Observation of Quantum ShockWaves Created with Ultra-

Compressed Slow Light Pulses ina Bose-Einstein Condensate

Zachary Dutton,1,2 Michael Budde,1,3 Christopher Slowe,1,2

Lene Vestergaard Hau1,2,3

We have used an extension of our slow light technique to provide a method forinducing small density defects in a Bose-Einstein condensate. These sub-resolution, micrometer-sized defects evolve into large-amplitude sound waves.We present an experimental observation and theoretical investigation of theresulting breakdown of superfluidity, and we observe directly the decay of thenarrow density defects into solitons, the onset of the “snake” instability, andthe subsequent nucleation of vortices.

Superfluidity in Bose condensed systems rep-resents conditions where frictionless flow oc-curs because it is energetically impossible tocreate excitations. When these conditions arenot satisfied, various excitations develop. Ex-periments on superfluid helium, for example,have provided evidence that the nucleation ofvortex rings occurs when ions move throughthe fluid faster than a critical speed (1, 2).Under similar conditions, shock waves wouldoccur in a normal fluid (3). Such discontinuitiesare not allowed in a superfluid, and insteadtopological defects (such as quantized vorticesand solitons) are nucleated when the spatialscale of induced density variations becomes onthe order of the healing length. This is thelength scale at which the kinetic energy, asso-ciated with spatial variations of the macroscop-ic condensate wave function, becomes on theorder of the atom-atom interaction energy (2,4). It is therefore also the minimum length overwhich the density of a condensate can adjust.

Bose-Einstein condensates (BECs) of al-kali atoms (5) have provided a system for thestudy of superfluidity that is theoreticallymore tractable than liquid helium and allowsgreater experimental control. An exciting re-cent development is the production of soli-tons and vortices. Experiments have usedtechniques that manipulated the phase of theBEC (6–9), or provided the system with ahigh angular momentum that makes vortexformation energetically favorable (10, 11).However, a direct observation of the forma-tion of vortices via the breakdown of super-fluidity has been lacking. Rather, rapid heat-ing from “stirring” with a focused laser beam

above a critical velocity was observed asindirect evidence of this process (12, 13).

In this context, it is natural to ask whatwould happen if one were to impose a sharpdensity feature in a BEC with a spatial scaleon the order of the healing length. Opticalresolution limits have prevented direct cre-ation of this kind of excitation. Here, wepresent an experimental demonstration of thecreation of such defects in sodium Bose-Einstein condensates, using a novel extensionof the method of ultraslow light pulse prop-agation (14) via electromagnetically inducedtransparency (EIT ) (15, 16).

Our slow light setup is described in (14). Byilluminating a BEC with a “coupling” laser, wecreate transparency and low group velocity fora “probe” laser pulse subsequently sent into thecloud. In a geometry where the coupling andprobe laser beams propagate at right angles, wecan control the propagation of the probe pulsefrom the side. By introducing a spatial variationof the coupling field along the pulse propaga-tion direction, we vary the group velocity of theprobe pulse across the cloud. We accomplishthis by blocking half of the coupling beam sothat it illuminates only the left side (z , 0) ofthe condensate, setting up a light “roadblock.”In this way, we compress the probe pulse to asmall spatial region at the center of the BECwhile bypassing the usual bandwidth require-ments for slow light (17). The technique pro-duces a short-wavelength excitation by sudden-ly removing a narrow disk of the condensate,with the width of the disk determined by thewidth of the compressed probe pulse.

We find that this excitation results in short-wavelength, large-amplitude sound waves thatshed “gray” solitons (18–20), in a process anal-ogous to the formation of shock waves in clas-sical fluids. The “snake” (Kadomtsev-Petviash-vili) instability is predicted to cause solitons todecay into vortices (21–24). This has been ob-

served with optical solitons (25), and recentlythe JILA group predicted and observed thatsolitons in a BEC decay into vortex rings (9).Here, we present a direct observation of thedynamics of the snake instability in a BEC andthe subsequent nucleation of vortices. The im-ages of the evolution are compared with numer-ical propagation of the Gross-Pitaevskii equa-tion in two dimensions.

Details of our Bose-Einstein condensationapparatus can be found in (26). We use con-densates with about 1.5 million sodium atomsin the state u1& [ u3S1/2, F 5 1, MF 5 21&,where F and MF are the total spin and spinprojection quantum numbers of the atoms,trapped in a 4-Dee magnet [described in(26)]. For the experiments presented here, themagnetic trap has an oscillator frequencyvz 5 (2p)21 Hz along its symmetry directionand a frequency vx 5 vy 5 3.8vz in thetransverse directions. The peak density of thecondensates is then 9.1 3 1013 cm23. Thetemperature is T ; 0.5Tc, where Tc 5 300 nKis the critical temperature for condensation,and so the vast majority (;90%) of the atomsoccupy the ground state.

Creation of the light roadblock. To cre-ate slow and spatially localized light pulses,the coupling beam propagates along the xaxis (Fig. 1B), is resonant with the D1 tran-sition from the unoccupied hyperfine state u2&[ u3S1/2, F 5 2, MF 5 22& to the excitedlevel u3& [ u3P1/2, F 5 2, MF 5 22&, and hasa Rabi frequency Vc 5 (2p)15 MHz (27).We inject probe pulses along the z axis, res-onant with the u1& 3 u3& transition and withpeak Rabi frequency Vp 5 (2p)2.5 MHz.The pulses are Gaussian-shaped with a 1/ehalf-width of t 5 1.3 ms. With the entireBEC illuminated by the coupling beam, weobserve probe pulse delays of 4 ms for prop-agation through the condensates, correspond-ing to group velocities of 18 m/s at the centerof the clouds. A pulse with a temporal half-width t is spatially compressed from a lengthL 5 2ct in vacuum (where c is the speed oflight in vacuum) to

L 5 2tVg 5 2tuVcu2

Gf13s0nc(1)

(14, 17, 28, 29) inside the cloud, where Vg

is the group velocity of the light pulse inthe medium, G 5 (2p)10 MHz is the decayrate of state u3&, nc is the cloud density,s0 5 1.65 3 1029 cm2 is the absorptioncross section for light resonant with a two-level atom, and f13 5 1/2 is the oscillatorstrength of the u1& 3 u3& transition. Theatoms are constantly being driven by thelight fields into a dark state, a coherentsuperposition of the two hyperfine states u1&and u2& (15). In the dark state, the ratio ofthe two population amplitudes is varying inspace and time with the electric field am-plitude of the probe pulse as

1Rowland Institute for Science, 100 Edwin H. LandBoulevard, Cambridge, MA 02142, USA. 2Lyman Lab-oratory, Department of Physics, 3Cruft Laboratory,Division of Engineering and Applied Sciences, HarvardUniversity, Cambridge, MA 02138, USA.

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c2 5 2Vp

Vcc1 (2)

where c1, c2 are the macroscopic condensatewave functions associated with the two statesu1& and u2&.

For the parameters listed above, the probepulse is spatially compressed from 0.8 km infree space to only 50 mm at the center of thecloud, at which point it is completely con-

tained within the atomic medium. The corre-sponding peak density of atoms in u2&, pro-portional to uc2u2, is 1/34 of the total atomdensity. The u1& atoms have a correspondingdensity depression.

From Eqs. 1 and 2, it is clear that to mini-mize the spatial scale of the density defect, weneed to use short pulse widths and low couplingintensities. However, for all the frequency com-ponents of the probe pulse to be contained with-in the transmission window for propagationthrough the BEC (17), we need a pulse with atemporal width t of at least 2D 1/2G/Vc

2 ' 0.3ms to avoid severe attenuation and distortion(D ' 520 is the column density of a condensatein the z direction times the absorption crosssection for on-resonance two-level transitions).Furthermore, we see from Eq. 2 that to maxi-mize the amplitude of the density depression, itwould be best to use a peak Rabi frequency forthe probe of Vp ; Vc. This also severely distortsthe pulse.

Both of these distortion effects accumulateas the pulse propagates through clouds withlarge D. This motivated us to introduce a road-block in the condensate for a light pulse ap-proaching from the left side (z , 0). By imag-ing a razor blade onto the right half of thecondensate, we ramp the coupling beam fromfull to zero intensity over the course of a 12-mmregion in the middle of the condensate, deter-mined by the optical resolution of the imagingsystem. In the illuminated region (z , 0), ourbandwidth and weak-probe requirements arewell satisfied and we get undistorted, unattenu-ated propagation through the first half of thecloud to the high-density, central region of thecondensate. As the pulse enters the roadblockregion of low coupling intensity, it is sloweddown and spatially compressed. The exactshape and size of the defects created with thismethod are dependent on when absorption ef-fects become important.

Theoretical model. To accurately modelthe pulse compression and defect formation,we account for the dynamics of the slow lightpulses, the coupling field, and the atoms self-consistently. At sufficiently low tempera-tures, the dynamics of the two-componentcondensate can be modeled with coupledGross-Pitaevskii equations (4, 5). Here, weinclude terms to account for the resonanttwo-photon light coupling between the twocomponents:

i\]

]tc1

5 F2 \2¹2

2m1 V1~r! 1 U11uc1u2 1 U12uc2u2Gc1

2 iuVpu2

2Gc1 2 i

Vp*Vc

2Gc2

2 iNcse\k2g

2muc2u2c1

i\]

]tc2

5 F2 \2~¹2 1 ik2g z ¹)

2m1 V2(r) 1 U22uc2u2

1 U12uc1u2Gc2 2 iuVcu2

2Gc2

2 iVpVc*

2Gc1 2 iNcse\

k2g

2muc1u2c2 (3)

where \ is the Planck constant divided by 2p, ¹is the spatial gradient operator, and V1(r) 51⁄2mvz

2[l2(x 2 1 y 2) 1 z 2] (where m is themass of the sodium atoms and l 5 3.8). Be-cause of the magnetic moment of atoms in stateu2&, V2(r) 5 22V1(r), and atoms in this state arerepelled from the trap. The EIT process involvesabsorption of a probe photon and stimulatedemission of a coupling photon, leading to arecoil velocity of 4.1 cm/s for atoms in state u2&.This is described by a term in the second part ofEq. 3, containing k2g 5 kp 2 kc, the differencein wave vectors between the two laser beams.(Here, we use a gauge where the recoil momen-tum is transformed away.) Atom-atom interac-tions are characterized by the scattering lengths,aij, via Uij 5 4pNc\

2aij/m, where a11 5 2.75nm, a12 5 a22 5 1.20a11 (30), and Nc is the totalnumber of condensate atoms. To obtain the lightcoupling terms in Eq. 3, we have adiabaticallyeliminated the excited-state amplitude c3 (31),because the relaxation from spontaneous emis-sion occurs much faster than the light couplingand external atomic dynamics driving c3. In ourmodel, atoms in u3& that spontaneously emit areassumed to be lost from the condensate, whichis why the light coupling terms are non-Hermi-tian. The last term in each equation accounts forlosses due to elastic collisions between high-momentum u2& atoms and the nearly stationaryu1& atoms (se 5 8pa12

2) (32).The spatial dynamics of the light fields are

described classically with Maxwell’s equa-tions in a slowly varying envelope approxi-mation, again using adiabatic elimination ofc3:

]

]zVp 5 2

1

2f13s0Nc~Vpuc1u2 1 Vcc1*c2!

]

]xVc 5 2

1

2f23s0Nc~Vcuc2u2 1 Vpc1c2*!

(4)In the region where the coupling beam isilluminating the BEC (z , 0), the light cou-pling terms dominate the atomic dynamics,and Eqs. 3 and 4 yield Eqs. 1 and 2 above.

We have performed numerical simulationsin two dimensions (x and z) to track the behav-ior of the light fields and the atoms. The probeand coupling fields were propagated accordingto Eq. 4 with a second-order Runge-Kutta al-gorithm (33) and the atomic mean fields werepropagated according to Eq. 3, with an Alter-

Fig. 1. (A) Experimental in-trap OD image of atypical BEC before illumination by the probe pulseand coupling field. The condensate contains 1.5 3106 atoms. The imaging beam was –30 MHzdetuned from the u1&3 u3P1/2, F 5 2, MF 5 21&transition. (B) Top view of the beam configurationused to create and study localized defects in aBEC. (C) Buildup of state u2& atoms at the road-block. In-trap OD images (left) show the transferof atoms from u1& to u2& as the probe pulsepropagates through the condensate and runs intothe roadblock. The atoms in u2& were imaged witha laser beam –13 MHz detuned from the u2& 3u3P3/2, F 5 3, MF 5 22& transition. To allowimaging, we stopped the probe pulse propagationat the indicated times by switching the couplingbeam off (29). The time origin t 5 0 is defined asthe moment when the center of the probe pulseenters the BEC from the left. The graphs on theright show the corresponding line cuts along theprobe propagation direction through the center ofthe BEC. The probe pulse had a Rabi frequencyVp 5 (2p)2.4 MHz, and the coupling Rabi fre-quency was Vc 5 (2p)14.6 MHz.

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nating-Direction Implicit variation of theCrank-Nicolson algorithm (33, 34). In this way,Eqs. 3 and 4 were solved self-consistently (35).Profiles of the probe pulse intensity along z,through x 5 0, are shown in Fig. 2A. As thepulse runs into the roadblock, a severe com-pression of the probe pulse’s spatial length oc-curs. When the probe pulse enters the lowcoupling region, the Rabi frequency uVpu be-comes on the order of uVcu. Thus, the density ofstate u2& atoms, Ncuc2u2, increases in a narrowregion, which is accompanied by a decrease inNcuc1u2 (Fig. 2B). The half-width of the defectis 2 mm. As the compression develops, absorp-tion and spontaneous emission events eventu-ally start to remove atoms from the condensateand reduce the probe intensity.

Observation of localized impurity for-mation in a Bose-Einstein condensate. Ex-perimental results are shown in Fig. 1. Figure1A is an in-trap image of the original con-densate of u1& atoms, Fig. 1B diagrams thebeam geometry, and Fig. 1C shows a series ofimages of state u2& atoms as the pulse propa-gates into the roadblock. The correspondingoptical density (OD) profiles along z throughx 5 0 are also shown. OD is defined to be2ln(I/I0) where (I/I0) is the transmission co-efficient of the imaging beam. All imaging isdone with near-resonant laser beams propa-gating along the y axis, and with a duration of10 ms. There is clearly a buildup of a dense,narrow sample of u2& atoms at the center ofthe BEC as the pulse propagates to the right.Note that the pulse reaches the roadblock atthe top and bottom edges of the cloud beforethe roadblock is reached at the center, whichis a consequence of the transverse variation inthe density of the BEC, with the largest den-sity along the center line. After the pulsecompression is achieved, we shut off thecoupling beam to avoid heating and phaseshifts of the atom cloud due to extendedexposure to the coupling laser, and we ob-serve the subsequent dynamics of the conden-sate. ( We found that exposure to the couplinglaser alone, for the exposure times used tocreate defects, caused no excitations of thecondensates.)

Formation of quantum shock waves. Inconsidering the dynamics resulting from thisexcitation of a condensate, it should be notedthat the roadblock “instantaneously” removesa spatially selected part of c1. The entire lightcompression happens in ;15 ms. After thepulse is stopped and the coupling laser turnedoff, the u2& atoms remaining in the condensatec2 have a recoil of 4.1 cm/s, and atoms thathave undergone absorption and spontaneousemission events have a similarly sized butrandomly directed recoil. Hence, the c2 com-ponent and the other recoiling atoms interactwith c1 for less than 0.5 ms before leavingthe region. Both of these time scales are shortrelative to the several-millisecond time scale

over which most of the subsequent dynamicsof c1 occur, as discussed below.

We first considered the one-dimensional(1D) dynamics along the z axis. Snapshots ofboth condensate components, obtained fromnumerical propagation in 1D according to Eqs.3 and 4, are shown in Fig. 3A for various timesafter the pulse is stopped at the roadblock (andthe coupling laser turned off ). In the linearhydrodynamic regime, where the density defecthas a relative amplitude A ,, 1 and a half-width d .. j [where j 5 1/(8pNcuc1u2a11)1/2 isthe local healing length, which is 0.4 mm at thecenter of the ground-state condensate in ourexperiment], one expects to see two densitywaves propagating in opposite directions at thelocal sound speed, cs 5 (U11uc1u2/m)1/2, as seen

experimentally in (36). However, for the pa-rameters used in our experiment, the soundwaves are seen to shed sharp features propagat-ing at lower velocities. Examination of thewidth and speed of these features and the phasejump across them shows that they are graysolitons. With c1

(0) describing the slowly vary-ing background wave function of the conden-sate, the wave function in the vicinity of a graysoliton centered at z0 is

c1~ z,t! 5 c1~0!~ z,t!H iÎ1 2 b2

1 b tanh F b

Î2j~ z 2 z0!GJ (5)

(18–20). The dimensionless constant b char-

Fig. 2. (A) Compression of a probe pulse at thelight “roadblock,” according to 2D numericalsimulations of Eqs. 3 and 4. The solid curvesindicate probe intensity profiles along z (at x 50), normalized to the peak input intensity. Thesnapshots are taken at the indicated times,where t 5 0 is defined as in Fig. 1C. Forreference, the atomic density profile of theoriginal condensate is plotted (in arbitraryunits) as a dashed curve. The gray shadingindicates the relative coupling input intensityas a function of z, with white corresponding tofull intensity and the darkest shade of graycorresponding to zero. The spatial turnoff ofthe coupling field is centered at z 5 0 andoccurs over 12 mm, as in the experiment. Thenumber of condensate atoms is 1.2 3 106

atoms, the peak density is 6.9 3 1013 cm23,and the coupling Rabi frequency is Vc 5(2p)8.0 MHz. The probe pulse has a peak Rabifrequency of Vp 5 (2p)2.5 MHz and a 1/ehalf-width of t 5 1.3 ms. (B) Creation of anarrow density defect in a BEC. Density profilesof the two condensate components, Ncuc1u2(dashed) and Ncuc2u2 (solid), are shown along zat x 5 0 for a sequence of times. Note that thez range of the plot is restricted to a narrowregion around the roadblock at the cloud cen-ter. The densities are normalized relative to thepeak density of the original condensate indicat-ed by the red dashed curve. The other curvescorrespond to times 1 ms (green), 4 ms (blue),and 14 ms (black). The width of the probe pulseis t 5 4 ms and the other parameters are thesame as in (A). An animated version is providedin (40).

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acterizes the “grayness,” with b 5 1 corre-sponding to a stationary soliton with a 100%density depletion. With b Þ 1, the solitonstravel at a fraction of the local sound velocity,cs(1 2 b2)1/2. As seen in Fig. 3A, after ashedding event, the remaining part of thesound wave continues to propagate at a re-duced amplitude. Our numerical simulationsshow that the solitons eventually reach apoint where their central density is zero andthen oscillate back to the other side, in agree-ment with the discussions in (18, 20).

In Fig. 3A, each of the two sound wavessheds two solitons. By considering the avail-

able free energy created by a defect, one findsthat when the defect size is somewhat largerthan the healing length, and the defect ampli-tude A is on the order of unity, the number ofsolitons that can be created is approximatelyA1/2d/(2j).

One obtains a simple physical estimate ofthe conditions necessary for soliton shedding bycalculating the difference in sound speed asso-ciated with the difference in atom density be-tween the center and back edge of the soundwave. As confirmed by our numerical simula-tions, this difference leads to development of asteeper back edge and an increasingly sharp

jump in the phase of the wave function. This isthe analog of shock wave formation from large-amplitude sound waves in a classical fluid (3).When the spatial width of the back edge hasdecreased to the width of a soliton with ampli-tude b 5 (A/2)1/2 (according to Eq. 5), such asoliton is shed off the back. Its subsonic speedcauses it to separate from the remaining soundwave. Furthermore, by creating defects withsizes on the order of the healing length, weexcite collective modes of the condensate, withwave vectors on the order of the inverse healinglength. In this regime, the Bogoliubov disper-sion relation is not linear (4, 5) and, according-ly, some of the sound wave will disperse intosmaller ripples, as seen in Fig. 3A.

Considering the evolution of a defect ofrelative density amplitude A and half-width din an otherwise homogeneous medium, weestimate that solitons of amplitude b 5 (A/2)1/2 will be created after the two resultingsound waves have propagated a distance

zsol 52d

A 1 1 2j

2d

1 2p2j2

d2A2 (6)

This is in agreement with our numerical cal-culations. We conclude that the minimumsoliton formation length is obtained for large-amplitude defects with a width just a fewtimes the healing length. This dictates thedefect width chosen in the experiments pre-sented here. Narrower defects disperse,whereas larger defects, comparable to thecloud size, couple to collective, nonlocalizedexcitations of the condensate.

Observation of a localized condensatevacancy. We explored the soliton formationexperimentally by creating defects in a BECwith the light roadblock. We controlled thesize of the defects by varying the intensity ofthe probe pulses, which had a width t 5 1.3ms. OD images of state u1& condensates areshown for one particular case in Fig. 4. Im-mediately after the defect is created, the trapis turned off and the cloud evolves and ex-pands for 1 ms and 10 ms, respectively. Asseen from the 1-ms picture, a single deepdefect is formed initially, which results increation of five solitons after 10 ms of con-densate dynamics and expansion. The initialdefect created in the trap could not be re-solved with our imaging system, which has aresolution of 5 mm. By varying the probeintensity, we find a linear relation betweenthe number of solitons formed and the probepulse energy, as expected.

Dynamics of quantum shock waves. Tostudy the stability of the solitons, we firstperformed 2D numerical simulations of Eqs.3 and 4. Figure 3B shows density profiles,Ncuc1u2, obtained for the same parameters asused in Fig. 2B. Again, the profiles are shown

Fig. 3. (A) Formation of solitons from a densitydefect. The plots show results of a 1D numer-ical simulation of Eqs. 3 and 4. The light road-block forms a defect, and the subsequent for-mation of solitons is seen. The defect is set upwith the same parameters for the light fields asin Fig. 2B. The number of condensate atoms isNc 5 1.2 3 106 and the peak density is 7.5 31013 cm23. The times in the plots indicate theevolution time relative to the time when theprobe pulse stops at the roadblock and thecoupling beam is switched off (at t 5 8 ms,with t 5 0 as defined in Fig. 1C). The solid anddashed curves show the densities of u1& atoms(Ncuc1u2) and u2& atoms (Ncuc2u2). The phase ofc1 is shown in each case with a dotted curve(with an arbitrary constant added for graphicalclarity). In the first two frames, the u2& atomsquickly leave as a result of the momentumrecoil, leaving a large-amplitude, narrow defectin c1 (because this is a 1D simulation, themomentum kick in the x direction is ignored).(B) The snake instability and the nucleation ofvortices. The plots show the density Ncuc1u2from a numerical simulation in 2D, with whitecorresponding to zero density and black to thepeak density (6.9 3 1013 cm23). The parame-ters are the same as in Fig. 2B, and the timesindicated are relative to the coupling beamturnoff at t 5 21 ms. The solitons curl abouttheir deepest point, eventually breaking andforming vortex pairs of opposite circulation(seen first at 3.5 ms). Several vortices areformed, and the last frame shows the vorticesslowly moving toward the edge of the conden-sate. At later times, they interact with soundwaves that have reflected off the condensateboundaries. Animated versions are provided in(40).

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at various times after the pulse is compressedand stopped. The deepest soliton (the oneclosest to the center) is observed to quicklycurl and eventually collapse into a vortexpair. The wave function develops a 2p phaseshift in a small circle around the vortex cores,which shows that they are singly quantizedvortices. Also, the core radius is approxi-mately the healing length. (Upon collapse, asmall sound wave between the two vorticescarries away some of the remaining solitonenergy.) This decay can be understood asresulting from variation in propagation speedalong the transverse soliton front. As dis-cussed in (22–25), a small deviation will beenhanced by the nonlinearity in the Gross-Pitaevskii equation, and thus the soliton col-lapses about the deepest (and therefore slow-est) point.

Observation of quantum shock waves,snake instability, and nucleation of vorti-ces. Figure 5 shows experimental images ofstate u1& condensates. After the defect wascreated, the condensate of u1& atoms was leftin the trap for varying amounts of time (asindicated in the figures). The trap was thenabruptly turned off and, 15 ms after release,we imaged a selected slice of the expandedcondensate (37), with a thickness of 30 mmalong the y direction. The release time of 15ms was chosen to be large enough that thecondensate structures are resolvable with ourimaging system (38). The slice was opticallypumped from state u1& to the u3S1/2, F 5 2&manifold for 10 ms before it was imaged withabsorption imaging by a laser beam nearlyresonant with the transition from u3S1/2, F 52& to u3P3/2, F 5 3&. The total pump andimaging time was sufficiently brief that nosignificant atom motion due to photon recoils

occurred during the exposure. The slice wasselected at the center of the condensate byplacing a slit in the path of the pump beam.

In Fig. 5A, the deepest soliton is seen to curlas it leaves certain sections behind, and at 1.2ms it has nucleated vortices. This is a directobservation of the snake instability. In Fig. 5B,at 0.5 ms, the snake instability has caused acomplicated curving structure in one of thesolitons, and vortices are observed after 2.5 ms.The vortices are seen to persist for many milli-seconds and slowly drift toward the condensateedge. We observed them even after 30 ms oftrap dynamics, long enough to study the inter-action of vortices with sound waves reflectedoff the condensate boundaries. Preliminaryresults, obtained by varying the y position of

the imaged condensate slices, indicate a com-plicated 3D structure of the vortices. In addi-tion, the defect has induced a collective motionof the condensate, whereby atoms originallyin the sides of the condensate fill in the defect.This leads to a narrow and dense central region,which then slowly relaxes (Fig. 5B).

We performed the experiment with a varietyof Rabi frequencies for the probe pulses, andsaw nucleation of vortices only for the peakVp . (2p)1.4 MHz. The free energy of avortex is substantially smaller near the borderof the condensate where the density is smaller;thus, smaller (and hence lower energy) defectswill form vortices very near the condensateedges, seen as “notches” in Figs. 3B and 5.

Conclusions. We have studied and ob-

Fig. 4. Experimental OD images and line cuts(at x 5 0) of a localized defect (top) and theresulting formation of solitons (bottom) in acondensate of u1& atoms. The imaging beamwas detuned 230 MHz and 220 MHz, respec-tively, from the u3S1/2, F 5 2&3 u3P3/2, F 5 3&transition. Before imaging, the atoms were op-tically pumped to u3S1/2, F 5 2& for 10 ms. Theprobe pulse had a peak Rabi frequency Vp 5(2p)2.4 MHz. The coupling laser had a Rabifrequency Vc 5 (2p)14.9 MHz, was turned on6 ms before the probe pulse maximum, and hada duration of 18 ms.

Fig. 5. Experimental OD images of a u1& con-densate, showing development of the snakeinstability and the nucleation of vortices. Ineach case, the BEC was allowed to evolve in thetrap for a variable amount of time after defectcreation. (A) The deepest soliton (nearest thecondensate center) curls as a result of thesnake instability and eventually breaks, nucle-ating vortices at 1.2 ms. Defects were producedin BECs with 1.9 3 106 atoms by probe pulseswith a peak Vp 5 (2p)2.4 MHz and a couplinglaser with Vc 5 (2p)14.6 MHz. The imagingbeam was 25 MHz detuned from the u3S1/2,F 5 2& 3 u3P3/2, F 5 3& transition. (B) Thesnake instability and behavior of vortices atlater times. The parameters in this series arethe same as in (A), except that the peak Vp 5(2p)2.0 MHz, the number of atoms in the BECswas 1.4 3 106, and the pictures were takenwith the imaging beam on resonance.

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served how short-wavelength excitationscause a breakdown of superfluidity in a BEC.Our results show how localized defects in asuperfluid will quite generally either disperseinto high-frequency ripples or end up in theform of topological defects such as solitonsand vortices, and we have obtained an ana-lytic expression for the transition between thetwo regimes. By varying our experimentalparameters, we can create differently sizedand shaped defects and can also control thenumber of defects created, allowing studiesof a myriad of effects. Among them are soli-ton-soliton collisions, more extensive studiesof soliton stability, soliton–sound wave col-lisions, vortex-soliton interactions, vortex dy-namics, interaction between vortices, and theinteraction between the BEC collective mo-tion and vortices.

References and Notes1. R. J. Donnelly, Quantized Vortices in Helium II (Cam-

bridge Univ. Press, Cambridge, 1991).2. R. M. Bowley, J. Low Temp. Phys. 87, 137 (1992).3. L. D. Landau, E. M. Lifshitz, Fluid Mechanics (Perga-

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35. For propagation of the Gross-Pitaevskii equation in 1D,we typically used a spatial grid with 4000 points anddz 5 0.040 mm. In 2D simulations, we typically used a750 3 750 grid with dz 5 0.21 mm and dx 5 0.057 mm.To solve the equations self-consistently, we kept track ofthe wave functions at previous time points and projectedforward to second order. Smaller time steps and gridspacing were also used to assure convergence of theresults. To mimic the nonlinear interaction strength atthe center of a 3D cloud, we put in an effective conden-sate radius [calculated with the Thomas-Fermi approxi-mation (39)] in the dimensions that were not treateddynamically. In all calculations, the initial condition was

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41. Supported by a National Defense Science and Engi-neering Grant sponsored by the U.S. Department ofDefense (C.S.), the Rowland Institute for Science, theDefense Advanced Research Projects Agency, the U.S.Air Force Office of Scientific Research, the U.S. ArmyResearch Office OSD Multidisciplinary University Re-search Initiative Program, the Harvard Materials Re-search Science and Engineering Center (sponsored byNSF), and the Carlsberg Foundation, Denmark.

15 May 2001; accepted 11 June 2001Published online 28 June 2001;10.1126/science.1062527Include this information when citing this paper.

The Composite Genome of theLegume Symbiont Sinorhizobium

melilotiFrancis Galibert,1 Turlough M. Finan,2

Sharon R. Long,3,4* Alfred Puhler,5 Pia Abola,6 Frederic Ampe,7

Frederique Barloy-Hubler,1 Melanie J. Barnett,3 Anke Becker,5

Pierre Boistard,7 Gordana Bothe,8 Marc Boutry,9 Leah Bowser,6

Jens Buhrmester,5 Edouard Cadieu,1

Delphine Capela,1,7† Patrick Chain,2 Alison Cowie,2

Ronald W. Davis,6 Stephane Dreano,1

Nancy A. Federspiel,6‡ Robert F. Fisher,3 Stephanie Gloux,1

Therese Godrie,10 Andre Goffeau,9 Brian Golding,2

Jerome Gouzy,7 Mani Gurjal,6 Ismael Hernandez-Lucas,2

Andrea Hong,3 Lucas Huizar,6 Richard W. Hyman,6 Ted Jones,6

Daniel Kahn,7 Michael L. Kahn,11

Sue Kalman,6§ David H. Keating,3,4 Erno Kiss,7 Caridad Komp,6

Valerie Lelaure,1 David Masuy,9 Curtis Palm,6 Melicent C. Peck,3

Thomas M Pohl,8 Daniel Portetelle,10 Benedicte Purnelle,9

Uwe Ramsperger,8 Raymond Surzycki,6\ Patricia Thebault,7

Micheline Vandenbol,10 Frank-J. Vorholter,5 Stefan Weidner,5

Derek H. Wells,3 Kim Wong,2 Kuo-Chen Yeh,3,4¶ Jacques Batut7

The scarcity of usable nitrogen frequently limits plant growth. A tight metabolicassociation with rhizobial bacteria allows legumes to obtain nitrogen com-pounds by bacterial reduction of dinitrogen (N2) to ammonium (NH4

1). Wepresent here the annotated DNA sequence of the a-proteobacterium Sinorhi-zobium meliloti, the symbiont of alfalfa. The tripartite 6.7-megabase (Mb)genome comprises a 3.65-Mb chromosome, and 1.35-Mb pSymA and 1.68-MbpSymB megaplasmids. Genome sequence analysis indicates that all three el-ements contribute, in varying degrees, to symbiosis and reveals how thisgenome may have emerged during evolution. The genome sequence will beuseful in understanding the dynamics of interkingdom associations and of lifein soil environments.

Symbiotic nitrogen fixation is profoundly im-portant for the environment. Most plants as-similate mineral nitrogen only from soil or

added fertilizer. An alternative source pow-ered by photosynthesis, rhizobia-legumesymbioses provide a major source of fixed

R E S E A R C H A R T I C L E S

27 JULY 2001 VOL 293 SCIENCE www.sciencemag.org668

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