Probing spin superfluidity in a spinor Bose-Einstein condensate with a moving magnetic obstacle
PProbing spin superfluidity in a spinor Bose-Einstein condensate with a movingmagnetic obstacle
Joon Hyun Kim, Deokhwa Hong,
1, 2
Kyuhwan Lee,
1, 2 and Y. Shin
1, 2, ∗ Department of Physics and Astronomy, and Institute of Applied Physics, Seoul National University, Seoul 08826, Korea Center for Correlated Electron Systems, Institute for Basic Science, Seoul 08826, Korea (Dated: February 22, 2021)We study the critical energy dissipation in a spin-1 antiferromagnetic Bose-Einstein condensateby an oscillating magnetic obstacle, which is formed by focusing a near-resonant laser beam. Whenthe magnetic obstacle is weak not to saturate local spin polarization, a sudden onset of spin-waveexcitations is observed as the oscillation frequency increases, demonstrating the spin superfluidbehavior of the system. For a strong obstacle that causes density perturbations via local spinsaturation, half-quantum vortices (HQVs) are created by the moving magnetic obstacle, whichshows that the critical dissipative dynamics evolves from spin-wave emission to HQV shedding.Critical HQV shedding is further investigated using a pulsed linear motion of the obstacle, and weidentify two critical velocities to create HQVs with different core magnetization.
Spin superfluidity, the absence of energy dissipationin a spin current, is a fascinating macroscopic quantumphenomenon. It was first observed in liquid He [1]and recently investigated in various magnetic materi-als [2–4], suggesting its potential applications in spin-tronics [5]. One minimal setting allowing the remarkablephenomenon is a binary superfluid system, which con-sists of two symmetric superflowing components. Ow-ing to the Z symmetry, the system has two Goldstonemodes corresponding to pure phonons and magnons [6],which are associated with mass and spin superfluidity,respectively [7, 8]. In cold atom experiments, such a sym-metric binary superfluid system was realized with spin-1antiferromagnteic Bose-Einstein condensates (BECs) of Na. Its spin superfluid behavior was demonstrated byobserving the absence of damping in spin dipole oscilla-tions of trapped samples [9, 10]. Two sound modes inthe mass and spin sectors were also observed [11].One of the key characteristics of a superfluid is thecritical velocity for its frictionless flow against externalperturbations. In a conventional scalar superfluid withbroken U (1) symmetry, it is known that when it flowspast an obstacle, energy dissipation occurs above a cer-tain critical velocity via phonon radiation [12] and nucle-ation of vortices [13], arising from the local accumulationof superfluid phase slippages [14]. An interesting ques-tion on a spin superfluid is how it responds to a moving magnetic obstacle, i.e., an obstacle that induces differentperturbations to each spin component [15]. Based on theanalogy between the mass and spin sectors, it is expectedthat magnon excitations would be generated above a cer-tain critical velocity. However, the situation is differentfor vortex nucleation because its fundamental topologicalexcitations are vortices with fractional circulation, whichare called half-quantum vortices (HQVs) [16]. An HQVcontains both mass and spin circulations, and therefore,its nucleation cannot be fulfilled by a pure phase slip ∗ [email protected] process in the spin sector.In this paper, we investigate the critical dissipative dy-namics in a symmetric binary superfluid by an oscillatingmagnetic obstacle. Pertaining to a weak obstacle, whichdoes not saturate local spin polarization, a sudden onsetof spin-wave excitations is observed with increasing theoscillation frequency, which demonstrates the spin super-fluidity of the system. Surprisingly, the creation of HQVsis not observed for the weak magnetic obstacle whosespeed exceeds the spin sound velocity. On the other hand,with a strong magnetic obstacle, which can produce massdensity perturbations by inducing local saturation of spinpolarization, HQVs can be created by moving the obsta-cle above a certain critical velocity. Furthermore, we findthat the critical velocities are different for the two typesof HQVs with different core magnetization, which origi-nates from the magnetic property of the obstacle. Thisstudy demonstrates spin superfluid behavior of a binarysuperfluid system against external magnetic perturba-tions, and furthermore, reveals the evolution of criticaldissipative dynamics from spin-wave emission to HQVshedding in a spin superfluid.Our experiment starts with a BEC of Na inthe | F =1 , m F =0 (cid:105) hyperfine ground state in an opti-cal dipole trap [9]. The condensate contains about2 . × atoms and its Thomas-Fermi radii are( R x , R y , R z ) ≈ (162 , , . µ m for trapping frequen-cies of ( ω x , ω y , ω z ) = 2 π × (5 . , . , |↑(cid:105) ≡| m F =1 (cid:105) and |↓(cid:105) ≡ | m F = − (cid:105) , by applying a π/ | m F =0 (cid:105) state. The twospin components are miscible [17] and constitute a sym-metric binary superfluid. The intercomponent interac-tion strength, g ↑↓ , is comparable to the intracomponentinteraction strength, g , given as ( g − g ↑↓ ) /g ≈
7% [18],so the mass and spin sectors of the binary system are en-ergetically well separated. For the peak atomic densityat the condensate center, the density and spin healinglengths are ξ n ≈ . µ m and ξ s ≈ . µ m, respectively,and the speed of spin sound is c s = 0 . a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b Positionn ↓ n ↑ D en s i t y s V V c (b)(a) FIG. 1. Oscillating magnetic obstacle in a symmetric binarysuperfluid. (a) Schematic of the experiment. A focused near-resonant Gaussian laser beam, which provides a repulsive (at-tractive) potential for the spin– ↑ ( ↓ ) component, undergoes si-nusoidal oscillations along a linear path at the central regionof the trapped sample. (b) Spin and mass density variations,∆ n s and ∆ n , as functions of the obstacle strength V . For V > V c , the spin polarization is saturated due to the den-sity depletion of the spin– ↑ component. The insets displaythe representative density profiles of the spin– ↑ (yellow solid)and spin– ↓ (green dashed) components for weak (left) andstrong obstacles (right). the highly oblate condensate [11]. During the experi-ment, spin-changing collisions are suppressed by a largenegative quadratic Zeeman energy via microwave fielddressing [19]. The external magnetic field is 50 mG, andits gradient on the xy plane is cancelled to be less than0 . |↑(cid:105) and |↓(cid:105) states, respectively,with same peak magnitude V [11]. The beam propagatestoward the central region of the condensate along the z axis and its 1 /e radius is about 7 ξ s . We adiabaticallyramp up the obstacle beam for 300 ms and hold it for100 ms to stabilize the beam intensity. Then, we sinu-soidally oscillate the obstacle by manipulating a piezo-driven mirror for 1 s at variable oscillation frequency f .The obstacle position is given by x ( t ) = A [1 − cos(2 πf t )]with x = 0 denoting the sample center. The sweep dis-tance is 2 A ≈ µ m, over which the atomic column den-sity varies less than 5 %. After the stirring process, weramp down the obstacle beam for 300 ms and take a spin-sensitive phase-contrast image of the sample along the z direction to measure the spatial magnetization distribu-tion [16]. We let the condensate expand for 19 ms beforeapplying the imaging light, which facilitates the obser-vation of magnon excitations via their self-interferenceeffect [21] as well as HQVs with their expanded ferro-magnetic cores [16].Perturbations generated by the magnetic obstacle de- arb.unitsM Z +1 −1 | ↑ 〉
100 µm
FIG. 2. Generation of spin excitations in a spinor Bose-Einstein condensate (BEC) by an oscillating magnetic ob-stacle. Magnetization ( M z ) images of the condensate stirredwith an obstacle of V /V c ≈ . ≈ . f . The images were obtainedafter a 19-ms time-of-flight. Fully magnetized pointlike do-mains in (g) and (h) indicate half-quantum vortices (HQVs)with ferromagnetic cores. (i)-(l) Images of the spin– ↑ com-ponent for the same stirring conditions in (e)-(h), taken afterStern-Gerlach spin separation. The HQVs are distinguishableas density-depleted holes in (k) and (l). pend on the obstacle strength V . Figure 1(b) shows thespin and mass density variations, ∆ n s and ∆ n , inducedat the center of a stationary obstacle as a function of V , where ∆ n s ≡ n ↓ − n ↑ and ∆ n ≡ n ↓ + n ↑ − n with n ↑ ( ↓ ) being the density of spin– ↑ ( ↓ ) component and n being the total density without the obstacle. When V is small, the density profiles of the two spin componentsvary antisymmetrically, yielding ∆ n s = 2 V / ( g − g ↑↓ )with ∆ n = 0, i.e., only spin perturbations are generatedby the magnetic obstacle. However, when V is increasedover a certain critical strength V c , the spin– ↑ componentis locally depleted, resulting in ∆ n >
0, and thus, massperturbations are also induced by the magnetic obsta-cle. The critical strength is given by V c = ( g − g ↑↓ ) n/ n s = n and in our experiment, V c /µ ≈ .
5% with µ = ( g + g ↑↓ ) n/ V /V c < >
1) weak (strong).In Fig. 2, we display a series of magnetization images ofthe perturbed condensate for various stirring frequencies f with weak and strong magnetic obstacles of V /V c ≈ . f increases, spin fluctu-ations in the condensate are observed to be enhanced,indicating that energy dissipation occurs by the oscillat-ing magnetic obstacle. In the case of strong obstacle, itis noticeable that fully spin-polarized, pointlike domainsappear in the condensate at high f > f in our experiment, wherethe obstacle’s speed reaches over 3 c s . This suggests thatHQV nucleation requires both spin and mass currents,which is consistent with the spin-mass composite natureof the HQVs. HQVs can be indirectly generated viadissociation of spin vortices that have pure spin circu-lation [16, 22], but the spin vortices are energetically toocostly because of their density-depleted cores.To quantitatively characterize the energy dissipationby the oscillating magnetic obstacle, we measure the spa-tial variance of magnetization, σ M , at the central regionof the condensate. Figure 3 displays the growth of σ M as a function of the stirring frequency f . For the weakobstacle, we observe a sudden increase of σ M above acertain critical frequency of f c ≈ v c = 2 πAf c ≈ . c s . As f further increases, σ M is observed tobe saturated and eventually decrease above f = 16 Hz.We checked that the stirring time, 1 s, remains in thelinear regime with respect to the growth of σ M [23]. InRefs. [24–26], it was discussed that the excitations aresuppressed for a supersonic obstacle due to its finite size.For the strong obstacle, we observe that σ M startsgrowing slowly from low f > f c,v ≈ σ M indicates the generation of spin waves,while the later rapid increase is due to the HQV shedding,where the magnitude of σ M is significantly enhanced ow-ing to the fully magnetized vortex cores. The two-stepgrowth of σ M reveals that the critical dissipative dynam-ics evolves from spin-wave emission to HQV sheddingin the binary superfluid under the perturbations of thestrong magnetic obstacle. The critical velocity for theHQV shedding is measured to be v c,v ≈ . c s . As f increases over 10 Hz, σ M gradually de-creases. At the extreme case of f = 50 Hz, σ M ≈ . v c ,it gradually accumulates energy in the form of local cur-rents and density compression around the moving ob-stacle [29, 30]. In the case of oscillating motion, if theamount of the energy accumulated over the oscillationperiod falls short of the energy cost of a vortex dipole,it is likely to dissipate through wave emission. It was f (Hz) v max (mm/s) σ M ( a r b . un i t ) /V c FIG. 3. Critical energy dissipation in the binary superfluid.Magnetization variance σ M as a function of the oscillation fre-quency f for the weak (red circles) and strong (blue squares)obstacles. At the top axis, v max denotes the maximum speedof the oscillating obstacle. σ M was measured from the area of157 × µ m at the central region of the condensate. Eachdata point is the mean value of five to seven measurements ofthe same experiment, and its error bar represents their stan-dard deviation. The inset shows an expanded view on theboxed region at low f . also theoretically shown that the accelerated motion canstimulate the radiation of waves [27, 31]. Here we notethat the relation between phonon emission and vortexshedding was not elucidated in previous stirring experi-ments with atomic superfluid gases, although the criticalvelocities were identified by observing a sudden increaseof sample temperature [32–34], the onset of a pressuregradient [35], and the critical vortex shedding [36–38]. Inour experiment, spin-wave excitations as well as HQVsare directly detected using the magnetization imaging inthe effective 2D sample ( ξ s > R z ), which allows to de-cipher the two-step evolution of the critical dissipativedynamics.There are two types of HQVs according to the coremagnetization and it is an intriguing query which one ismore favorable to be nucleated for the given magneticobstacle. To examine the detailed aspects of the criti-cal HQV shedding, we perform a modified experiment,where the strong obstacle with V /V c ≈ . ≈ µ m with a constant velocity v to shed a fewpairs of vortices. In Fig. 4(a), representative magneti-zation images of the condensate after the linear sweepof the obstacle are shown for various velocities v . As v increases, we observe that the HQV shedding dynam-ics develops in four stages: (i) no excitation arises in thesample, (ii) a HQV dipole with spin– ↓ core begins to shedabove a critical velocity, (iii) a HQV dipole with spin– ↑ core is also created, and (iv) many HQVs of both typesare irregularly generated. The first-shed HQV dipole hascores of the same magnetization as the spin polarization
100 µm | ↑ 〉 v = 1.4 mm/s v (mm/s) O cc u rr en c e p r obab ili t y EitherSpin-↑ coreSpin-↓ core arb.unitsM Z −1+1100 µm (c)(b)(a) FIG. 4. Critical HQV shedding. (a) Magnetization images ofthe condensate after a linear sweep of the strong obstacle forvarious obstacle’s moving velocity v . (b) Occurrence proba-bilities for spin– ↓ -core HQV (red squares), spin– ↑ -core HQV(blue triangles), and either of them (black circles) as functionsof v . The probabilities for each v were obtained from four-teen to eighteen measurements of the same experiment. Thesolid lines denote a guide for the eyes to each data set basedon the sigmoid functions. (c) Examples of the magnetizationimages for v = 1 . ↑ -core HQVs appear.The two images on the right side are the images of the spin– ↑ component for the same stirring condition. induced by the obstacle.The occurrence probability P ↑ ( ↓ ) of the HQVs withspin– ↑ ( ↓ ) core is plotted in Fig. 4(b) as a function of v . The onset of vortex generation occurs with the spin– ↓ -core HQVs at v ≈ . v c,v in Fig. 3, probably due to the dif-ference of the obstacle’s motion. We find that HQVs withspin– ↓ core are always present when spin– ↑ -core HQVsappear at v ≤ . ↑ -core HQVs requires higher v , and that P ↑ increases more slowly than P ↓ . In the supersonic regime, v > . P ↓ begins tobe suppressed prior to P ↑ . At high v > ↑ -core HQVs appeared inthe condensate [Fig. 4(c)].The nucleation of spin– ↑ -core HQVs is notable be-cause the circulation is formed by the spin componentwhich experiences an attractive potential from the mag-netic obstacle. The quantum vortex shedding by an at-tractive obstacle was not observed in previous experi-ments [33, 34] and the role of the attractive stirrer is stilldebatable in numerical studies [31, 39]. To clarify theissue, we carried out the oscillating obstacle experimentwith a scalar condensate containing only the spin– ↓ com-ponent, where V /µ ≈ . ↑ componentand it was found that the critical velocity of the attractiveobstacle is higher than that of the repulsive one with thesame potential magnitude V . This observation seems tobe accounted for by the local Landau criterion at the ob-stacle position [27] and provides a qualitative explanationof the measured critical velocities in Fig. 4. Nevertheless,it is important to note that the HQV shedding dynamicscannot be fully described as the sum of the two indepen-dent vortex shedding processes. For example, no HQVswere created by the weak obstacle, whereas a penetra-ble moving obstacle can generate a vortex dipole in asingle-component condensate [28]. Note that HQVs haveshort-range interactions for different core magnetizationsand they are also dynamically coupled to magnons [40].In conclusion, we have studied the critical dissipativedynamics in an antiferromagnetic spinor BEC by movinga magnetic obstacle. The onset of spin-wave excitationswas observed for the weak obstacle, directly probing thespin superfluidity of the binary superfulid. The criticalHQV shedding was demonstrated with the strong obsta-cle and the two-step evolution of the critical dissipativedynamics provided insight on the hierarchy between waveemission and vortex generation in the superfluid. Aninteresting extension of this work is to investigate thespinor superfluid near the quantum critical point withzero quadratic Zeeman energy. Spin superfluidity waspredicted to vanish due to the full recovery of spin rota-tion symmetry [9] and novel topological objects such asmerons and skrymions may exist stably [41, 42]. ACKNOWLEDGMENTS
This work was supported by the Samsung Sci-ence and Technology Foundation (SSTF-BA1601-06),the National Research Foundation of Korea (NRF-2018R1A2B3003373, NRF-2019M3E4A1080400), andthe Institute for Basic Science in Korea (IBS-R009-D1). [1] A. S. Borovik-Romanov, Y. Bun’kov, V. V. Dmitriev, andY. Mukharskii, Observation of phase slippage during the flow of a superfluid spin current in He- B , JETP Lett. , 124 (1987).[2] D. A. Bozhko, A. A. Serga, P. Clausen, V. I. Vasyuchka,F. Heussner, G. A. Melkov, A. Pomyalov, V. S. L’vov,and B. Hillebrands, Supercurrent in a room-temperatureBose–Einstein magnon condensate, Nat. Phys. , 1057(2016).[3] W. Yuan, Q. Zhu, T. Su, Y. Yao, W. Xing, Y. Chen,Y. Ma, X. Lin, J. Shi, R. Shindou, X. C. Xie, and W. Han,Experimental signatures of spin superfluid ground statein canted antiferromagnet Cr O via nonlocal spin trans-port, Sci. Adv. , 1098 (2018).[4] P. Stepanov, S. Che, D. Shcherbakov, J. Yang, R. Chen,K. Thilahar, G. Voigt, M. W. Bockrath, D. Smirnov,K. Watanabe, T. Taniguchi, R. K. Lake, Y. Barlas,A. H. MacDonald, and C. N. Lau, Long-distance spintransport through a graphene quantum Hall antiferro-magnet, Nat. Phy. , 907–911 (2018).[5] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono,and Y. Tserkovnyak, Antiferromagnetic spintronics,Rev. Mod. Phys. , 015005 (2018).[6] Y. Kawaguchi and M. Ueda, Spinor Bose-Einstein con-densates, Phys. Rep. , 253 (2012).[7] E. B. Sonin, Spin currents and spin superfluidity, Adv.Phys. , 181 (2010).[8] J. Armaitis and R. A. Duine, Superfluidity and spin su-perfluidity in spinor Bose gases, Phys. Rev. A , 053607(2017).[9] J. H. Kim, S. W. Seo, and Y. Shin, CriticalSpin Superflow in a Spinor Bose-Einstein Condensate,Phys. Rev. Lett. , 185302 (2017).[10] E. Fava, T. Bienaim´e, C. Mordini, G. Colzi, C. Qu,S. Stringari, G. Lamporesi, and G. Ferrari, Obser-vation of Spin Superfluidity in a Bose Gas Mixture,Phys. Rev. Lett. , 170401 (2018).[11] J. H. Kim, D. Hong, and Y. Shin, Observation of twosound modes in a binary superfluid gas, Phys. Rev. A , 061601(R) (2020).[12] G. E. Astrakharchik and L. P. Pitaevskii, Motion ofa heavy impurity through a Bose-Einstein condensate,Phys. Rev. A , 013608 (2004).[13] E. Varoquaux, Anderson’s consideration on the flow ofsuperfluid helium: Some offshoots, Rev. Mod. Phys. ,803 (2015).[14] B. Jackson, J. F. McCann, and C. S. Adams, VortexFormation in Dilute Inhomogeneous Bose-Einstein Con-densate, Phys. Rev. Lett. , 3903 (1998).[15] J. H. Jung, H. J. Kim, and Y. Shin, Spin andmass currents near a moving magnetic obstacle in atwo-component Bose–Einstein condensate, J. KoreanPhys. Soc. , 19-26 (2021).[16] S. W. Seo, S. Kang, W. J. Kwon, and Y. Shin, Half-Quantum Vortices in an Antiferromagnetic Spinor Bose-Einstein Condensate, Phys. Rev. Lett. , 015301(2015).[17] J. Stenger, S. Inouye, D. M. Stamper-Kurn, H.-J. Mies-ner, A. P. Chikkatur, and W. Ketterle, Spin domainsin ground-state Bose-Einstein condensates, Nature (Lon-don) , 345 (1998).[18] S. Knoop, T. Schuster, R. Scelle, A. Trautmann, J. App-meier, M. K. Oberthaler, E. Tiesinga, and E. Tiemann,Feshbach spectroscopy and analysis of the interaction po-tentials of ultracold sodium, Phys. Rev. A , 042704(2011).[19] F. Gerbier, A. Widera, S. F˝olling, O. Mandel, and I. Bloch, Resonant control of spin dynamics in ultracoldquantum gases by microwave dressing, Phys. Rev. A ,041602(R) (2006).[20] J. H. Kim, D. H. Hong, S. Kang, and Y. Shin, Metastablehard-axis polar state of a spinor Bose-Einstein conden-sate under a magnetic field gradient, Phys. Rev. A ,023606 (2019).[21] S. W. Seo, J. Choi, and Y. Shin, Scaling behavior of den-sity fluctuations in an expanding quasi-two-dimensionaldegenerate Bose gas, Phys. Rev. A , 043606 (2014).[22] F. Manni, K. G. Lagoudakis, T. C. H. Liew, R. Andr,V. Savona, and B. Deveaud, Dissociation dynamics ofsingly charged vortices into half-quantum vortex pairs,Nat. Commun. , 1309 (2012).[23] See Supplemental Material for further details on the mag-netic obstacle, the stirring time dependence of σ M , andvortex generation by an attractive obstacle.[24] A. Radouani, Soliton and phonon production by an oscil-lating obstacle in a quasi-one-dimensional trapped repul-sive Bose-Einstein condensate, Phys. Rev. A , 013602(2004).[25] P. Engels and C. Atherton, Stationary and Nonstation-ary Fluid Flow of a Bose-Einstein Condensate Through aPenetrable Barrier, Phys. Rev. Lett. , 160405 (2007).[26] F. Pinsker, Gaussian impurity moving through a Bose-Einstein superfluid, Phys. B: Condens. Matter ,36–42 (2017).[27] B. Jackson, J. F. McCann, and C. S. Adams, Dissipationand vortex creation in Bose-Einstein condensed gases,Phys. Rev. A , 051603(R) (2000).[28] W. J. Kwon, S. W. Seo, and Y. Shin, Periodic sheddingof vortex dipoles from a moving penetrable obstacle ina Bose-Einstein condensate, Phys. Rev. A , 033613(2015).[29] T. Frisch, Y. Pomeau, and S. Rica, Transition to Dissipa-tion in a Model of Superflow, Phys. Rev. Lett. , 1644(1992).[30] T. Winiecki, J. F. McCann, and C. S. Adams, PressureDrag in Linear and Nonlinear Quantum Fluids, Phys.Rev. Lett. , 5186 (1999).[31] V. P. Singh, W. Weimer, K. Morgener, J. Siegl, K. Hueck,N. Luick, H. Moritz, and L. Mathey, Probing super-fluidity of Bose-Einstein condensates via laser stirring,Phys. Rev. A , 023634 (2016).[32] C. Raman, M. K¨ohl, R. Onofrio, D. S. Durfee, C. E. Kuk-lewicz, Z. Hadzibabic, and W. Ketterle, Evidence fora Critical Velocity in a Bose-Einstein Condensed Gas,Phys. Rev. Lett. , 2502 (1999).[33] R. Desbuquois, L. Chomaz, T. Yefsah, J. Leonard,J. Beugnon, C. Weitenberg, and J. Dalibard, Superfluidbehaviour of a two-dimensional Bose gas, Nat. Phys. ,645 (2012).[34] W. Weimer, K. Morgener, V. P. Singh, J. Siegl, K. Hueck,N. Luick, L. Mathey, and H. Moritz, Critical Velocity inthe BEC-BCS Crossover, Phys. Rev. Lett. , 095301(2015).[35] R. Onofrio, C. Raman, J. M. Vogels, J. R. Abo-Shaeer, A. P. Chikkatur, and W. Ketterle, Observationof Superfluid Flow in a Bose-Einstein Condensed Gas,Phys. Rev. Lett. , 2228 (2000).[36] T. W. Neely, E. C. Samson, A. S. Bradley, M. J. Davis,and B. P. Anderson, Observation of Vortex Dipoles in anOblate Bose-Einstein Condensate, Phys. Rev. Lett. ,160401 (2010). [37] W. J. Kwon, G. Moon, S. W. Seo, and Y. Shin, Criticalvelocity for vortex shedding in a Bose-Einstein conden-sate, Phys. Rev. A , 053615 (2015).[38] J. W. Park, B. Ko, and Y. Shin, Critical Vortex Shed-ding in a Strongly Interacting Fermionic Superfluid,Phys. Rev. Lett. , 225301 (2018).[39] T. Aioi, T. Kadokura, T. Kishimoto, and H. Saito, Con-trolled Generation and Manipulation of Vortex Dipolesin a Bose-Einstein Condensate, Phys. Rev. X , 021003(2011).[40] S. W. Seo, W. J. Kwon, S. Kang, and Y. Shin, Colli-sional Dynamics of Half-Quantum Vortices in a SpinorBose-Einstein Condensate, Phys. Rev. Lett. , 185301(2016).[41] J. Choi, W. J. Kwon, and Y. Shin, Observation of Topo-logically Stable 2D Skyrmions in an AntiferromagneticSpinor Bose-Einstein Condensate, Phys. Rev. Lett. ,035301 (2012).[42] A. P. C. Underwood, D. Baillie, P. B. Blakie, andH. Takeuchi, Properties of a nematic spin vortex inan antiferromagnetic spin-1 Bose-Einstein condensate,Phys. Rev. A , 023326 (2020).[43] R. Grimm, M. Weidem¨uller, and Y. B. Ovchin-nikov, Optical dipole traps for neutral atoms,Adv. At. Mol. Opt. Phys. , 95 (2000). Supplemental Material
Magnetic obstacle
For Na in the F =1 hyperfine ground state, the dipolepotential generated by a laser beam is given as U m F ( r ) = 3 πc Γ2 ω I ( r ) (cid:16) − g F m F P + 2 + g F m F P (cid:17) , (S1)where c is the speed of light, ω is the resonance fre-quency for the 3 S → P transition, Γ is the decay rateof the excited state, I ( r ) is the intensity of the laser beam, g F = − is the Land´e g -factor, m F = 0 , ± F on the quantization axis set by the laserbeam propagation direction, P = 0 , ± π – and σ ± –polarization, and ∆ , is the frequency detuning of thelaser beam with respect to the D , transition line [43].Here, the hyperfine structures of the excited state are ne-glected, assumed that their gaps are small enough com-pared with the frequency detunings ∆ , .In the experiment of the main text, we used a 589-nmnear-resonant laser beam to produce a magnetic obstaclefor Na [11]. The frequency of the laser beam was setto have ∆ = − ∆ /
2, providing U − = − U and U = 0regardless of P [Fig. S1(a) and (b)]. Such antisymmetricpotentials can be implemented as a magnetic obstacle by
100 µm m F =+1m F = 0m F =−1 m F =+1m F = 0m F =−1−u U m F S Δ −Δ P P (c) (b)(a) (d) FIG. S1. Spin-dependent optical potential for Na. (a) D line doublet of Na with denoting the optical frequency con-ditions for the magnetic obstacle laser beam. ∆ , indicatesthe frequency detuning from the D , transition line. (b)Schematic of the dipole potential U m F of the laser beamwith σ − -polarization. The frequency detuning condition of∆ = − ∆ / π ×
172 GHz yields U − = − U and U = 0. u = (3 πc Γ I ) / (8 ω | ∆ | ). (c) The focused laser beam pene-trates the condensate to form a magnetic obstacle. (d) In-situ absorption images of the condensate in each spin state. In thebottom, the density profiles along the horizontal line crossingthe obstacle position are shown. focusing and penetrating the laser beam to the conden-sate [Fig. S1(c)]. The sign of U ± can be inverted bychanging the sign of P . In our experiment, we used themagnetic obstacle beam that is repulsive for the m F =1state and attractive for the m F = − V was calibrated from the in-situ densityprofiles of the spin components [Fig. S1(d)]. stir (s)0.020.030.04 σ M ( a r b . un i t ) arb. unitsM Z −1+1100 µm FIG. S2. Evolution of σ M as a function of the stirring time t stir . The weak magnetic obstacle of V /V c ≈ . f = 10 Hz,which is above the critical frequency ≈ σ M wasfound to increase linearly with t stir up to 1 s. Each data pointwas obtained from five measurements of the same experimentand its error bar indicates their standard deviation. In theupper row, the magnetization images of the condensate aredisplayed for t stir = 0, 0.5, and 1 s, respectively. Vortex generation by an attractive obstacle
In order to address the question whether quantum vor-tices can be generated by a moving attractive obsta-cle, we performed the same stirring experiment with acondensate prepared to contain only the spin– ↓ com-ponent, where the magnetic obstacle acted as an at-tractive obstacle. As in the main experiment, the ob-stacle beam sinusoidally oscillates along a linear pathfor 1 s with 2 A ≈ µ m. The obstacle strength is V /µ ↓ ≈ .
7, where µ ↓ = gn is the chemical potentialof the condensate. We indeed observed that quantumvortices were generated in the condensate by the oscil-lating attractive obstacle above a certain stirring fre-quency (Fig. S3) [31, 39]. In previous stirring experi-ments using attractive optical obstacles [33, 34], vortexgeneration was not observed and it was attributed to the O cc u rr en c e p r obab ili t y arb. unitsOD02
10 Hz 15 Hz 20 Hz
100 µm | ↓ 〉 FIG. S3. Vortex occurrence probability for an oscillating at-tractive obstacle as a function of the stirring frequency f .The solid line denotes a sigmoidal function fit to the data.Each data point was obtained from five measurements of thesame experiment. In the upper row, the optical density im-ages of the condensate are shown for f = 10, 15, and 20 Hz,respectively. small size of the obstacles. In our experiment, the 1 /e radius of the obstacle was ≈ ξ n , where ξ n is the densityhealing length of the condensate. In Fig. S3, we displaythe occurrence probability P ( v ) for quantum vortices inthe condensate for various stirring frequencies f . From P ( f ) = 0 .
5, the threshold frequency was estimated to be ≈
16 Hz, where the maximal speed of the oscillating ob-stacle is ≈ . ↑↑