In this paper we demonstrate a technique that can create out-of-equilibrium
vortex configurations with almost arbitrary charge and geometry in a
Bose-Einstein condensate. We coherently transfer orbital angular momentum from
a holographically generated light beam to a Rubidium 87 condensate using a
two-photon stimulated Raman process. Using matter wave interferometry, we
verify the phase pattern imprinted onto the atomic wave function for a single
vortex and a vortex-antivortex pair. In addition to their phase winding, the
vortices created with this technique have an associated hyperfine spin texture.Comment: 4 pages, 5 figure

Bakr, Waseem S.

Brachmann, Johannes F. S.

Gillen, Jonathon

Greiner, Markus

Peng, Amy

Optics Express

English

arXiv

In this paper we demonstrate a technique that can create non-equilibrium vortex configurations with almost arbitrary charge and geometry in a Bose-Einstein condensate. We coherently transfer orbital angular momentum from a holographically generated light beam to a 87Rb condensate using a two-photon stimulated Raman process. Using matter wave interferometry, we verify the phase pattern imprinted onto the atomic wave function for a single vortex and a vortex-antivortex pair. In addition to their phase winding, the vortices created with this technique have an associated hyperfine spin texture.Physic

Brachmann, Johannes

Bakr, Waseem

Harvard University - DASH 

 Inducing Vortices in a Bose-Einstein Condensate UsingHolographically Produced Light Beams  (Article begins on next page)The Harvard community has made this article openly available.Please share how this access benefits you. Your story matters.Citation Brachmann, Johannes, Waseem S. Bakr, Jonathon Gillen, AmyPeng, and Markus Greiner. 2011. Inducing vortices in a Bose-Einstein condensate using holographically produced light beams.Optics Express 19(14): 12984-12991.Published Version doi:10.1364/OE.19.012984Accessed February 19, 2015 9:24:01 AM ESTCitable Link http://nrs.harvard.edu/urn-3:HUL.InstRepos:8160885Terms of Use This article was downloaded from Harvard University's DASHrepository, and is made available under the terms and conditionsapplicable to Open Access Policy Articles, as set forth athttp://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAParXiv:1102.3373v1  [cond-mat.quant-gas]  16 Feb 2011Inducing vortices in a Bose-Einstein condensate usingholographically produced light beamsJ. F. S. Brachmann,1,2,3 W. S. Bakr,1,2 J. Gillen,1,2,4 A. Peng,1,2,5 and M. Greiner1,21MIT-Harvard Center for Ultracold Atoms, Cambridge, MA 02138, USA2Physics Department, Harvard University, Cambridge, MA 02138, USA3Ludwig-Maximilians-Universit¨ at, Schellingstraße 4, 80799 M¨ unchen, Germany4Physics Department, Massachusetts Institute of Technology, Cambridge, MA 02139, USA5Washington University School of Medicine, Saint Louis, MO 63110, USA(Dated: February 17, 2011)In this paper we demonstrate a technique that can create out-of-equilibrium vortex conﬁgurationswith almost arbitrary charge and geometry in a Bose-Einstein condensate. We coherently transferorbital angular momentum from a holographically generated light beam to a87Rb condensate usinga two-photon stimulated Raman process. Using matter wave interferometry, we verify the phasepattern imprinted onto the atomic wave function for a single vortex and a vortex-antivortex pair.In addition to their phase winding, the vortices created with this technique have an associatedhyperﬁne spin texture.Light ﬁelds that carry orbital angular momentum(OAM), have attracted a lot of attention from researchersin ﬁelds as diverse as biophysics [1], micromechanics and-fabrication [2], astronomy [3], quantum communication[4] and the ﬁeld of ultracold atoms [5]. While spin an-gular momentum is associated with the polarization ofthe light ﬁeld, orbital angular momentum is associatedwith the spatial mode of the ﬁeld. In the paraxial ap-proximation, both angular momentum components canbe measured separately [6]. It has been shown [7] thatparaxial Laguerre-Gaussian laser beams, which are feasi-ble to create experimentally, carry a well-deﬁned orbitalangular momentum associated with their spiral wave-fronts. More generally, light beams that contain a phasesingularity, carry OAM parallel or anti-parallel to theirpropagation direction depending on the phase variationaround the singularity. If more than one singularity ispresent, (qleft − qright)¯ h OAM quanta are contained inthe beam. Here the quantum numbers qleft and qrightdetermine the charge and chirality (left- or right handedspiraling of the wavefront) of the optical vortices.The torque produced through momentum exchange be-tween an OAM light beam and matter can be used to in-duce rotation of the so illuminated object. In biophysicsand micromechanics this torque has been proposed tobe utilizable in driving molecular motors and microma-chines, which can at the same time be trapped at thelocation of the singularity, where the beam intensity hasto vanish [1, 8–10]. Also, in the ﬁeld of ultracold atoms,direct transfer of OAM from light to atoms in an ultra-cold gas cloud oﬀers interesting possibilities [11–14].An analogue of optical vortices can be realized in su-perﬂuids, where orbital angular momentum can only beadded in the form of quantized vortices. In liquid helium,vortices have been studied experimentally since the 1960s[15]. With the achievement of Bose-Einstein condensa-tion (BEC) in dilute atomic gases, a very controllablemacroscopic quantum object is available, which has al-ready been used extensively for further study of quan-tized vortex states[16–19]. In the regime where the num-ber of vortices is much smaller than the number of atomsin the condensate, the manybody wavefunction of theBEC can be approximated using mean ﬁeld theory. Inthis description, every particle in the condensate carriesthe same angular momentum, quantized in units of ¯ h.Bose-Einstein condensates have opened the path tostudying many new regimes in vortex physics. For ex-ample, because of the analogy to quantum Hall physics,the low ﬁlling factor regime of atoms to vortices is inter-esting to reach [20–22]. Also, the Berezinskii-Kosterlitz-Thouless phase transition characterized by the bind-ing/unbinding of thermally activated vortex-antivortexpairs in two-dimensional gases is subject of current re-search [23, 24].Although the ﬁrst vortices in BECs were producedby a phase imprinting technique [25], vortices are mostcommonly produced by stirring with an anisotropic trap-ping potential. This can be created for example by theuse of a rotating detuned laser beam [16]. In this way,Abrikosov lattices of vortices with the same circulationdirection are created. With this technique, it is not easyto create vortices with diﬀerent circulation directions,non-equilibrium vortex states such as multiply chargedvortices, arbitrary vortex lattices, or superpositions ofstates with diﬀerent vorticity. However, other methodsof producing vortices have demonstrated some of thesepossibilities [26–28].Recently, a diﬀerent approach to create vortices hasbeen demonstrated: Orbital angular momentum wastransfered in units of ¯ h from photons in a Laguerre-Gaussian beam to atoms in a BEC using a two-photonRaman process [11–14]. In this way, superpositions ofvortex states could be created in a controlled manner.To be able to really take advantage of this method, oneneeds the ability to create OAM beams in a versatile way.Since a hologram is, in the paraxial approximation, capa-ble of creating any desired wavefront, it can be the righttool for OAM beam creation if manufactured in good2FIG. 1: (a) Raman scheme for87Rb, D1 line, using a Gaussianbeam and the ﬁrst order diﬀracted beam from a hologram. ∆is the common detuning of both beams from the intermediate|52P1/2;F = 2  state. (b) interference pattern of a Gaussianbeam with the ﬁrst order diﬀracted beam from a hologramfor producing optical vortices with charges 1,2,4 and 8. Thekey structures from the original binary hologram producingthe respective phase windings are shown as cut outs and arerelated to the interference pattern.enough quality. We make use of standard lithographictechniques for chrome mask manufacturing to obtain bi-nary approximations of greyscale holograms for diﬀerentOAM beams in very good quality. Figure 1(b) showscut-outs of a hologram used to produce multiply chargedoptical vortices as an example.In this paper we present results from inducing vorticesby transferring OAM from a holographically producedlight beam to a 87Rb BEC using a two-photon Ramanprocess shown in Figure 1(a). With a suitable hologram,we are able to directly produce a rotational state of theBEC with a vortex-antivortex pair [28]. The coherence ofthe process is shown by atom interferometry, with whichit can be seen that the phase of the holographically pro-duced light beam is imprinted onto the condensate wave-function.Our holograms do not create Laguerre-Gaussianbeams, but rather beams with a Gaussian proﬁle ex-cept at the points of the phase singularities. The sizeof the optical vortices is determined by the resolution ofthe projection optics (about 5µm). By using Gaussianbeams with a large waist, the intensity proﬁle away fromthe vortex core becomes uniform over the condensate, en-suring a homogeneous transfer rate to the vortex state.Since the beam intensity close to the phase singularityis nevertheless close to zero, atoms in the vortex core donot undergo the Raman transition. As a result, the vor-tices that we produce have cores ﬁlled with a diﬀerentspin state. Both spin states can be imaged separately.In the experiment, the atoms start in the F = 1 stateand are coherently transferred to the F = 2 state by illu-minating them with two copropagating beams, a Gaus-sian beam and a beam carrying OAM that is producedby a hologram (Figure 1(a)). An atom transferring aphoton from the hologram beam to the Gaussian beamgains q¯ h orbital angular momentum quanta, where q isthe number of OAM quanta in the hologram beam. Sincethe beams are copropagating, no net linear momentumis transferred to the atoms by this process.The diﬀerence in frequency of the two light ﬁelds isclose to 6.8GHz corresponding to the hyperﬁne splittingbetween the initial F = 1 and ﬁnal F = 2 state of thetransition. σ+ - σ− polarizations are chosen, resulting ina change of the magnetic state by ∆mF = +2. A weakhomogeneous magnetic ﬁeld pointing in the same direc-tion as the beams is applied during the Raman pulses forthe deﬁnition of a quantization axis.Our experiments begin with a 87Rb BEC of approxi-mately 3 × 105 atoms in the |52S1/2;F = 1,mF = −1 state. The condensate is in a spherical quadrupole-Ioﬀeconﬁguration (QUIC) trap with a trapping frequency ofω = 2π × 20Hz and has a Thomas-Fermi radius of ap-proximately 15µm. 14ms after releasing the atoms fromthe magnetic trap, when the radius of the cloud is about30µm, we pulse on the two Raman beams to project thevortex pattern onto the expanded cloud. The beams,travelling horizontally, are applied as square pulses tothe vertically falling cloud and change the internal atomicstate to |52S1/2;F = 2,mF = 1 . Beam powers of 10µWand waists of 131µm are used with square pulse durationsof 8µs. The common detuning of both Raman beamsfrom the |52P1/2;F = 2  state is set to ∆ = 4.18GHz.Spontaneous emission during the Raman pulse durationis negligible and does not introduce decoherence relevantto the experiment. Absorption images of the condensateare taken 3ms after applying the Raman pulses.Figures 2(a),(b) show absorption images of BECs witha single vortex and two counter-rotating vortices re-3spectively. In the pictures, only atoms projected onto|52S1/2;F = 2,mF = 1  are imaged, although a fractionof the condensate in |52S1/2;F = 1,mF = −1  is stillpresent. The vortex core sizes in these pictures is limitedby the resolution of the projection optics.We emphasize here that this technique produces spinvortices [29], as opposed to stirring techniques which pro-duce charge (density) vortices. To produce the vortexin Figure 2(a), the intensity/pulse length is adjusted todrive a π Raman transition far enough from the singu-larity. If we regard F = 1,2 as a two level system, thespin vector rotates from (F = 1) at the center of thevortex to (F = 2) at the edge of the core region, wherethe OAM beam intensity becomes constant. At the sametime, the projection of the spin vector in a plane perpen-dicular to the direction of the beam propagation rotatesazimuthally by 2π due to the phase winding in the op-tical vortex. This spin texture, shown in Figure 2(c),describes a skyrmion. Skrymions have been used as amodel for baryonic particles in nuclear physics [30, 31].Multi-component vortices have also been studied previ-ously in BECs by other authors [13, 14, 26, 32].To show the coherence of the process, the atomic vor-tex state is interfered with an atomic plane uniform phasestate. This is done by applying two separate π2 Ramanpulses as shown in Figure 3(a): the ﬁrst pulse uses thehologram beam and the Gaussian beam to transfer afraction of atoms into a vortex state as before. Imme-diately afterwards, a second pulse is executed where thehologram beam is replaced with a second copropagatingGaussian beam at the same frequency to transfer an-FIG. 2: (a) Absorption image of a condensate with a singlevortex and (b) two counter-rotating vortices. (c) A spin tex-ture similar to a skyrmion corresponding to the core regionof the vortex in (a).FIG. 3: (a) Pulse sequence for atom interferometry with twoRaman pulses (details in text) (b) Absorption images for dif-ferent phases of a vortex state of the BEC interfering with auniform phase state.other fraction of atoms to the same internal state, thistime with a uniform phase. The pulse lengths used are4µs each.Figure 3(b) shows absorption images for diﬀerent rel-ative phases of the wavefunctions of the vortex state in-terfering with the uniform phase state. The phase diﬀer-ence between the hologram beam and the phase referenceGaussian beam maps directly onto the atoms.In Figure 4, we show the results of atom interferom-etry on a BEC state with a vortex-antivortex pair, to-gether with simulated data. From these images, we ver-ify that the two vortices are counter-rotating, as theirphases wind in opposite directions as the phase betweenthe light beams is varied.If the hologram beam and the reference Gaussian beamused for the interference experiments are not perfectly co-propogating, the structure of the original hologram canbe reproduced in the atomic cloud as an interference be-tween the two states. Figure 5 shows an interferenceimage for a single vortex state obtained in this way. Therelative linear momentum between the two interferingcondensates appears as a sinusoidally varying pattern ofFIG. 4: Top row: absorption images for a vortex-antivortexstate interfering with a uniform phase state. Bottom row:calculated interference patterns. The results are shown forphase diﬀerences of (a) 0π/5, (b) 4π/5, (c) 5π/5 and (d) 9π/5.The phases of the vortex state in the simulated data wherechosen to match the experimental data.4FIG. 5: A matter wave hologram (b) resembling the originalhologram (a) (microscopy image of the chrome mask) is cre-ated if linear momentum is transferred in the second pulse ofthe described sequence.stripes on top of the interference pattern due to the vor-tex [33].In this paper, we have demonstrated the controlled co-herent preparation of vortex states in Bose-Einstein con-densates. The phase of the classical electromagnetic ﬁeld,which can be shaped holographically in a versatile way, isimprinted onto the atomic cloud, allowing great freedomin the phase pattern to be transferred. Vortex patternswith arbitrary charge, circulation and conﬁgurations canbe created.We have recently implemented high resolution opticsin our apparatus (∼ 600nm resolution)[34]. 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Inducing Vortices in a Bose-Einstein Condensate Using Holographically Produced Light Beams

Brachmann, Johannes F S

Bakr, Waseem S

Gillen, Jonathon I

Digital Commons@Becker

Inducing vortices in a Bose-Einstein condensate using holographically produced light beams

Inducing vortices in a Bose-Einstein condensate using holographically
  produced light beams

Inducing vortices in a Bose-Einstein condensate using holographically produced light beams

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