T. S. Iskhakov, I. N. Agafonov, M. V. Chekhova, andG. Leuchs, Phys. Rev. Lett. , 150502 (2012).[34] T. Meˇznarˇsiˇc, R. ˇZitko, T. Arh, K. Gosar, E. Zupaniˇc, andP. Jegliˇc, Phys. Rev. A , 031601(R) (2020).[35] J. S. Bell, Physics Physique Fizika , 195 (1964).[36] C. Mathy, M. Zvonarev, and E. Demler, Nat. Phys. , 881(2012).[37] C. Simon and D. Bouwmeester, Phys. Rev. Lett. , 053601(2003).[38] J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, Rev. Mod.Phys. , 1 (1999).[39] J. M. Gerton, C. A. Sackett, B. J. Frew, and R. G. Hulet, Phys.Rev. A , 1514 (1999).[40] W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J.Heinzen, F. L. Moore, M. G. Raizen, and D. J. Wineland, Phys.Rev. A , 3554 (1993).[41] J. Est`eve, C. Gross, W. A., G. S., and M. K. Oberthaler, Nature(London) , 1216 (2007).[42] M. F. Reidel, P. B¨ohi, Y. Li, T. W. H¨ansch, A. Sinatra, andP. Treutlein, Nature (London) , 1170 (2010).[43] W. Muessel, H. Strobel, M. Joos, E. Nicklas, I. Stroescu,J. Tomkovi˘c, D. H. Hume, and M. K. Oberthaler, Appl. Phys.B , 69 (2013).[44] K. Hueck, N. Luick, L. Sobirey, J. Siegl, T. Lompe, H. Moritz,L. W. Clark, and C. Chin, Opt. Express , 8670 (2017).[45] M. V. Chekhova, G. Leuchs, and M. Zukowski, Optics Com-munications , 27 (2015).[46] A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (Dover Publications, 2003) p. 640.
Supplement Material for:“Emission of Spin-correlated Matter-wave Jets from Spinor Bose-Einstein Condensates”
I. EMISSION OF THE MATTER-WAVE JETS IN TWO DIMENSIONS
Like the super-radiant light scattering experiment, the directionality of the matter-wave jet is determined by the geometry ofthe condensates. This is because the gain factor for the collision amplification process is given by the wavefunction overlapbetween the source condensate and scattered pair state [5, 21]. When the condensate size is much larger than the characteristicwavelength of the pair state, a sufficiently large gain can be obtained, displaying matter-wave amplification. In our experiment,the condensates are prepared in two dimensions since the chemical potential (320 Hz) is about half of the trapping frequencyalong the vertical direction ( ω z = π ×
680 Hz). The axial motions of the condensates are restricted to the harmonic ground statewith the spatial extent of l z = (cid:112) ¯ h / M ω = . µ m. In the xy -plane, it roughly follows the Thomas-Fermi profile with R TF (cid:39) µ m. The generated atom pairs after spin-mixing collisions have a kinetic energy of few kHz, which have a characteristicwavelength of λ mw ∼ µ m. In such parameters, we satisfy the following conditions, l z < λ mw (cid:28) R TF , and therefore, thecollisional amplification processes are dominantly occurring in the planar direction (In FIG. S1). μ m0 ms 12 ms 18 ms 100 μ m 0 ms9 ms15 ms24 ms (a) (b) FIG. S1. in-situ image of the matter-wave jets. (a) Top and (b) side images during the matter-wave jet emission under the quadratic Zeemanenergy q = − . q = . II. CALIBRATION OF QUADRATIC ZEEMAN ENERGY
To characterize the quadratic Zeeman energy q , we first calibrate the external bias field B from the Rabi spectroscopy on the | F = , m F = (cid:105) → | F = , m F = (cid:105) state transition. The field uncertainty is about 0.2 mG at 1 G of the magnetic field, and thequadratic Zeeman energy is determined by q = h ×
610 Hz/G × B . The energy scale will be compared to the kinetic energy ofmatter-wave jets of spin | (cid:105) state in the next section.For the matter-wave jets of opposite spin states, | F = , m F = (cid:105) ( |↑(cid:105) ) and | F = , m F = − (cid:105) ( |↓(cid:105) ) , we apply a microwavedressing field to the atoms through a single-loop coil inside of our vacuum chamber so that the quadratic Zeeman energy can betuned to have negative value ( q <
0) [25]. The microwave field is red-detuned to the | , − (cid:105) ↔ | , − (cid:105) transition with a detuning δ d under the bias field B = . ( ) mG. This scheme minimizes the population in the F = | , (cid:105) ↔ | , − (cid:105) .Probing the Zeeman sub-level transitions ( | (cid:105) → |↑(cid:105) , |↓(cid:105) ) under the microwave field, we directly measure the magnitude of q ,instead of characterizing all experimental parameters, such as microwave polarization, detuning, and power. FIG. S2(a) showsthe frequency difference between the two transitions, which corresponds to 2 q , and we control the q by changing the detuning[FIG. S2(b)]. The measurements can be well described by an expression, q = q + (cid:16) f − f − (cid:112) ( f − f ) + Ω (cid:17) /
2, where f is the microwave frequency, and q , f , and Ω are the characteristic fit parameters of the dressing scheme. The measurementuncertainty of q under the dressing field is less than 100 Hz. -60 -50 -40 -30 -20 -10 0-4-3-2-10 q ( k H z ) -10 -5 0 5 10 Relative RF frequency (kHz) no i t a l upop l ano i t c a r F (a) (b) (kHz) q Detuning,
FIG. S2.
Calibration of negative quadratic Zeeman energy. (a) Typical spectrum of the Zeeman sub-level spectroscopy. We probe | , (cid:105) → | , ± (cid:105) transitions while the dressing field is on. The solid lines are fitted curve. Half of the frequency difference between thetwo transitions corresponds to the quadratic Zeeman energy, q . (b) We interpolate the data (the distance between two spectrums in (a)) tocalibrate q as a function of the microwave frequency, f . The solid line is the fitted curve we used for the interpolation. See the text for theexpression. The experiments are mostly performed around q = − III. KINETIC ENERGY AND QUADRATIC ZEEMAN ENERGY
The radial momentum of the atomic beam is measured by taking images with various expansion time (time-of-flight, TOF),and we calculate its average kinetic energy in radial direction ( E k ). The E k is plotted as a function of the quadratic Zeemanenergy in FIG. S3, which is well described by E k (cid:39) | q | and is consistent with the Bogoliubov theory ( E k (cid:39) | q | − | c | ) [24]. Thesmall spin-dependent interaction energy, c = − h ×
160 Hz (about 10% of q ), is hindered by the measurement uncertainties suchas limited TOF, trap curvature, and the finite size of the jets. This results indicate that the kinetic energy of the jets for all spinstates are originated from the internal Zeeman energy. -4 -2 0 2 4 6 Quadratic Zeeman energy, q (kHz) K i ne t i c ene r g y , E k ( k H z ) FIG. S3.
Kinetic energy of the jets.
By varying the free expansion time, we measured the kinetic energy of the emitted atom. The solidline is E k = | q | , representing the kinetic energy source is the excess internal Zeeman energy. The error bars show the uncertainty (1- σ ) of themomentum fit. The color code is for the sign of q , and we used a different quench protocol for each sign (see the main text). We keep the holdtime short (about 10% of the harmonic period) in order to reduce the effect of trapping potential. IV. THERMAL-LIKE DISTRIBUTION OF THE JETS
Figure S4 shows the histogram of the atom number for each spin state in the bin size of about 1 ◦ . The distribution of eachspin state is characterized by an exponentially decaying function convolved with a Gaussian function (accounting the detectionnoise), which is the thermal-like distribution as pointed out in [10, 32]. The slight difference in the distributions is attributed tothe atom loss in |↓(cid:105) state because of the microwave dressing, and its details are explained in the following section. Atom numbers in a bin P r obab ili t y FIG. S4.
Thermal like distribution of the jets.
The atom number is sampled with the angular bin size of 1 ◦ . Solid lines are fitted curves, anexponentially decaying function convolved with a Gaussian. We used the same data as FIG. 3(b) in the main text. V. DISCUSSION ON THE SEPARABILITY ANALYSIS
In this section, we discuss our experimental imperfections that could increase entanglement witness. We adopt the separabilitycriterion of the collective spin vectors that have been developed for bright squeezed vacuum state in quantum optics [33]. W = ∑ i = x , y , z Var (cid:0) ˆJ i (cid:1) (cid:10) ˆJ (cid:11) ≥ , (S1) (cid:10) ˆJ (cid:11) ≡ (cid:42) ∑ m = ↑ , ↓ [ N m ( φ ) + N m ( φ + π )] (cid:43) , ˆJ i ≡ ˆ P i ( φ ) ± ˆ P i ( φ + π ) , ˆ P i ( φ ) ≡ (cid:90) ∆ φ / − ∆ φ / [ ˆ n ↑ i ( φ + φ (cid:48) ) − ˆ n ↓ i ( φ + φ (cid:48) )] d φ (cid:48) . The index i = ( x , y , z ) refers to the spin axis, and the evaluation of the spin vector depends of the target Bell states. That is, threetriplet Bell pair states take a minus (plus) sign to calculate the transverse (longitudinal) spin length and the witness W triplet . Forthe singlet Bell pair state | Ψ S (cid:105) = (cid:0) ˆ a k , ↑ ˆ a − k , ↓ − ˆ a k , ↓ ˆ a − k , ↑ (cid:1) † | vac (cid:105) , the witness W singlet should be calculated with minus sign in all( x , y , and z ) directions of the spin vectors. Investigating the separability criteria for both quantum states, we present three mainsources that disturb entanglement certification besides the near field effect. Unbalanced atom losses.
We observe the variance of longitudinal spin vector linearly increases as a function of the angularbin size (atom number), Var ( ˆJ z ) ∝ N [FIG. S5(a)], which indicates spin selective atom losses [37]. Measuring the populationimbalance ( I = N ↑ − N ↓ ) after rotation of the spin axis, we observe the atom loss occurs in |↓(cid:105) state, which is estimated to about20% during the hold time t h = . | , − (cid:105) under the dressing field [38, 39], where its fraction is measured to 20% when we turn off the optical trap and reduce the atomicdensity. Detection uncertainty.
The calibration of atom numbers can be done by studying fluctuations of atom number differencein a coherent superposition state [14, 40–42]. This method allows one to count several hundreds of Rb atoms with few atomuncertainty even with absorption images [43]. However, it is not directly applicable to Li atoms because of its light mass (and0large doppler shift during imaging) and unresolved level structures of the D transition lines [44]. With our best efforts, thefluctuations are minimized at low probe beam intensity ( (cid:39) . I sat , I sat = . is the saturation intensity), where thevariance of number imbalance for the superposition state [( |↑(cid:105) − |↓(cid:105) ) / √ ] ⊗ N is 100 times larger than the total particle number:Var ( N ↑ − N ↓ ) ∼ × ( N ↑ + N ↓ ) . The contribution of the detection uncertainty in the entanglement witness would be around ∼
50 at the most [FIG. S5(a)].
Triplet-singlet mixing from the field gradient.
Under inhomogeneous magnetic field, the triplet Bell state | Ψ T (cid:105) = (cid:0) ˆ a k , ↑ ˆ a − k , ↓ + ˆ a k , ↓ ˆ a − k , ↑ (cid:1) † | vac (cid:105) can be transformed into the singlet Bell state | Ψ S (cid:105) after a hold time because of different phaseaccumulation speed in the spin-momentum pair states. The field gradient in our setup is estimated to 4 mG/cm, which leads topopulate ∼
30% (maximal) of the singlet component in the matter-wave jets during the 7.5 ms of hold time. This could increasethe entanglement witness for W triplet and reduce W singlet for the singlet Bell state [FIG. S5(b)]. Angular bin size, (°) W i t ne ss W triplet , expW singlet , expW triplet , theoryW singlet , theory0 90 180 Angular bin size, (°) e c na i r a v de z il a m r o N (a) (b) FIG. S5. (a) Normalized variances of the collective spin vectors ˆJ z and ˆJ n as a function of bin size. The index n stands for the normal to z -axis.We calculate the observables with two angular bins with a 180 ◦ difference (back-to-back propagating) as two modes for the separability test.The unbalanced loss leads quadratically increasing trend for z -directional collective spin variance (linear growth of the normalized variance).(b) Witness with different target Bell configurations. We calculate the witnesses for different target states using the same data. As the stateoscillates between singlet and triplet state, the calculated value of W triplet (or W singlet ) also oscillates in between the two theoretical lines. VI. THEORETICAL MODEL
We employ the Bogoliubov theory of inhomogeneous spinor condensate [24, 31] to understand the experiment. We try tofollow the notations introduced in the Ref. [24]. Here, { + , , −} denotes m F = { + , , − } states of F = H = ˆ H + ˆ V , where the non-interacting part is given asˆ H = (cid:90) d r ∑ m , m (cid:48) ∈{ + , , −} ˆ ψ † m ( r ) (cid:20) − ¯ h ∇ M + U trap ( r ) + qm δ mm (cid:48) (cid:21) ˆ ψ m (cid:48) ( r ) ≡ (cid:90) d r ∑ m , m (cid:48) ∈{ + , , −} ˆ ψ † m ( r ) ˆ h m ( r ) ˆ ψ m (cid:48) ( r ) . ˆ ψ m is the Bose field operator, M is the mass of the atom, U trap is the harmonic trapping potential, and q is the quadratic Zeemanenergy. The interaction part isˆ V = (cid:90) d r c : ˆ n ( r ) : + c : ˆ F ( r ) : = (cid:90) d r (cid:0) ˆ n − ˆ n ˆ n + (cid:1) g g c − c g c g c − c g g ˆ n − ˆ n ˆ n + + c ( ˆ ψ † − ˆ ψ † + ˆ ψ ˆ ψ + ˆ ψ †0 ˆ ψ †0 ˆ ψ + ˆ ψ − ) . c = g + g ( c = g − g ) is the spin independent (dependent) interaction coefficient. g = π ¯ h M a ( g = π ¯ h M a ), a = a B ( a = a B ) for Li, where a B is the Bohr radius [23]. The colons denote normal ordering. ˆ n = ∑ m ˆ n m = ∑ m ˆ ψ † m ( r ) ˆ ψ m ( r ) is thedensity operator, ˆ F ν = ∑ m , m (cid:48) ˆ ψ † m ( r )( f ν ) mm (cid:48) ˆ ψ m (cid:48) ( r ) with spin-1 matrices, f x = √ , f y = i √ − −
10 1 0 , f z = − . The first term describes the density-density interaction. The second term corresponds to the spin-mixing channel relevant for thejet formation.
Analytic expression for homogeneous system
In a homogeneous system, we can obtain an analytic expression of matter-wave jets. The procedure is similar to [24, 32]. Weexpand the field operator with the plain wave basis.ˆ ψ m ( r ) = √ Ω ∑ k ˆ a k , m e i k · r , where Ω is for the normalization. When the mean field part is Φ = ( , , ) T , the Bogoliubov Hamiltonian can be written asfollows. ˆ H B = ∑ k (cid:54) = , m Re ( E k , m ) (cid:18) ˆ b † k , m ˆ b k , m + (cid:19) + (cid:48) ∑ k (cid:54) = , m Im ( E k , m ) (cid:16) ˆ b k , m ˆ b − k , m + ˆ b † k , m ˆ b † − k , m (cid:17) , where E k , ± = (cid:112) ( ε k + q )( ε k + q + c n ) ˆ b k , ± = − (cid:115) ε k + q + c n + E k , ± E k , ± ˆ a k , ± + (cid:115) ε k + q + c n − E k , ± E k , ± ˆ a † − k , ∓ . E k is the complex energy eigenvalue, ε k ≡ ¯ h k / M is the single particle kinetic energy. The prime symbol ( (cid:48) ) reminds not todouble-count k . The second term represents the instability that is in our main interest. The energy eigenvalues for different q with a shorthand, c ≡ c n , is presented in FIG. S6.2 p k = j c j E k = j c j q = -10|c|q = 0|c|q = 2|c|q = 4|c| FIG. S6.
Bogoliubov dispersion.
Solid (dashed) lines are the real (imaginary) part of E k . For q larger than 2 | c | , we see the excitation gap.As q decreases, the gap decreases and the instability (imaginary energy) emerges. The instability dome gets narrower as the magnitude of the q increases. The width of the dome is simply 2 | c | for | q | (cid:29) | c | regime. For high quadratic Zeeman energy, | q | (cid:29) | c | , instability sharply peaks at ε k = | q | − | c | ≈ | q | . The energy eigenvalue becomespure imaginary at this point and Hamiltonian is further reduced to the following form.ˆ H B = (cid:48) ∑ ε k ≈| q | c (cid:16) ˆ a † − k , − ˆ a † k , + + ˆ a † − k , + ˆ a † k , − (cid:17) + H.c . Neglecting the condensate depletion effect, this Hamiltonian is time-independent. We obtain the equation of motion with the aidof SU(1,1) algebra. The final state of the jet, | f ( t ) (cid:105) is | f ( t ) (cid:105) = exp (cid:8) − i ˆ H B t / ¯ h (cid:9) | vac (cid:105) = (cid:48) (cid:79) ε k ≈| q | ∞ ∑ j = λ j ( t ) (cid:2) ( ˆ a − k , − ˆ a k , + + ˆ a − k , + ˆ a k , − ) † (cid:3) j | vac (cid:105) , where λ j ( t ) = ( − i tanh ( | c | t )) j j !cosh ( | c | t ) . Note that this state is analogous to the (one of the triplet form) bright squeezed vacuum states [45]. Time-dependent Bogoliubov theory
We perform numerical calculations that include experimental situations, such as the system’s finite size, condensate depletion,and harmonic curvature. Here, we summarize the setup for the numerical calculations we showed in the main text. In orderto describe the dynamics of the system, we consider a grand canonical ensemble with the potential, ˆ K = ˆ H − µ ˆ N , where µ is the chemical potential and ˆ N is the total particle number operator. We also apply the Bogoliubov prescription: ˆ ψ m ( r , t ) → e i ˆ Kt ( Φ m ( r )+ ˆ ϕ m ( r , )) e − i ˆ Kt , Φ m is the mean field stationary solution that describe the condensates and ˆ ϕ m is the non-condensatesoperator. We obtain Φ m ( r ) and µ using Gross-Pitaevskii equation under Thomas-Fermi approximation. The equation of motion,up to the first order of ˆ ϕ , is [31, 46] i ¯ h ∂∂ t ˆ ϕ m ( r , t ) = (cid:0) ˆ h m ( r ) − µ (cid:1) ˆ ϕ m + ∑ m , m (cid:48) , m (cid:48) (cid:16) C mm (cid:48) m m (cid:48) + C mm (cid:48) m (cid:48) m (cid:17) Φ ∗ m (cid:48) Φ m (cid:48) ˆ ϕ m + C mm m (cid:48) m (cid:48) Φ m (cid:48) Φ m (cid:48) ˆ ϕ † m , where C m m m (cid:48) m (cid:48) = c δ m m (cid:48) δ m m (cid:48) + c ∑ ν = x , y , z ( f ν ) m m (cid:48) ( f ν ) m m (cid:48) . We neglect the excitations of spin-0 component (spin- ± q < q >
0) case.3We continue with the q < q > ϕ + = ∑ l , n r (cid:16) u + , l , n r ( r , t ) ˆ α n r , l + v ∗ + , l , n r ( r , t ) ˆ β † l , n r (cid:17) , ˆ ϕ − = ∑ l , n r (cid:16) u − , l , n r ( r , t ) ˆ β l , n r + v ∗− , l , n r ( r , t ) ˆ α † l , n r (cid:17) . u and v are the eigenfunction for single-particle Hamiltonian, ˆ h m . ˆ α and ˆ β satisfy the bosonic commutation relation. Theindices ( l , n r ) is for the quantum numbers of the eigenfunctions. Because the system is in 2D regime (chemical potentialis half of the tight trapping frequency), dynamics can be neglected along the tight confinement direction. We use 2D polarcoordinates, r = ( ρ , θ ) and assume the trapping potential to be a simple harmonic oscillator. We assumed that the system isperfectly symmetric (with 8 Hz harmonic frequency); we neglect the effect from the slight ellipticity of the system. The resultingquantum numbers are ( l , n r ) for angular and radial quantum number with the following eigenfunction, φ ln r . φ ln r ( r ) = R ln r ( r ) e il θ √ π , R ln r ( r ) = (cid:115) n r ! ( n r + l ) ! k α l + l e − α x / x l L ln r (cid:18) α x (cid:19) , α = n r + l + . Here, L ln r is the associated Laguerre polynomial and x = k · r = (cid:112) M ω ( n r + l + ) / ¯ h · r . We can write u and v with thecoefficients U and V . u m , l , n r = ∑ n (cid:48) r U lm , n r n (cid:48) r φ ln (cid:48) r ( r ) , v m , l , n r = ∑ n (cid:48) r V lm , n r n (cid:48) r φ ln (cid:48) r ( r ) Due to the symmetry of the system, different l modes are decoupled. We construct the matrix equation of U and V for each l andevaluate the matrix form Hamiltonian using the orthonormality of the eigenfunction. Corresponding instability rate distributionsfor several q can be found in FIG. S7. -300 -200 -100 0 100 200 300 Angular quantum number, l N o r m a li z ed i n s t ab ili t y r a t e q = -0.4 kHzq = -1.0 kHzq = -1.6 kHz FIG. S7.
Instability for different angular mode.
Normalized instability rate as a function of the angular quantum number, l . As the internalenergy ( q ) increases, the distribution expand because of the increased density of states for higher energy. In FIG. 4(c) on the main text, we include the effect of field gradient. We add spin dependent phase term to ˆ h m → ˆ h m + pm ,where p = h × β for the interaction strength c → β c and β = ..