We have studied the dynamics of large vortex lattices in a dilute-gas
Bose-Einstein condensate. While undisturbed lattices have a regular hexagonal
structure, large-amplitude quadrupolar shape oscillations of the condensate are
shown to induce a wealth of nonequilibrium lattice dynamics. When exciting an m
= -2 mode, we observe shifting of lattice planes, changes of lattice structure,
and sheet-like structures in which individual vortices appear to have merged.
Excitation of an m = +2 mode dissolves the regular lattice, leading to randomly
arranged but still strictly parallel vortex lines.Comment: 5 pages, 6 figure

E. A. Cornell

I. Coddington

J. R. Abo-Shaeer

P. C. Haljan

P. Engels

English

arXiv

We have studied the dynamics of large vortex lattices in a dilute-gas Bose-Einstein condensate. While undisturbed lattices have a regular hexagonal structure, large-amplitude quadrupolar shape oscillations of the condensate are shown to induce a wealth of nonequilibrium lattice dynamics. When exciting an m=−2 mode, we observe shifting of lattice planes, changes of lattice structure, and sheetlike structures in which individual vortices appear to have merged. Excitation of an m=+2 mode dissolves the regular lattice, leading to randomly arranged but still strictly parallel vortex lines

Engels, Peter

Cornell, Eric A.

Haljan, P. C.

Coddington, I.

CU Scholar Institutional Repository

arXiv:cond-mat/0204449v1  20 Apr 2002Nonequilibrium Effects of Anisotropic Compression Applied to Vortex Lattices inBose-Einstein CondensatesP. Engels, I. Coddington, P. C. Haljan, and E. A. Cornell∗JILA, National Institute of Standards and Technology and Department of Physics,University of Colorado, Boulder, Colorado 80309-0440(February 1, 2008)We have studied the dynamics of large vortex lattices in a dilute-gas Bose-Einstein condensate.While undisturbed lattices have a regular hexagonal structure, large-amplitude quadrupolar shapeoscillations of the condensate are shown to induce a wealth of nonequilibrium lattice dynamics.When exciting an m = -2 mode, we observe shifting of lattice planes, changes of lattice structure,and sheet-like structures in which individual vortices appear to have merged. Excitation of an m =+2 mode dissolves the regular lattice, leading to randomly arranged but still strictly parallel vortexlines.03.75.Fi,67.90.+z,67.40.Vs,32.80.PjThe experimental study of vorticity in a dilute-gasBose-Einstein condensate (BEC) provides an useful per-spective on superfluidity. While the focus of early workwas on the study of single or few vortices [1,2], recent ad-vances in technique have made it possible to create BECscontaining large amounts of vorticity [3–6]. In a rapidlyrotating BEC, a lattice of vortices is the lowest energystate in the rotating frame [7]. Experimental studies oflattices to date have concentrated on formation and decayprocesses [2–6,8,9]. In this Letter we study the nonequi-librium behavior of vortex lattices under large-amplitude,anisotropic strain.Experimental technique For creating large amountsof vorticity in condensates, we use a modified version ofthe spin-up technique described in our previous paper[5]. The starting point is a magnetically trapped cloudof 17 × 106 87Rb atoms in the |F = 1, mF = −1〉 statewith a temperature approximately three times above thecritical temperature for the BEC phase transition, Tc =67 nK. By manipulating the ellipticity of the confiningpotential, we can resonantly pump angular momentuminto the as yet uncondensed cloud [10]. We then tunethe trapping potential to near perfect axial symmetrywith trap frequencies {ωρ, ωz} = 2π{8.35, 5.45}Hz. Inthis round trap the cloud continues to rotate, and byperforming a nearly one-dimensional evaporation alongthe axis of rotation (z-axis) as described in Ref. [5], wecool down and further spin up the cloud until a highlyrotating condensate is born out of the rotating normalcloud. For the experiments described in this paper, theevaporation is continued to a point where little or nothermal fraction remains. As a result of this procedure,we obtain highly rotating condensates with typically 1×106 atoms, containing 130 vortices or more.To study these condensates we can image along thex, y, or z directions. When the condensate is held in thetrap, the core size of the vortices is close to the resolutionlimit of our optical detection system so that individualvortices are not resolved by in-trap imaging. However, bynondestructively imaging a trapped condensate from theside (along the x-direction) and measuring its aspect ra-tio, the rotation rate of the condensate can be determined[5,8]. For our highly rotating condensates we determinetypical aspect ratios of 0.5, which is markedly differentfrom the static aspect ratio of the magnetic trap, 1.53,because of centrifugal forces. Typical condensate rota-tion rates of 0.95ωρ are inferred.To resolve individual vortices, we can release a con-densate from the trap and image it after it has expandedby a factor of 5 [2,12]. Such expansion pictures takenalong the z-direction reveal a hexagonal lattice of vor-tex cores in highly rotating condensates [Fig. 1(a)]. Thelattice structure is remarkably regular even at the outerregions of the condensate, and is observed with very goodcontrast as shown by the cross section in Fig. 1(b).When looking at such vortex lattices in expansion fromthe side (i.e., along the x or y direction), good contrastis only obtained if the lattice is oriented in such a waythat vortices line up behind each other along the direc-tion of view, or in other words when the direction ofview is parallel to lattice planes. The vortex lattices arenot stationary in the lab frame, but are rotating withthe condensate, and furthermore the initial orientationof the lattice in the xy plane cannot be controlled incurrent experiments. Therefore the times when vorticesline up along the direction of sight are unpredictable,and images revealing lattice planes are only obtained ina random subset of all trials. One example is shown inFig. 1(c), where the vortex planes are clearly visible asdark vertical lines. These images are direct verificationsof the theoretical prediction that vortex lines are onlymarginally bent for our experimental parameters [13,14],as is also confirmed by the good contrast seen in thetopview picture of Fig. 1(a).Vortex lattices such as the one shown in Fig. 1 are thestarting point for our studies of the effects of dynami-1cally generated strain. We apply the distortion by reso-nantly driving quadrupolar shape modes in the vortex-filled condensate [15,16]. In the large-amplitude limit,the mz = +2 and mz = −2 modes distort the circu-lar cross-section of the condensate into an ellipse andcause the major axis of the ellipse to rotate along with,or against, the sense of the condensate rotation, respec-tively. The generated strain is considerable – the ratio ofmajor to minor axis of the condensate can be larger thanthree to one. The frequencies of the mz = ±2 modesmay be calculated using the sum rule argument given byRefs. [17,18]. Our condensates are rotating typically at0.95ωρ, just 5% under the centrifugal limit. At the cen-trifugal limit, the major axis of the mz = +2 ellipse isfixed in the rotating condensate frame, and the mz = −2shape is fixed in the lab frame. At our rotation rate of0.95ωρ, the distortion, as seen in the frame of the lattice,rotates at only 0.4 Hz for the mz = +2 mode, comparedwith 8.34 Hz for the mz = −2 mode. For this reason [19],we expect very different resulting lattice dynamics in thepresence of either of the two modes.Lattice dynamics in the presence of an mz = −2surface mode Because the mz = −2 mode is almoststationary in the lab frame for the parameters of ourexperiment, this mode can be excited conveniently andnearly resonantly by a static trap deformation. Indeedwe observe that by jumping from a round trap to a trapthat has an ellipticity of 3.6% [20], the ellipticity of thecondensate increases over a timescale of 300 ms from ap-proximately 0 to 40%, thereby exceeding the trap ellip-ticity by more than an order of magnitude. Since thevortex lattice is rotating quickly with respect to this al-most static deformation, it is an interesting and nontriv-ial question to ask whether or not a lattice structure ismaintained, and, if so, how it rearranges in the presenceof the excitation.Experimentally, vortex lattices can indeed be observedin expansion images up to 400 ms after the start of thecontinuously applied trap deformation (Fig. 2). Subse-quently, long range order is lost and vortex visibility be-comes low, presumably due to tilting [16] or bending ofthe vortex lines [4]. A closer look at the first 400 ms ofthis evolution reveals a wealth of intriguing vortex dy-namics. As the vortex lattice rotates in the deformedcondensate, the lattice planes must continuously shift rel-ative to each other to accommodate to the elliptical shapeof the condensate. One consequence of this is transientchanges of the lattice structure. In Fig. 3 the lattice haschanged from the hexagonal structure of an undisturbedlattice to a near orthorhombic structure.The most striking observation, however, is of conden-sates containing sheet-like structures rather than individ-ual vortex cores, as shown in Fig. 4(a). We interpret thesesheets as rows of vorticity along which individual vortexcores have essentially merged. Given the near perfectcontrast of these sheets, as shown by the cross section inFig. 4(b), we can exclude the possibility that the sheetsare merely formed by a collective tilting of vortices alonga lattice plane, which would result in an intermediatecontrast. Similarly, we can rule out that this is an effectmerely of imaging resolution - had the vortices retainedtheir original structure but come so close together thatwe could not spatially resolve them, we again would seewhite stripes set off by grey troughs, rather than the near100% alternating stripes of white and black. If there con-tinue to exist wispy fingers of condensate that cross thestripes and differentiate individual vortices, they must bevery tenuous indeed to be consistent with the observedcontrast. The clouds have been continuously distortedfrom their original hexagonal symmetry, so the observa-tion of 16 stripes along the major axis of the BEC inFig. 4(a) tells us that there must also be the equivalentof 16 units of vorticity along the length of the stripesthrough the center of the cloud. By measuring the in-trap height of the condensate, the ellipticity in the xy-plane and the number of atoms, we can infer the ratio ofunit vorticity spacing (along the minor axis) to healinglength (given by 1√8πna, where n is the mean density anda the scattering length) to be on the order of 5.In fact the contrast of this sheet-like phase is so deepthat the structure can be detected by looking at conden-sates in trap (along the x or z direction), even thoughthe structure is close to our optical resolution limit. Bytaking a time series of ten nondestructive images of asingle condensate along the x direction, we see that thesheet structure appears and disappears periodically witha period of about 21.2 ms for a condensate rotating at afrequency of 0.95ωρ [Fig. 5].This periodicity can be understood as follows: Thesheet structure forms when a lattice vector lies along aminor axis of the cloud, presumably since this is whenthe vortices are closest together. As that vector rotatespast the minor axis, structure along the vector is reestab-lished, the two-dimensional (2D) pattern of individualvortices reemerges, and the high-contrast stripes disap-pear from the in-trap images. A lattice vector alignswith the minor axis six times per lattice rotation perioddue to the sixfold symmetry of the unperturbed lattice,consistent with the observed frequency of periodic stripeformation. This interpretation is bolstered by the ex-pansion images, which show high contrast sheets whena lattice vector lies close to the minor axis, but 2D lat-tices (albeit distorted) for other orientations. We find itremarkable that this lattice-to-sheet-to-lattice sequencepersists through multiple cycles before long range orderand visibility are compromised.Using a trap with a stronger distortion of 34% as inFig. 4(c-e), we also observe sheets breaking apart in thecenter. In Fig. 4(c), the density distribution along the lat-tice vector that we see near-parallel with the minor axisis nearly featureless. Figure 4(d,e) hint at the succeeding2lattice evolution: Fig. 4(d) represents a small rotationfrom Fig. 4(c), and we see that the sheets have begun toreconnect across the cloud. Figure 4(e) appears to be acontinuation of that process, with considerable evidenceof uniformity in the repeating structure.Lattice dynamics in the presence of an mz =+2 surface mode We excite the mz = +2 mode byelliptically deforming the magnetic trap and rotating thatellipse around in the xy plane. When the mz = +2 modeis driven approximately 1 Hz away from its resonance,the ellipticity of the condensate increases from 0 to 50%over a timescale of 250 msec. Under the influence of anon-resonant drive, the ellipticity increases continuously,even passing beyond 90% after 700 msec and approaching100% [21]. As discussed above, the distortion rotatesonly very slowly with respect to the lattice, and thus weexpect it to be correspondingly “gentle” with respect toits action on the lattice.Indeed, as we show in Fig. 6(a), we do not see thelattices squeezed into sheets for the mz = +2 mode. In-stead we observe a gradual increase of disorder in thelattice. 400 msec after we start the excitation, we seea complete loss of long-range order in the vortex struc-ture. The imaging contrast of individual vortices remainshigh, so we know they remain parallel to the line of sight(in contrast for instance to the case of Fig. 2), but oth-erwise they appear to have liquified (see the correlationhistogram of Fig. 6). This behavior confirms our expec-tation of the mz = +2 mode being a gentler disturbancefor the lattice than the mz = −2 mode.Conclusions In conclusion we have succeeded in ob-serving a wealth of nonequilibrium vortex lattice dynam-ics in a BEC under anisotropic compression. Key ob-servations include the shifting of lattice planes relativeto each other, leading to transient changes of the lat-tice structure, the formation of an unexpected and verypronounced sheet structure, and finally, a gradual tran-sition from an ordered lattice to a rather liquid-like ar-rangement in which vortex cores remain visible, but theirrelative positions are irregular. The observed sheet struc-ture is of particular interest: at present the behavior ofdensely packed vortex lattices is in the center of theo-retical discussions about vortices [22–24]. While most ofthese theoretical discussions propose to enter this regimeby having high rotation rates almost inaccessible to ex-periment or inconveniently low atom numbers in the con-densate, this paper demonstrates an alternative and un-expected way of bringing vortices close together. Wehope that the results presented in this Letter will stimu-late further theoretical discussions in the field of vortexlattices.The work presented in this paper was funded by NSFand NIST. PE acknowledges support by the Alexandervon Humboldt Foundation.∗ Quantum Physics Division, National Institute of Stan-dards and Technology.[1] M. R. Matthews et al., Phys. Rev. Lett. 83, 2498 (1999);B. P. Anderson, P. C. Haljan, C. E. Wieman, and E. A.Cornell, Phys. Rev. Lett. 85, 2857 (2000).[2] K.W. Madison, F. Chevy, W. Wohlleben, and J. Dal-ibard, Phys. Rev. Lett. 84, 806 (2000).[3] K.W. Madison, F. Chevy, V. Bretin, and J. Dalibard,Phys. Rev. Lett. 86, 4443 (2001).[4] J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ket-terle, Science 292, 476 (2001).[5] P. C.Haljan, I. Coddington, P. Engels, and E. A. Cornell,Phys. Rev. Lett. 87, 210403 (2001).[6] E. Hodby, G. Heckenblaikner, S.A. Hopkins, O.M.Maragó, and C. J. Foot, Phys. Rev. Lett. 88, 010405(2001).[7] D. A. Butts and D. S. Rokhsar, Nature 397, 327 (1999).[8] C. Raman, J. R. Abo-Shaeer, J. M. Vogels, K. Xu, andW. Ketterle, Phys. Rev. Lett. 87, 210402 (2001).[9] J. R. Abo-Shaeer, C. Raman, and W. Ketterle, Phys.Rev. Lett. 88, 070409 (2002).[10] We impart angular momentum to the noncondensedcloud by stepwise driving a quarter cycle of a scissorsmode [11].[11] O.M. Maragó, S.A. Hopkins, J. Arlt, E. Hodby, G. Heck-enblaikner, and C.J. Foot, Phys. Rev. Lett. 84, 2056(2000).[12] B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder,L. A. Collins, C. W. Clark, and E. A. Cornell, Phys. Rev.Lett. 86, 2926 (2001).[13] J. J. Garćia-Ripoll and V. M. Pérez-Garćia, Phys. Rev.A 64, 053611 (2001).[14] D. L. Feder and C. W. Clark, Phys. Rev. Lett. 87, 190401(2001).[15] F. Chevy, K. W. Madison, and J. Dalibard, Phys. Rev.Lett. 85, 2223 (2000).[16] P. C. Haljan, B. P. Anderson, I. Coddington, and E. A.Cornell, Phys. Rev. Lett. 86, 2922 (2001).[17] F. Zambelli and S. Stringari, Phys. Rev. Lett. 81, 1754(1998).[18] S. Stringari, private communication. For an alternativeapproach, see also A. A. Svidzinsky and A. L. Fetter,Phys. Rev. A 58, 3168 (1998).[19] We emphasize as well that the two different modes re-sult in very different fluid-flow patterns. The mz =+2 mode very closely approximates rigid-body rotation,while much of the mz = −2 velocity field is provided byan irrotational, quadrupole flow pattern. This distinctionmay provide a parallel explanation for the correspond-ingly distinct observed effects.[20] We define ellipticity asω2x−ω2yω2x+ω2ywith ωx and ωy being thetrap frequency along the minor and major axes of theellipse, repectively.[21] For related work on the evolution of the ellipticity seealso P. Rosenbusch, D. S. Petrov, S. Sinha, F. Chevy,3V. Bretin, Y. Castin, G. Shlyapnikov, and J. Dalibard,con-mat/0201568.[22] T. L. Ho, Phys. Rev. Lett. 87, 060403 (2001).[23] U. R. Fischer and G. Baym, condmat/0111443.[24] B. Paredes, P. Zoller, and J. I. Cirac, condmat/0203061.DEFµPµPFIG. 1. (a) Expansion picture of a vortex lattice seen alongthe rotation axis. (b) One pixel wide cross section along thewhite line in (a). (c) Expansion picture of a different conden-sate rotating more slowly than in (a), seen from the side.D EµPFIG. 2. Lattice evolution after an mz = −2 excitation.Pictures taken (a) 173 ms and (b) 873 ms after start of trapdeformation.D EµP µPFIG. 3. Change of lattice structure. (a) Hexagonal struc-ture in an undisturbed lattice. (b) Near orthorhombic struc-ture seen transiently during lattice evolution in the presenceof an mz = −2 quadrupolar surface mode.DEFGHµPµPFIG. 4. (a) Sheet-like structure as seen during latticeevolution in the presence of an mz = −2 quadrupolar sur-face mode. (b) Cross section integrated over the white boxin (a). Even though the box is wider than the calculatedvortex core spacing, the observed contrast is nearly perfect.(c-e) Same as (a), but observed in a more deformed trap with{ωx,y,z} = 2π{6.0, 8.6, 13.8}Hz.D E FG H IµPFIG. 5. (a-e) Nondestructive in-trap images of thesheet-like structures seen along the x-direction [conditionssimilar to Fig. 4(a)]. Spacing between images 10.6 ms. Notethe very different spatial scale from expansion images (e.g.,Fig. 1). (f) Cross section of (a), integrated over the conden-sate.D EPP     FRXQWVGLVWDQFH>PP@F4FIG. 6. (a) Irregular assembly of vortex cores as observedafter 500 ms in the presence of an mz = +2 quadrupolarsurface mode. (b) Histogram of distances between each pairof vortices in the regular hexagonal lattice shown in Fig. 1a.The visibility of separate peaks reveals the high degree oflong-range order. (c) Same as (b) but calculated for the BECof Fig. 6a. Peaks are barely visible here, meaning that thelattice has almost reached a random disorder.5

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Nonequilibrium Effects of Anisotropic Compression Applied to Vortex Lattices in Bose-Einstein Condensates

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Nonequilibrium effects of anisotropic compression applied to vortex
  lattices in Bose-Einstein condensates

Nonequilibrium effects of anisotropic compression applied to vortex lattices in Bose-Einstein condensates

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