Atomic Bose-Einstein condensates and Rydberg gases

Sebastian Wüster

Summer-term 2013 Information

contact details: Sebastian Wüster, • Max Planck Institute for the Physics of Complex Systems, Noethnitzer Str. 38, Dresden Office: 2B8

email://[email protected]

Phone: +49 351 871 2208 web: http://www.mpipks-dresden.mpg.de/~sew654/teaching.html

• Lecture: Wednesdays: 16:40 - 18:10 Room BZW/ A120/ P 13 lectures (10.4. - 17.7., not 22.5, 5.6.) beamer introduction, then chalk and blackboard + movies and images Literature

(1) C.J. Pethick and H. Smith, “Bose-Einstein Condensation in Dilute Gases” , Cambridge University Press (2002) (2) L.P. Pitaevskii and S. Stringari “Bose-Einstein Condensation”, Oxford University Press (2003) (3) F. Dalfovo, S. Giorgini, L.P. Pitaevskii, and S. Stringari “Theory of Bose- Einstein condensation”, Rev. Mod. Phys. 71, 463 (1999) (4) T. F. Gallagher, “Rydberg atoms”, Cambridge University Press (1994) Outline

1. Introduction 1.1. Motivation 1.2. Revision 2. Ultra-cold atomic gases 2.1. Quantum statistical physics 2.2. Trapping and cooling of atoms 2.3. Interactions between atoms 3. Bose-Einstein condensates 3.1. Field operator 3.2. Gross-Pitaevskii theory 3.3. Excitations of the condensate

4. Rydberg atoms 4.1. Quantum defects 4.2. Rydberg interactions and external fields 4.3. Rydberg excitation and decay 4.4. Dipole blockade 4.5. Field ionization 4.6. Applications in quantum information 1. Introduction • Purpose of 1.1. motivation: ★ get you excited about the field of ultra-cold atoms (quantum-atom-optics)

★ show for what the material presented later is important

★ personal view: highlight use of BEC for inter-disciplinary physics (quantum simulation) • Purpose of 1.2. revision: ★ remind you of required material from quantum mechanics and atomic physics

★ make sure everyone has similar starting point

★ will not replace prior study of these topics, if you notice that you are not familiar with some of the physics, please recap. Survey: Please indicate for the following physics topics how familiar you feel that you are with them: never not had in remember what it is, heard and can fully heard lecture but no details revise the details familiar quantum oscillator:

second quantisation:

quantum field theory: quantum statistical physics:

Bose-Einstein statistics:

Zeeman effect:

Stark effect:

two-level atom interacting with light field:

hyperfine splitting:

quantum scattering-theory/ partial wave expansion: 1.1. Motivation (part1) Bose-Einstein Condensation

Bose 1924: Identical quantum particles (e.g. photons) have to be treated as indistinguishable. This affects the counting of the possible ways to realize a given state. S. N. Bose, Z. Phys. 26 (1924) 178. Albert Einstein Satyendra Nath Bose

Einstein 1925: This results at T=0 in a case where all bosons occupy the same quantum state. A.Einstein, Sitzungsber. Kgl. Preuss. Akad. Wiss. page 261 (1924). A.Einstein, Sitzungsber. Kgl. Preuss. Akad. Wiss. page 3 (1925). 1 => Bose-Einstein Statistics: Occupation of state with energy E: n = e(E µ)/kB T 1 − − => Bose-Einstein Condensation: At T=0, all Bosons will occupy the same lowest energy quantum state cloud was determinedfrom the absorption tion to characterizing any deviations from fractions, respectively. (Figs. 2B and 4). As of a 20-is, circularlypolarized laser pulse thermal equilibrium. The measurement the cooling progresses (Fig. 4), the noncon- resonantwith the 5S1/2,F - 2 to 5P3/2, F - process destroys the sample, but the entire densate fraction is reduced until, at a value

3 transition.The shadowof the cloud was load-evaporate-probe cycle can be repeat- of vevap of 4.1 MHz,little remainsbut a pure imagedonto a charge-coupleddevice array, ed. Our data represent a sequence of evap- condensate containing 2000 atoms. digitized,and storedfor analysis. orative cycles performed under identical The condensate first appears at an rf This shadow image (Fig. 2) contains a conditions except for decreasing values of frequency between 4.25 and 4.23 MHz. The large amount of easily interpreted infor- vevap, which gives a corresponding de- 4.25 MHz cloud is a sample of 2 X 104 mation. Basically, we did a 2D time-of- crease in the sample temperature and an atoms at a number density of 2.6 x 1012 flight measurementof the velocity distri- increase in phase-space density. cm-3 and a temperature of 170 nK. This bution. At each point in the image, the The discontinuous behavior of thermo- represents a phase-space density pps of 0.3, optical densitywe observedis proportional dynamic quantities or their derivatives is which is well below the expected value of to the column density of atoms at the always a strong indication of a phase tran- 2.612. The phase-space density scales as the correspondingpart of the expandedcloud. sition. In Fig. 3, we see a sharp increase in sixth power of the linear size of the cloud. Thus, the recorded image is the initial the peak density at a value of vevap of 4.23 Thus, modest errors in our size calibration velocity distribution projected onto the MHz. This increase is expected at the BEC could explain much of this difference. Be- plane of the image. For all harmoniccon- transition. As cooling proceeds below the low the transition, one can estimate an fining potentials, includingthe TOP trap, transition temperature, atoms rapidly accu- effective phase-space density by simply di- the spatial distributionis identical to the mulate in the lowest energy state of the 3D viding the number of atoms by the observed velocity distribution,if each axis is linear- harmonic trapping potential (23). For an volume they occupy in coordinate and ve- ly scaled by the harmonic oscillator fre- ideal gas, this state would be as near to a locity space. The result is several hundred, quency for that dimension (22). Thus, singularity in velocity and coordinate space which is much greater than 2.6 and is con- from the single image we obtained both as the uncertainty principle permits. sistent with a large occupation number of a the velocity and coordinate-spacedistri- Thus, below the transition we expect a single state. The temperatures and densities butions, and from these we extracted the two-component cloud, with a dense central quoted here were calculated for the sample temperatureand central density, in addi- condensate surrounded by a diffuse, non- in the unexpanded trap. However, after the condensate fraction.Experimental This behavior is clear- adiabatic expansion Condensation stage, the atoms are in dilute ly displayed in sections taken horizontally still in good thermal equilibrium, but the

4I I II through the center of the distributions, as temperatures and densities are greatly re- 4~~~~~~~~~~~~~~~~~~~~~~~~~~shown in Fig. 4. For values of vevap above duced. The 170 nK temperature is reduced 4.23 MHz, the sections show a single, to 20 nK, and alkalithe number density gases is reduced smooth, Gaussian-like distribution. At 4.23 from 2.6 X 1012 cm3 to 1 x 1011 cm-. theoreticalMHz, a sharp central prediction peak in the of distribu- BEC 1925There is no obstacle to adiabatically coolingNobel Prize in Physics (2001), E 3 tion begins to appear. At frequencies below and expanding the cloud further when it is en) 1111 experimental4.23 MHz, two distinct realization components to 1995 the desirable to reduce the atom-atom interac-W. Ketterle ,E.A. Cornell and C.E. Wieman E 0 cloud are visible, the smooth broad curve tions, as discussed below (24). and a narrow central peak, which we iden- A striking feature evident in the images ' 2 tify as the noncondensate and condensate shown in Fig. 2 is the differing axial-to- radial aspect ratios for the two components 7,_I of the cloud. In the clouds with no conden- 4.71 sate (vevap > 4.23 MHz) and in the non- condensate fraction of the colder clouds, the velocity distribution is isotropic (as ev- 01 idenced by the circular shape of the yellow 4.25 to green contour lines in Fig. 2, A and B). But the condensate fraction clearly has a 4.23 4.0 4.2 4.4 4.6 4.8 larger velocity spread in the axial direction than in the radial direction (Fig. 2, B and Vevap(MHZ) 4.21 C). This difference in aspect ratios is readily Fig. 3. Peak density at the center of the sample as 4.19 explained and in fact is strong evidence in a function of the finaldepth of the evaporative cut, support of the interpretation that the cen- vevap. As evaporation progresses to smaller values tral peak is a Bose-Einstein condensate. The of vevap' the cloud shrinks and cools, causing a f 4.16 modest increase in peak density untilvevap reach- noncondensate atoms represent a thermal es 4.23 MHz. The discontinuity at 4.23 MHz indi- distribution across many quantum wave cates the first appearance of the high-density functions. In thermal equilibrium, velocity condensate fraction as the cloud undergoes a 4.11 distributions of a gas are always isotropic phase transition.When a value for Vevap of 4.1 Mhz regardless of the shape of the confining is reached, nearly all the remaining atoms are in potential. The condensate atoms, however, the condensate fraction. Below 4.1 MHz,the cen- are all describedK.B. by theDavis same wave et function,al., Phys. Rev. Lett. 75 (1995) 3969. traldensity decreases, as the evaporative "rfscal- which will have an anisotropy reflecting 4.06 pel" begins to cut into the condensate itself. Each that of the confining potential. The veloc- data point is the average of several evaporative M. H. Anderson et al. , Science 269 (1995) 198. 300 pm of the wave function cycles, and the error bars shown reflect only the ity spread ground-state scatter in the data. The temperature of the cloud is Fig. 4. Horizontalsections taken through the ve- for a noninteracting Bose C.C.gas should Bradley be 1.7 et al., Phys. Rev. Lett. 75 (1995) 1687. a complicated but monotonic function of vevap. At locity distributionat progressively lower values of (81/4) times larger in the axial direction Vevap =4.7 MHz,T = 1.6 pK,and for vevap= 4.25 vevap show the appearance of the condensate than in the radial direction. Our observa-crucial techniques: laser-cooling, magneto-optical MHz, T= 180 nK. fraction. tionsT arecrit in = qualitative 170 nK agreement with this 200 SCIENCE * VOL. 269 * 14 JULY 1995 trap, evaporative cooling => chapter 2 BEC in other physical systems (some examples) Cooper-pair condensates in superconductors

http://www.supraconductivite.fr

•more complicated than direct BEC of alkali-gases, since we first have to bind two Fermions to make a Boson. •Can see similar effect in degenerate Fermi gases. liquid helium, superfluidity

•Helium is strongly interacting, only shows very small condensed fractions

exciton-polaritons in semi-conductor micro cavities PHYSICAL REVIEW LETTERS week ending PRL 95, 143201 (2005) 30 SEPTEMBER 2005

axial field curvature, and Cartesian coordinates x; y; z are passing close to the line of zero field, which extends all chosen so that z is the axial coordinate. The magnetic † field around the ring, may flip their spins and be expelled from magnitude falls to zero in the x^-y^ plane along a circle of the trap. Extending the treatment by Petrich et al. [22] to 1=2 radius 0 2 B0=Bz00 centered at the origin. This is the Q this scenario, we estimate a Majorana loss rate of @ ˆ m3=4  ring, a ring-shaped magnetic trap for weak-field seeking 3=2 B0 1 p † 6s for our trap at a temperature of 60 K. In a atoms. Near the field zeros, the magnetic field has the form k T 5=4 ÿ B † ˆ of a transverse (radial and axial directions) two- tilted Q ring, the zero-field region is reduced to just two points at opposite sides of the ring. Majorana losses in a dimensional quadrupole field with gradient B0 B0Bz00. Such traps can also be obtained using differentˆ electro- tilted Q ring are thus similar to those in spherical quadru- p magnet configurations [12]. pole traps and much smaller than in a balanced Q ring. We In our apparatus, the Q ring is formed using a subset of confirmed this qualitative behavior by measuring the life- the coils (the curvature and antibias coils; see Fig. 1) used time of trapped atoms in balanced and tilted Q-ring traps. 1 in our recently demonstrated millimeter-scale Ioffe- In the balanced Q ring, the measured 0:3sÿ Majorana loss Pritchard magnetic trap [21]. Our work is aided, in par- rate was thrice that in a tilted Q ring, while falling far short 1 ticular, by the large axial curvatures produced in this trap of the predicted 6sÿ loss rate, presumably due to residual and by the vertical orientation of the trap axis. These azimuthal fields. features are relevant for the operation of a Q ring in the The high loss rates in the Q ring can be mended in a presence of gravity, for two reasons. First, trapping atoms manner similar to the time-orbiting potential (TOP) trap by in the Q ring requires transverse confinement sufficient to which Majorana losses in a spherical quadrupole field were Variousovercome the force of gravity;atom this places a lowertraps bound on overcome [22]. As proposed by Arnold [12], a TORT with nonzero bias field can be formed by displacing the ring of the radius of the Q ring of 0 > min 2mg=  Bz00 with m the atomic mass, g the acceleration due’ to gravity,j j and  field zeros away from and then rapidly rotating it around the trapped atoms [Fig. 1(d)]. From Eq. (1), the Q ring can the atomic magnetic moment. Indeed, if min exceeds the •Experimental techniques to coolrange down over which to Eq. the (1) is valid, nK typically regime the distance are to anywaybe displaced quite radially by sophisticated application of an axial bias => field, the field-producing coils, the formation of a Q ring may be and displaced along z^ by a cylindrically symmetric spheri- 2 cal quadrupole field. The TORT provides transverse qua- They then also allow a tremendousprecluded degree entirely. Withof Bcontrolz00 5300 G= cmoverin our the experi- BEC/ ultra-cold gas, once it has ˆ dratic confinement with an effective field curvature of ments, min 115 m is much smaller than the millime- ˆ B B 2=2B , where B is the magnitude of the rotat- been created. ter dimensions of the electromagnets used for the trap. eff00 ˆ 0 rot rot Second, the vertical orientation of the Q-ring axis allows ing field seen at the trap minimum. Just as the TOP trap cold atoms to move slowly along the nearly horizontal depth is limited by the ‘‘circle of death,’’ the TORT trap magneto optical trap opticalwaveguide ratherlattice than being confined in a deep gravita- ring trap tional well. Atoms can be localized to a particular portion of the Q ring by application of a uniform sideways (in the x^-y^ plane) magnetic field; e.g., a weak bias field Bsx^ tilts the Q ring by
z= B =B = about the y^ axis. This adjustment 0 ˆ s 0† 0 also adds an azimuthal field of magnitude Bs sin , split- ting the Q ring into two trap minima at oppositej sidesj of the ring, with  being the azimuthal angle conventionally defined. We loaded cold atoms into the Q ring using a procedure similar to previous work [21]. Briefly, about 2 109 87Rb  atoms in the F 1;mF 1 hyperfine ground state were loaded intoj oneˆ of twoˆÿ adjacenti spherical quadrupole magnetic traps. Using these traps, atoms were transported FIG. 2 (color). Atoms in a ring-shaped magnetic trap. Shown 3 inches from the loading region to the Q-ring trap region. are top (a)–(f) and side (g)–(i) absorption images of ultracold 87S. Gupta et al., Phys. Rev. Lett. 95 (2005) 143201. During this transport, rf evaporative cooling was applied, Rb clouds in either a Q ring (a)–(c) or TORT (d)–(i) with 7 applied side field B 9:2 (left), 0 (middle), and 2:5G(right yielding 2:5 10 atoms at a temperature of 60 K in a s ˆ ÿ spherical quadrupole trap with an axial field gradient of column), respectively, in the x^ direction. Images were taken 2 ms afteratom turning off chip the traps. The applied field tilts the Q ring or 200 G=cm. Within 1 s, we then converted the spherical TORT and favors atomic population in one side or another of the quadrupole to a tilted Q-ring trap produced with Bz00 trap. For B 0, the trap lies nearly in the horizontal plane and 2 ˆ s 5300 G=cm , B0 22 G, and a side field of magnitude its azimuthal potential variation is minimized. For the Q ring, I. Bloch, Natureˆ Phys. 1 (2001) 23.7 Bs 9:2G. This process left 2 10 atoms trapped in the B0 22 G; for the TORT, B0 20 G and Brot 17 G; and ˆ  ˆ 2 ˆ ˆ http://www.physics.otago.ac.nz/research/ Q ring (Fig. 2). Bz00 5300 G=cm for both. The temperature of trapped atoms is ˆ jackdodd/resources/ResourceFiles/galleryimages/ The trapping lifetime of atoms in the Q ring is limited by 90 K in the Q ring, and 10 K in the TORT. Resonant Majorana losses. In a balanced Q ring, trapped atoms absorption ranges from 0 (blue) to >80% (red).

143201-2 •Can confine atoms using magnetic field, electrical fields (from lasers) or both. http://www.physics.uq.edu.au/BEC/images/ experimental/research_combinedtrap.jpg R ESEARCH A RTICLES optic communications (13). Solitons may be density at its center, a width on the order of ed phase distribution of the condensate wave either bright or dark, depending on the details the healing length ␰ϭ(2nMg/ប2)–1/2 (15), function with a Mach-Zehnder matter-wave in- of the governing nonlinear wave equation. A and a discontinuous phase step. As ␦ decreas- terferometer that makes use of optically in- bright soliton is a peak in the amplitude; a es, the speed increases and approaches the duced Bragg diffraction (25, 26). Our Bragg dark soliton is a notch with a characteristic speed of sound. The solitons become shal- interferometer differs from previous ones in phase step across it. lower and wider and have a more gradual that we can independently manipulate atoms in A weakly interacting BEC obeys a non- phase step (15). They travel opposite to the the two arms (because of their large separation) linear wave equation that supports solitons, direction of the phase gradient. Because a and can resolve the output ports to reveal the as shown by recent theoretical studies (14– soliton has a characteristic phase step, opti- spatial distribution of the condensate phase. In 17). At zero temperature, this wave equation cally imprinting a phase step on the BEC our interferometer, a Bragg pulse splits the is known as the Gross-Pitaevskii equation wave function should be a way to create a initial condensate into two states, ͉A͘ and ͉B͘, (18), soliton. differing only in their momenta (Fig. 2). After

2 2 2 Phase imprinting. We performed our ex- they spatially separate, the phase step (Fig. 1A) t)␺ϭ[Ϫ(ប /2M )ٌ ϩ V ϩ g!␺! ]␺ periments with a condensate having ϳ2 ϫ is imprinted on ͉A͘,while͉B͘ is unaffected andץ/ץ)iប (1) 6 10 sodium atoms in the 3S1/2, F ϭ 1, mF ϭ serves as a phase reference. When recombined, where ␺ is the condensate wave function Ϫ1 state, with no discernible thermal fraction they interfere according to their local phase normalized to the number of atoms, V is the (7). The condensate was held in a magnetic difference. Where this phase difference is 0, ͌ trapping potential, M is the atomic mass, ប is trap with trapping frequencies ␻x ϭ 2␻y ϭ atoms appear in port 1, and where it is ␲ atoms the Planck constant divided by 2␲ and g 2␻z ϭ 2␲ϫ28 Hz. The Thomas-Fermi appear in port 2. Imaging the density distribu- describes the strength of the atom-atom inter- diameters (18) were 45, 64, and 90 ␮m, tions of ports 1 and 2 displays the spatially action (19). Solitons propagate without respectively. Initially the BEC, described by varying phase (27). The image in Fig. 2 shows spreading (dispersing) because the nonlinear- the ground-state solution of Eq. 1, had a the output of the interferometer when a phase of ity balances the dispersion; for Eq. 1, the uniform phase (21, 22). ␲ was imprinted on the upper half of ͉A͘ (28). corresponding terms are the nonlinear inter- We modified the phase distribution of the The high-contrast “half moons” are direct evi- action g!␺!2 and the kinetic energy – (ប2/ BEC by exposing it to pulsed, off-resonant laser dence that we can control the condensate spatial 2M)ٌ2, respectively. Our sodium condensate light with an intensity pattern I(x, y)(Fig.1).In phase distribution and, in particular, imprint the only supports dark solitons because the atom- this process, the atoms experience a spatially phase step appropriate for a soliton (29). atom interactions are repulsive (g Ͼ 0). varying light-shift potential U(x, y) ϭ (ប⌫2/ Soliton propagation. To observe soliton

A distinguishing characteristic of a dark 8⌬)[I(x, y)/I0]andacquireacorresponding propagation, we did not use interferometry soliton is that its speed is less than the Bogo- phase ␾(x, y) ϭ –VUOLUME(x, y)T81,/ប.Here NUMBER⌫8is the PHYSICALREVIEWLETTERS(30) but instead measured BEC density dis- 24AUGUST 1998 articles 1/2 liubov speed of sound, ␷0 ϭ (gn/M) (18, transition line width, I0 is the saturation inten- tributions with absorption imaging (1, 27) 2 R ESEARCH20), whereA RTICLESn ϭ !␺0! is the unperturbed sity, ⌬ is the detuning of the laser from the after imprinting a phase step (31). Figure 3, A about 80% of the atoms were lost. This, coupled with the stability monic oscillator potential one finds n ￿ NðNaÞ ￿ 3=5 (ref. 23) (We condensate￿ density.￿ Thepi soliton speed ␷ can atomic resonance, and T is the laser pulse du- to E, shows the evolution of the condensate and finite programmingtion hasspeed a largeof the velocitypower supplies, in the ϩlimitedx direction.the notevalue.that, for Thea experimentalgeneral power-la uncertaintyw potential isSi main-cix i , one obtainsand other excitationss near x ϭ 20 ␮mmoving Experimentalk ￿ 1 control ramp rates to those given above. n ￿ NðNaÞ , where k ¼ 1=ð1 þ S 1=p Þ). Thus, the value of the This velocity, which can be understood as ly due to the difficultybe expressed ini determiningi in terms the of eitherrapidly the to phasethe right. step Most ofration these features (23). Weare chose T to be short enough so after the top half was phase-imprinted with The observation of twin resonances separated by 54 ￿ 1 G, with ￿scattering￿ length scales as: arising from the impulse imparted by the position of the initial␦ (0 phaseϽ␦v5 step.Յ ␲) or the solitonnot well “depth” resolved inn the, experimentalthat the atomicimages motion was negligible during ␾ Ϸ 1.5␲, a phase for which we observed a the weaker one at lower field, exactly matches the theoretically a ￿ rms ð3Þ d 0 predicted patternopticaland thus dipolestrongly force,confirms resultsour interpretation. in a positiveN den-o We can alsowhich compare isN the the results difference of the between(Fig. 3, A ton E).and We the observedthe both pulse experimen- (Raman-Nath regime). In this limit, single deep soliton (the reason for imprinting resonance phenomenasity disturbancewere observed thatin the travelsjm ¼ ￿ at1 orstate aboveat any the Bothnumericalv and N can 3Dbe solutionsdirectly evaluated of Eq. 1from toabsorption the ana- imagestallyof and theoretically that when the imprinted F ￿ rms density at the bottom of the notch (14, 15): the effect of the pulse can be expressed as a a phase step larger than ␲ is discussed be- field up to 1,000speedG, in ofagre sound.ement Awith darktheor notchy which is leftpredicted behind; freelylyticalexpanding predictionscondensates. of Eq.For 2, whicha cigar-shaped describescondensate a phasethe step is increased, the weak soliton on the resonancFeshbaches for thisthisstate isonly a solitonat much resonancemovinghigher fields. slowly in the –x free traditionalexpansion is darkpredominantly solitonPhase in aradial, homogeneous,and soimprintingthe c 1Dontributionleftof becomes 1/2 deeper; whensudden the phase phase step is imprint,transitions which modifies the ini- low). Immediately after the phase imprint, 5 8 the axial dimension to v could␷s/␷0beϭnegcos(lected.␦The/2) quantitϭ [1yϪv (/nNd/n)] (2) direction (opposite to the direction of the geometry. We calculatedrms the soliton speed lowered,rms both solitons becometial shallower wave function: and ␺VOLUME3 ␺ 81,exp[ NiUMBER␾(x, y)]8 (24). PHYSICALREVIEWLETTERSthere is a steep phase gradient across the 24AUGUST 1998 Changing the scattering length (equation (3)), normalized to unity outside the resonance, should The trap(interactionloss measurappliedements force).easily located control)the Feshbach resonances. be identicalusing a localto a/a˜ density(equationFor ␦ϭ␲(motion approximation(1)).,This thequantit soliton in Eq.ycontrol)was 2hasmeasured zeropropagate velocity, faster. zero Interferometry.Wemeasuredtheimprint-(internal state control)middle of the condensate such that this por- 2 To measure the variationWe haveof the numericallyscattering length solvedar Eq.ound 1 inthese three around[n ϭthe!resonanc␲0(r)!e ,at where907 G and␲0is(rshown) is thein Fig ground-. 2b together wWeith could avoid the uncertainty in the resonances, we determineddimensionsthe throughinteraction theenerg applicationy of a trapped of real- the statetheoreti solutioncal prediction of Eq.of 1]a reson fromance eitherwith thewidth phase∆ ¼ 1 G.positionThe of the initial phase step and improve condensate. This was done by suddenly switching off the trap, data clearly displays the predicted dispersive shape and shows FIG. 3. The relative motion of the centers of mass of the two space product formulas (32) and by using a or depth of the solitonsA obtained in the 3D our measurement of the soliton speed by condensates under the same conditions as those in Fig. 2. allowing the storeddiscreteinteraction variableenerg representationy to be converted ofinto the wavethe evidencesimulations.for a variation In everyin the casescattering examined,length b thisy more replacingthan a the step mask (Fig. 1A) with a thin kinetic energy of a freely expanding condensate and measuring it by factor of ten. time-of-flight absorptionfunctionimaging (33)1,2,21 based. The oninteraction Gauss-Chebyshevenergy is Wspeede now discuss is in excellentthe assumptions agreementfor equation with(3) theand re-showslitthat (Fig.it 1B). The thin slit produced a stripe 2 proportional to thequadraturescattering length with 50and tothe 400average spatialdensit gridy of pointsthe is approsultsximately of 3Dvalid numericalfor our simulations.conditions. (1) We assumed thatofthe light with a Gaussian profile (1/e full The two states almost completely separate (Figs. 2a–2c) condensate n : in each dimension; in the latter approach, the condensateFigurerem 4ains showsin equilibrium the theoreticalduring the densitymagnetic and fieldwidthramp. Ϸ 15 ␮m). With this stripe in the center after 10 ms; they then “bounce” back until, at T ￿ 25 ms, ￿ ￿ This is the case if the adiabatic condition a˙=a p q holds for the time dependence2p 2 of the solution was obtained phase profile along the x axisz through the centeri of the condensate, numerical simulations pre- the centers of mass are once more almost exactly superim- = temporal change of the scattering length16, and a similar condition by Runge-KuttaEI N ¼ integration.a n Figure 3, Fð to2Þ J, of the condensate 5 ms after they ␾ ϭ 1.5␲ dict the generation of solitons that propagate posed (Fig. 3), although a distinctive (and reproducible) m ￿ ￿ for the loss of atoms (the q are the trapping0 frequencies). For the shows the results of the simulations where the phase imprint (Fig. 3H).i The dark soliton notch symmetrically outward. We selected experi- vertical structure has formed (Figs. 2d–2f). By T ￿ where N is the number of condensed atoms of mass m. For a large condensate’s fast radial dynamics (qr ￿ 2p ￿ 1:5 kHz) this con- condensate the kineticexperimentalenergy in phasethe trap imprintis negligible is approximated(Thomas– ditionandis its phasefulfilled, stepwher areeas centeredfor the atxx ϭϪslower8 ␮axialm. motionmental images with solitons symmetrically 65 ms, the system has apparently reached a steady state

Fermi limit), andasEI is␾(equalx, y) toϭthe(␾kinetic0/2)[1 energϩ tanh(y EK ofx/lthe)],freely where (qz This￿ 2p ￿ phase0:1 kHz) step,it ␦ϭbreaks0.58down␲ isclose lessto thanor within the thelocatedreso- about the middle of the condensate (Figs. 2g, 2h, and 3) in which the separation of the cen- = 2 = expanding condensate␾0 ϭ E1.5K N␲,¼ andmvrmsl ϭ2, 2wher␮me correspondsvrms is the root- to an nance.imprintedIn this case phasethe densit of 1.5y wou␲.Thedifferenceisld approach the two-dimensionaland measured the distance between them. ters of mass is 20% of the extent of the entire condensate. mean-square velocity of the atoms. For a three-dimensional har- scaling N(Na)!1/2, but the values for a/a˜ (Fig. 2b) would differ by at imprinting resolution of ϳ4.4 ␮m(27, 34). caused by the mismatchBC between the phase Figure 5A shows the separation of the pair of FIG.From 3. these The images relative wemotion observe of the (i) centers the fractional of mass of steady- the two The calculated and experimental images are mostimprint50%. (2) andThe thesec phaseond andassumption depth ofwas thea solitonthree-dimensionalsolitons as a function of time. For a small state separation of the expanded image is large compared harmonic trap. If the axial potential has linear contributions, the condensates under the same conditions as those in Fig. 2. in very good agreement. solution of Eq. 1: Our imprinting!2/3 resolution phase imprint of 0.5 at Gaussian density scales instead like N(Na) resulting in at most a 50% ␾0 Ϸ ␲ A striking feature of the images is the change(27)islargerthanthesolitonwidth,whichisonfor a/a˜. (3) We assumed that contributions of collectivmaximum,e we observed solitons moving at curvature of the soliton. This curvature arises excitationsthe orderto the of thereleas healinged energ lengthy were (small.␰Ϸ0.7Axial␮m),striationsthewere Bogoliubov speed of sound within exper- The two states almost completely separate (Figs. 2a–2c) from the 3D geometry of the trapped conden- obserandved wein free doexpansion not controlfor theboth amplitudejmF ¼ þ1 ofand thejmF ¼imental￿ 1 uncertainty. For a larger phase im- after 10 ms; they then “bounce” back until, at T ￿ 25 ms, ￿ ￿ sate and occurs for two reasons. First, the atomswave(prob function.ably crea Theted mismatchby the changing producespotential featuresduring theprintfast of ␾ Ϸ 1.5␲,weobservedamuchslowerFig. 2. Space-time diagram of the matter-wave interferometer used tothe measure centers of the mass spatial are once phase more almost exactly superim- magnetic field ramp).Fig.However 1. ,(Athe)small Writingscatter aof phasepoints outside step onto0 the con- distribution imprinted on the BEC. Three optically induced Bragg diffraction pulses (7) formed the speed of sound ␷0 is largest at the trap center, in addition to the deep soliton, such as a shallow soliton propagation, in agreement with numer- posed (Fig. 3), although a distinctive (and reproducible) the resonance in Fig. 2b, which do not show any evidence of vertical structure has formed (Figs. 2d–2f). By T where the density is greatest, and decreases oscillations,dark solitonsuggests at xdensate.ϭϪthat the14 ␮contributionmmovingtotheleft A far-detunedof excitations uniformicalto the simulations. light Anpulse even largerinterferometer. phase imprint Each pulse consisted of two counterpropagating laser beams detuned by Ϫ2 GHz ￿ toward the condensate edge. Second, as the released energy is negligible.projects(4) W ae assumed maska (asudden razorswitch-off blade)of onto the con- from atomic resonance (so that spontaneous emission is negligible), with65 theirms, the frequencies system has differing apparently reached a steady state soliton moves into regions of lower conden- the trap and ballisticdensate.expansion. BecauseThe inhomogeneous of the lightbias shift,field this imprints by 100 kHz. The first Bragg pulse had a duration of 8 ␮s and coherently(Figs. split 2g, the 2h, condensate and 3) in which into the separation of the cen- sate density, we find numerically that the during the first1 ms1– 2ams phaseof free distribution2expansion ms accelerated that5 isthems proportionalaxial 7 to ms the two components10 ms ͉A͘ and ͉B͘ with equal numbers of atoms; ͉A͘ remainedters at rest of mass and is͉B 20%͘ received of the extent two of the entire condensate. expansion,ABCDEbut had a negligible effect on the expansion of the From these images we observe (i) the fractional steady- density at its center (n Ϫ nd) approaches zero, x light intensity distribution. A lens (not shown) photon recoils of momentum. When they were completely separated, we applied the 500-ns phase approaches , and decreases to zero condensate in the radial direction, which was evaluated for Fig. 2b. state separation of the expanded image is large compared ␦ ␲ ␷s None of the correctionsis used(1)– to(4) discussed image theabove razoraffect our blade onto the imprint pulse to the top half of ͉A͘, which changed the phase distribution of ͉A͘ while ͉B͘ served as before reaching the edge. The soliton stops conclusion that the condensate.scattering length Thevaries maskdispersively in (nearB) writesa a phase a phase reference. A second Bragg pulse (duration 16 ␮s), 1 ms after the first pulse, brought ͉B͘ to because its depth n , rather than its phase d Feshbach resonance. stripeMore accurate ontoexperim the condensate.ents should be done The mask in (C) rest and imparted two photon momenta to ͉A͘. When they overlapped again, 1 ms later, a third offset ␦, appears to be a conserved quantity in with a homogeneousimprintsbias field. In anaddition, azimuthallyan optical varyingtrap with phase pattern pulse (duration 8 ␮s) converted their phase differences into density distributions at ports 1 and 2. a nonuniform medium. larger volume and lower density would preclude the need to ramp FGHI J Soliton speed. The subsonic propagation the field quickly becausethatthr canee-body be usedrecombination to createwou vortices.ld be The image shows the output ports 1 and 2 as seen when we imprinted a phase step of ␲ (29). reduced. speed of the notches seen in Fig. 3 shows that The trap losses observed around the Feshbach resonances merit they are solitons and not simply sound waves. further study as98they might impose practical limits on the possi- 7 JANUARY 2000 VOL 287 SCIENCE www.sciencemag.org To determine this speed, we measured the bilities for varying the scattering length. An increase of the dipolar distance after propagation between the notch relaxation rate near Feshbach resonances has been predicted6,7, but and the position of the imprinted phase step for Fig.atoms 3.inExperimentalthe lowest hy (perfineA to E)state andno theoreticalsuch inelastic (F tobinarJ) imagesy of the integrated BEC density for along the direction indicated in Fig. 3H. Be- collisionsvariousare timespossible. afterTher weefore, imprintedthe obser ave phased trap steploss ofis probablyϳ1.5␲ on the top half of the condensate with a due1-to␮thrs pulse.ee-body Thecollisions. measuredIn this numbercase the ofloss atomsrate is incharacterized the condensate was 1.7 (Ϯ0.3) ϫ 106, and this value cause the position of our condensate varied by thewascoefficient used in theK , defined calculations.as N˙ =N A¼ positive￿ K n2 density. So far, disturbancethere is no moved rapidly in the ϩx direction, and 3 3￿ ￿ randomly from one shot to the next (presum- theoretia darkcal work solitonon movedK3 near a oppositelyFeshbach resonanc at significantlye. An analysis lessbased than the speed of sound. Because the imaging FIG. 4. Vertical cross sections of the density profiles at T ￿ Figure 2 Observationablyof the becauseFeshbach r ofesonance stray,at time-varying907 G using time-of- fields),flight on Figpulse. 2 shows (27) isthat destructive,K3 increased eachon imageboth sides showsof the a differentresonance, BEC. The width of each frame is 70 ␮m. FIG. 2(color). Time evolution of the double-condensate sys- 65 ms for different relative numbers of atoms in the two states. absorption imaging. awe, Num couldber of atoms notin alwaysthe condensat applye ver thesus phasemagnetic stepfield. at because the loss rate increased while the density decreased or stayed tem with a relative sag of 0.4 mm (3% of the width of the The combined density distribution (solid line) is shown for Field values above the resonance were reached by quickly crossing the constant. In any case, the fact that we observed Feshbach resonances combined distribution prior to expansion) and a trap frequency comparison to the Thomas-Fermi parabolic fit (dashed line). the center. A marker for the location of the 15 !3 Fig. 4. Calculated density and phase resonance from below and then slowly approaching from above. b, The at high atomic densities (￿10 cm ) strongly enhanced this loss nz ￿ 59 Hz. The trap parameters are the same as those in Fig. 2. S. Inouye etinitial al., phaseNature5 step 151 is the (1998) intersection 392. of the J. Denschlag et al., Science 287along (2000) the x axis (dashed97. line in Fig. 3H) atD. S. Hall et al., Phys. Rev. Lett. 81 (1998) 1539. normalized scattering length a=a˜ ￿ vrms=N calculated from the released energy, process, which can be avoided with a condensate at lower density in together with the predsolitonicted shape with(eq theuation condensate(1), solid line). edge,The values becauseof the at a modified optical trap. Control of the bias field with a precision0 ms (thin lines) and at 5 ms (thick lines) 1541 !4 magnetic field in thethisupper pointscan relative theto solitonthe lower hasone have zeroan velocity.uncertainty of By better than ￿10 will be necessary to achieve negative or extremelyafter applying a phase step imprint of ￿0.5 G. using images taken 5 ms after the imprint, at large values of the scattering length in a stable way. 1.5␲. The soliton located at x ϭϪ8 ␮m which time the soliton had not traveled far has a phase step of 0.58␲ and a speed of Nature © Macmillan Publishers Ltd 1998 NATURE | VOL 392 | 12 MARCH 1998 1531.61 mm/s, which is much less than that from the BEC center, we obtained a mean of sound. soliton speed of 1.8 Ϯ 0.4 mm/s (35). This value is significantly less than the mean

Bogoliubov speed of sound, ␷0 ϭ 2.8 Ϯ 0.1 FIG. 4. Vertical cross sections of the density profiles at T ￿ mm/s. From the propagation of the notch in FIG. 2(color). Time evolution of the double-condensate sys- 65 ms for different relative numbers of atoms in the two states. the numerical simulations (Fig. 3, F to J), we tem with a relative sag of 0.4 mm (3% of the width of the The combined density distribution (solid line) is shown for combined distribution prior to expansion) and a trap frequency comparison to the Thomas-Fermi parabolic fit (dashed line). obtained a mean soliton speed, ␷ ϭ 1.6 s nz ￿ 59 Hz. The trap parameters are the same as those in Fig. 2. mm/s, in agreement with the experimental 1541 www.sciencemag.org SCIENCE VOL 287 7 JANUARY 2000 99 RESEARCHARTICLE probelight was only absorbedby a thin slice We observedthat the fringe periodbe- The resultingtime-of-flight image did not REPORTS of the cloud wherethe atomswere optically came smallerfor largerpowers of the argon exhibit interference,and the profile of a pumped. Because high spatial resolution ion laser-lightsheet (Fig.3A). Largerpower single expanded condensate1, m = matched-1) followed one by two BECs after process with Majoranatrasitions due to the are acceleratedaway from the trap, causing wasrequired from only the fractionof atoms increased the distance between the two side of the profile ofapplication a double condensate of a single Raman pulse. The TOP rotatingmagnetic field zero (13). Atoms them to move further. residingin the slice, a good signal-to-noise condensates(Fig. 1A). Fromphase-contrast (Fig. 4). The profileTOP of a trapsingle confining expanded fields were held on for 7 were firstoutput-coupled to the m = 0 stateand To produce quasi-continuous output cou- ratio requiredcondensates with millions of images,we determinedthe distance d be- condensateshowed somems after coarse application structure, of the Raman pulse be- imaged 10.6 ms later after having crossed the pling, we used multiple Raman pulses. The atoms. tween the density maximaof the two con- whichmost likely resulted forefrom being the switched nonpara- off. The system was then orbit of the zero of the magnetic field (27), laser intensitieswere reducedand the detuning Interferencebetween two Bose conden- densatesversus argon ion laserpower. The bolic shape of the confiningallowedpotential. to evolveWe freely for 1.6 ms before known as the "circleof death,"which orbitsin was again chosen to be 8&2Tr 6.4 MHz to sates. In general,the patternof interference fringeperiod versus maxima separation (Fig. found that the structurebeingbecame imaged. more Note pro- that the position of the the _-i plane (Fig. 2D). For this image the primarilycouple to the m = 0 state. For the fringes differs for continuous and pulsed 3B) is in reasonableagreement with the nounced when the focus of the argon ion output-coupledatoms in Fig. 2B is different rotatingbias field was reducedby a factorweekof 3, endingoptical depth inages of the condensateafter sources. Two point-like monochromatic predictionof Eq. 1, althoughthis equation laser had some weak secondary.intensity PHYSICAL REVIEW LETTERS VOLUME 93, NUMBERfrom that17 in Fig. 2C. In the the atoms which reducedthe distanceto the orbiting22 OCTOBERmag- 2004 and six Ramanpulses (Fig. A to continuous sources would produce curved strictly applies only to two point sources. maxima. In addition, the interferencebe- former, one, three, 4, that have undergonethe Ramantransition are netic field zero to 0.3 mm. As the atomscrossed C), the TOP was held on for a 9-ms window (hyperbolic) interferencefringes. In con- Wallis et al. (26) calculated the interfer- tweenBy two rf-evaporative condensates coolingdisappeared we reachwhen Bose-Einstein con- the Weizmann group [20], this allows a computerized in the state m = 0 and thereforeno longer feel this orbit they lost their quantizationaxis and during which time 6-p.s Raman pulses were trast, two point-like pulsed sources show ence patternfor two extended condensates the argondensationion laser-light with aboutsheet 10 was5 left atoms on for in each cloud and a tomography transformation [12] to reconstruct the radial the trapping whereas in 2C the were repeatedlyprojected to all three fired at a subharmonicof the rotatingbias fre- straightinterference fringes; if d is the sep- in a harmonic potential with a Gaussian condensate fraction of 60%potential,. Fig. density distribution of the halo in cylindricalmagnetic coordinates, aration between two point-like conden- barrier.They concludedthat Eq. 1 remains We then switchatoms off the are TOP still fields trapped. and rampThe positionB back of to the sublevels at the 20-kHz TOP frequency.The quency. The magnetic fields were then extin- 25 = I 0 =- sates, then their relativespeed at any point valid for the central fringesif d is replaced m 0 atoms correspondsto free flight with atoms in m 1 -1 10, and +1 states were, guished and the atoms were imaged 1.6 ms positive. values, thus accelerating~~~A the clouds until they n ;z n~2 x;z J0 x x dx: (1) in space is d/t, where t is the delay between by the geometricmean of the separationof momentum 2hk2 during the entire 8.6 ms, respectively, †ˆretarded, 4 unimpeded, † and † ejectedj j later.The intensityof the laser whose polariza- collide with opposite horizontal momenta at the location Zÿ1 pulsing on the source (switching off the the centers of mass and the distance be- 20 whereas the position of the atoms in Fig. 2C by the trappingpotential, giving rise to three tion was alignedwith x was 300, 150, and 100 of the trap center. The collision energies E 2BB0ÿ Here  x2 y2 1=2 and n~  ;z is the 1D Fourier trap) and observation.The fringeperiod is tween the density maximaof the two con- 2 2 correspondsto their classicalˆ tumingj pointj ˆ in spatiallyseparated stripes of 2 atoms.x At the time mW/cm2, respectively,to couple out different h k =m (with B the Bohr magneton and m the mass of ˆ ‡ † † the de Brogliewavelength X associated with densates.This predictionis also shown in transform along the x direction of the optical= density 87Rb) range fromthe138 trap (7K toms 1.23 is about mK with one-quarter an overallof the of imaging,the atomsthat ended in them -1 fractionsof the condensate(the intensityof the the relativemotion of atoms with massm, Fig. 3B. The agreementis satisfactorygiven 28.6-ms oscillation period along z). withstate respecthave already to z, andbeen Jpulled 0 % isback theto zero-orderthe circle BesselsecondRaman laser was twice thatof the first). uncertainty of 3% (rms). Approximately 0.5 ms before the function. The transformed plots† corresponding to the ht ourexperimental uncertainties in the deter- In the case where the detuningof the lasers of deathby the trap. In Fig. 4D the TOP trapwas held on during mination of the maxima separations(-3 Xcollision 0 we10 switch 20 off the30 trap. A few ms later we images of Figs. 1(a) and 1(d) are shown as Figs. 1(b) and A= d ~~~~(1) Powerfrom the excited state is large comparedwith Although it is not possible to drive the a 7-ms window duringwhich time 140 Raman P,m) and of the center-of-massseparations observe the scattering(mW) halo by absorption imaging. 1(e), respectively. Macroscopicthe excited-state matterhypeine structure splitting, wavesit Am = 2 transitionwith two photons,it is pos- pulseswere firedat the 20-kHz frequencyof the where h is Planck'sconstant. The ampli- (-20%). We conclude that the numerical Figure 1(a) (upper part) displays the s-wave-dominated To obtain the angular scattering distribution W  the is not possibleB to drive Am = 2 transitions sible with four them = -1 -> m = 0 rotatingbias field and the distributionof atoms tude and contrast of the interferencepat- simulationsfor extended interactingcon- scattering,L20 halo (averaged over 20 pictures) of fully en- tomography picturesby using are binned in 40 discrete angular † directlywith two photons because the ground -> m = ?I transition scheme (Fig. 3C). A was imaged 1.6 ms later. The Raman tem depends on the overlap between the densates (26) are consistent with the ob- tangled&L.. pairs (see [18]) obtained for a collision energy of sectors. For gas clouds much smaller than the diameter of pulse E=k10 138 4 ------_stateK. Inlooks Fig. like 1(d) a (upperspin 1/2 part), system taken for which at single 6-p.s Raman pulse with B/2Tr= 6.15 durationand intensitywere reducedto I V.sand two condensates. •servedsinglefringe quantumperiods. wave-function occupiedB ˆ † by many atoms can theeasily halo, W be is directlyimaged proportional to the differential The interferencepattern of two conden- We a of to E=kB 1:23 4 theremK,are theonly halo two is entirely states.Instead, different,we can show- couple MHz and the† same intensitiesused for Fig.2 2B 40 mW/cm2for Fig. 4D to ensurethat the total performed series tests support ˆ † cross section   2 f  f   . Here, the sates after40 ms time-of-flightis shown in our interpretationof matter-waveinterfer- ing a d-wave-dominatedto the m = pattern.+ 1 state The by combining lower halvesthe Raman of Bose-symmetrized was applied.In Fig.†ˆ scattering 3A,j the †‡TOP amplitude was ÿ switched† isj given integrated by a pulse time of 140 p.s was much less Fig. 2. A seriesof measurementswith fringe •ence.the To system demonstrate isthat analogous the fringepattem to manyFigs. 1(a) photons and 1(d) show in the theoreticala laser column cavity densities => atomoff immediately opticsafter the Ramanpulse and the than the spontaneousemission time of about 30 40 50 summation over the even partial waves, f  f  spacings of -15 ,um showed a contrast was caused by two condensates,we com- n2 x; z n x; y; z dy, where n x; y; z is the calculated atoms were imaged 5.6 msil later.The m = + †‡1 500ÿ p.s.The phaseof the output-coupledmatter †ˆ Separation † (jrm) †  2=k l even 2l 1 e Pl cos sinl. Note that varying between 20 and 40%. When the paredit with the pattem froma single con- [19] densityR of theA halo. - †ˆ atoms †receivedˆ 4hki‡ of† momentum † from four=2 wave evolves at about 100 kHz with respectto Fig. 3. (A)Fringe period versus power in the argon unlike in theP total elastic cross section [ 0   imagingsystem was calibratedwith a stan- densate (this is equivalentto performinga As the atoms are scattered by a central field, the scat- photons, and the2 m = 0 atoms received2 ˆ only the† condensateitself because of the kinetic en- dard test we found double-slit and of ion laser-light sheet. (B) Fringe period versus ob- sind 8=k l even 2l 1 sin l], theR interfer- optical pattem, two-40% condensatesexperiment coveringone teringscattering pattern must be partial axially symmetric waves around the 2hk:ˆ fromtwoatom photons;† ˆ laser thus ‡the † m = + I atoms ergy impartedby the two-photonRaman tran- contrast at the same spatial frequency. the slits). One condensatewas illuminated served spacing between the density maximaof ence between the partialP waves is prominent in the dif- the two(horizontal) condensates. scatteringThe solid line axis is the (z axis). As pointed out by moved twice as far as the m = 0 atoms. With sition (28). Because 100 kHz is an integer Hence, the contrastof the atomic interfer- with a focusedbeam of weak resonantlight depen- ferential cross section. Given the small collision energy in interfering dence given by Eq. 1, and the dashed line is the ourthe experiments, TOP switched only off the 4s -ms and afterd-wave the Raman scattering multiple am- of the -20-kHz outputcoupling rep- ence was between 50 and 100%. Because 20 ms beforerelease, causing it to disappear theoretical prediction of (26)B incorporating a con- plitudespulse contribute,and the systemf  imagedf 1.6 ms later2(Fig. =k e i0 sinetition rate, the interferenceof successivepuls- the condensatesare much largerthan the almost completely as a result of optical stant center-of-mass separation of 96 ,um, ne- s = s 0 a) d) 3B), the atoms in the m†‡ + 1 antitrappedÿ †ˆ i2 state †2 es is almostcompletely constructive (29). In the observedfringe spacing,they must have a pumpingto untrappedstates and evapora- glecting the small variation (?10%) with laser and fd  fd   2=k 5=2 e 3cos  1 x †‡ ÿ †ˆ † † ÿtime† between two Raman pulses each output- high degreeof spatialcoherence. tion afterheating by photonrecoit (Fig. 1B). power. sin2. Therefore the differential cross section is given by y z coupledwave packetmoves only 2.9 .m. much  A 8 2 25 2 less thanthe -50-p.m size of the condensate,so mm   sin 0 1 5 cos 0 2 u u ; (2) 1m †ˆ k2  ‡ ÿ † ‡ 4  the output-coupledatoms form a quasi-contin- Fig. 2. Interferencepattem of two 60. expandingcondensates observed 2 uous coherent matter wave. By varying the where u sin2= sin0 3cos  1 . after 40 ms time-of-flight,for two C  † ÿ † delay betweenpulses, the interferencebetween differentpowers of the 0 c Ie > To obtain the phase shifts, we plot the measured angu- argon ion b) e) 2 wave packets can be used to investigatethe sheet lar distribution W  as a function of 3cos  1 as laser-light (raw-dataimages). θ ρ B † ÿ coherence† properties of the condensate. The fringeperiods were 20 and 15 A suggested by Eq. (2). The results for Figs. 1(a) and 1(d) ,um,the powerswere 3 and 5 mW, z +25° are shown as the solid dots in Figs. 1(c) and 1(f), respec-It is apparentthat there is also coupling to the m = +1 state in 4D and the maximum absorptions 0O tively. A parabolic fit to W  directly yields a pair Fig. because some were 90 and 50%, respectively, for 0 200 400 −25° exp exp † output-coupledatoms have moved the distance Positlon (Rm) 0 k ; 2 k of asymptotic phase shifts (defined the leftand rightimages. The fields modulo † ) corresponding †† to the two partial wavesthat in- an atom with momentum4/ik moves in of view are 1.1 mm horizontallyby Fig. 4. Comparison between C c) time-of-flightf)images volved [21]. The absolute value of W  depends on quan-8.6 ms. Such coupling to the m = +I state 0.5 mm vertically.The horizontal for a single and double \r -0ver- condensate, showing) ) . . † occursin this case becausethe spectralwidth of u u tities that are hard to measure accurately (such as the . widths are compressed fourfold, tical profiles. through time-of-flight pictures similar a a ( ( the 1-p.sRaman pulse is sufficientlybroad (30)

atom number), so we leave it out of consideration. We ) whichenhances the effect of fringe to Fig. 2. The) solid line is a profileof two interfering θ θ ( ( a from = 0 to drive transition the state m curvature.For the determinationof condensates, and the dotted line is the profileof a rather emphasize that the measurement of the phase shifts W fringe spacing, the dark central single condensate,W both released from the same is a complete determination of the (complex) s-(momenturn and 2hik) to the state m = + 1 (mo- m=-1 D fringeon the leftwas excluded. double-well potential (argon2 ion laser power, 14 2 In the 3 cos θ -1 (O hk) _ 3 cos- -θ -1 d-wave scattering amplitudes at a given energy. mentum4hki). our experiment trajecto- mW; fringe period, 13 ,um; time of flight, 40 ms). The radial wave functions corresponding to scatteringries of the two output-coupledbeamns (m = 0 The profiles were horizontallyintegrated over 450 FIG. 1. (a) Optical density of the scatteringm=O halo of two 87Rb at different (low) collision energies and different (low)and m = + 1) are spatiallyseparated because ,um. Thecondensates dashed profile for collisionwas multiplied energyby E=k a factor138 4 K, measured of 1.5 to account for fewer atoms in the singleB ˆ(2 hk) † m=+1 angular momenta should all be in phase at small inter-the directionof momentumtransfer is orthogo- 0 0.5 condensate,2.4 ms most after the collision (upper half: measured; lower half: atomic distances [13]. This so-called accumulated phasenal to gravity.These two beamsappear to over- likelythe resultof loss during (4 hk) ~- 1 mm Absorption eliminationcalculatedof the [19]);second (b) half. radial density distribution obtained after common to all low-energy wave functions can belap ex- in Fig. 4D because the cameraviews them M.R. Andrews et al., Science 275 (1997) 637. Ch.tomography Buggle et transformational., Phys. Rev. ofLett. image 93 (a) (2004) (upper 173202. half: mea- E.W. Hagley et al., Science 283 (1999) 1706. Fig. 3. Atoms are coupledto both the m = 0 tractedFig. 4. from (A to the C) One, full datathree, set and sixexp 6-pLsk ;Raman exp k men-from above. Coupling to the m = +I state sured; lower half: calculated [19]); (c) the dots show the f 0 † 2 ††g SCIENCE * VOL.275 * 31 JANUARY1997 and m = +1 and639 magnetic sublevels using a tionedpulses, above. respectively, In practice,were we applied use theto experimental the con- phasecould be suppressedby using a larger bias angular scatteringsingle distribution RamanW pulse. obtained (A) Magnetic after binning trap is densate.exp (D) Firingexp1-p.s Ramanpulses at the † shifts  k and  k as boundary conditions tomagnetic in- field and a largerdetuning A to ex- plot (b); the lineswitched is the bestoff parabolicimmediatelyfit. after (d)–(f) the As Raman in full repetition0 † rate 2 of about† 20 kHzimposed by plots (a)–(c), butpulse. measured (B) Magnetic 0.5 mstrap after is aheld collision on for at4 ms tegratethe frequency inwards—forof the given rotatingE and biasl—the field (140 Schro ¨dingerploit the second-orderZeeman shift and to re- 1230 40 K. Theafterfield the of viewRaman of thepulse. images (C) Transition is 1 1 mmused 2. andequation pulsesh in2d 27 is)r =drproduces2 p2 r a quasi-continuousr 0 and obtain duce the the pulse bandwidh without excessive †   † ‡ † †ˆ laser polarizations. atomic beam. spontaneousemission. To completelysuppress 173202-2 173202-2 1708 12 MARCH 1999 VOL 283 SCIENCE www.sciencemag.org 7 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT

3. Results

3.1. Stationary states The low density region of a vortex ring in a BEC has a toroidal shape, which matches the shape of the high intensity region of a focussed optical vortex laser beam. In the presence of the optical vortex potential it becomes energetically favourable for the BEC ring singularity to

VOLUME 84, NUMBERbe located5 at PHYSICALREVIEWLETTERS the maximum of laser light intensity. 31J ThisANUARY prevents2000 the contraction of the vortex ring, a mechanism similar to the stabilization of line vortices within a harmonically trapped horizontal in our setup), while the transverse oscillation the free fall. The transverse xy and z sizes grow by a frequency is vЌ͑͞BEC2p͒ ෇ cloud219 Hz. For [24 a]. quasipure con- factor of 40 and 1.2, respectively [31]. In addition, the densate with 105 atoms, using the Thomas-Fermi approxi- core size of the vortex should expand at least as fast as mation, we find for the radialExamples and longitudinal of sizes stationary of the the transverse states size resulting of the condensate from [31– the33]. Therefore, imaginary time evolution are shown in condensate DЌ ෇ 2.6 mm and Dz ෇ 49 mm, respectively. a vortex with an initial diameter 2j ෇ 0.4 mm for our When the evaporationfigure radio1. frequency In a harmonicnrf reaches the trapexperimental with ω parametersr 7 is.8 expected2π toHz grow and to a sizeω ofz 0.5 2π Hz the BEC contains ͑min͒ 6 16 mm. = × = × value nrf 1 802kHz,.2 we10 switchatoms. on the stirring The laser subplots are for two different sets of optical vortex parameters and ring beam which propagates along the slow axis of the mag- At the end of the time-of-flight period, we illuminate the × 20 m netic trap. The beamflow waist charges: is ws ෇ 20.0 case͑6 1͒ (i)mm and caseatomic sample(ii), as with given a resonant in probe the laser caption. for s. The Case (ii) represents the more intense the laser power P is 0.4 mW. The recoil heating induced shadow of the atomic cloud in the probe beam is imaged by this far-detunedoptical beam (wavelength vortex.852 Itnm) can is be negli- seenonto in a figure CCD camera1(a) with that an optical the single resolution chargeϳ7 mm. 7 ring singularity that was present gible. Two crossed acousto-optic modulators, combined The probe laser propagates along the z axis so that the with a proper imagingin the system, initial then allow state for an has arbitrary brokenimage up reveals into the column a regular density of stack the cloud of after seven expan- unit charge ring singularities. For translation of thecase laser beam (i) axis they with are respect responsible to the sym- sion for along the the multi-peak stirring axis. The structure analysis of the of images, the Mach number shown in figure 1(c). metry axis of the condensate. which proceeds along the same lines as in [29], gives ac- The motion ofIf the the stirring parameters beam consists of are the super- alteredcess towards to the number those of condensed ofN case0 and uncondensed (ii), theN influence0 of the singularities on the position of a fast and a slow component. The optical atoms and to the temperature T. Actually, for the present spoon’s axis is toggledflow atin a the high tunnel frequency (100is reduced. kHz) data, This the uncondensed corresponds part of the to atomic the cloud transition is nearly from the solid to the dashed lines between two symmetric positions about the trap axis z. undetectable, and we can give only an upper bound for the The intersectionsin of the figures stirring beam1(b) axis and and the (c).z ෇ 0 temperature T , 80 nK. Vortices Figure 1 shows a series of five pictures taken at vari- plane are 6a͑cosu ux 1 Thesinu uy͒ integral, where the distance alonga is the z-axis of the flow velocity is directly connected to the ring vortex 8 mm. The fast toggle frequency is chosen to be much ous rotation frequencies V. They clearly show that for PHYSICAL REVIEW LETTERS week ending larger than the magneticcharge trap frequenciesq. Due so to that the the atoms quantizationfast enough of rotationRotating circulation frequencies condensates: we in can a generate BEC, Circulation one its or velocity is quantized, must obey VOLUMEv dl 90,h NqUMBER, 17 2 MAY 2003 experience an effective two-beam, time averaged potential. several (up to 4) “holes” in the transverse density distri- m condensate must rotate via vortices. C · = The slow componentwhere of the motiondenotes is a uniform any rotation closedbution contour corresponding threading to vortices. We through show for the 0-all and the ring singularities. As the BEC along the original axis of rotation. This technique allows of the angle u ෇ Vt. The valueC of the angular frequency 1-vortex cases a cross section of the column density of the ! V is maintainedhas fixed during a much the evaporation smaller at a cross-sectional value Vorticescloud along in a transverse areastirred available axis. BEC The (q=1) 1-vortex when state exhibits it flowsAbrikosov through the Lattice tunnel than when us to expand the condensate to an adjustable diameter that chosen between 0 and 250 rad s21. REPORTS can exceed 1.4 mm [18]. Since ws DitЌ, returns the dipole potential, on the proportional outer toshell of the cloud, the velocity in the tunnel is much higher. The above ¿ Fig. . Observationof If the removal of central atoms is driven strongly the power of the stirring beam, is well approximated by vortex lattices. The 2 2 equation2 thus approximately becomes examples shown con- mvЌ͑eX X 1eY Y ͒͞2. The X, Y basis is rotated with tain approximately enough, a giant vortex appears. The core is surrounded (A) 16, (B) 32, (C) 80, ' . "., ' respect to the fixed axes (x, y) by the angle u͑t͒, and and (D) 130 vortices. . by a ring-shaped superflow. The circulation of this super- e e The vortices have X ෇ 0.03 and Y ෇ 0.09 for the parameters givenL/2 above "crystallized"in a tri- h angular pattern. The flow is given by summing up all the ‘‘missing’’ phase [30]. The action of this beam is essentially a slight modi- diameterof the cloud fication of the transverse frequencies of the magnetic trap,vz(r 0, z) dz q, in (D) was 1 mm after (12) singularities in the core and can assume very large values. ballistic expansion, while the longitudinal frequency is nearly unchanged.L/2 The = = m which represents a Ell Using the fact that for large amounts of circulation the − magnification of 20. overall stability of the stirring beam on the condensate" ap- Slightasymmetries in the density distributionwere due to absorptionof the optical pumpinglight. pears to be a crucial element for the success of the experi- J.R. Abo-Shaeer et al., Science 292 (2001) 476. rotation of the velocity field can be obtained classically, ment, and we estimate that our stirring beam axis is fixed is constant V X v = 2h. For a superfluid, 1.0- Fig. 2. Density profile through a we determine that in some cases the BEC supports a ring- where L denotes the length of the optical tunnel in the z-direction. vortex lattice. The curve the circulation of the velocity field, v, is co repre- to and stable on the condensate axis to within 2 mm. We c sents a 5-p.m-widecut througha quantizedin units of K = hlM, where M is the two-dimensional similarto shaped superflow with a circulation of up to 60 quanta V,VThe presented stability analysis of stationary states isatomic not mass and completely h is Planck's constant. The general,.6 as we impose image checked that for c the stirring beam does not affect CU those in Fig. 1 and shows the high quantized vortex lines are distributedin the :3 contrast in the observationof the around the core. We can easily control the amount of this the evaporation. with vortex cores. The fluid a uniform area density (18) c peak absorption cylindrical symmetry throughout the imaginary time evolution. An important01) class of ring vortex in this is 90%. For the data presented here, the final frequency of the 0 image circulation by changing the intensity or the duration of n, = 212/KGiant vortices(1) ͑min͒ E evaporation ramp was chosen just above nrf (Dnrf [ In this way the quantum fluid achieves the the laser beam that removes atoms along the axis of excitations, the Kelvin modes (see [25, 26] and referencessame therein),(q>>1)as a couldU3 in principle break ͓3, 6͔ kHz). After the end of the evaporation ramp, we average vorticity rigidly rotating ,\ body, when "coarse-grained"over several rotation. let the system reachthis thermal cylindrical equilibrium in symmetry. this “rotating However, a full 3D stability analysisvortex lines. For a of uniform skyrmions density of vorti- [15] indicates that bucket” for a duration t 500 ms in the presence of an ces, the angular momentum per particle is Position(umn) In our experiment the formation of the giant vortex r ෇ N,h/2, where N, is the numberof vortices in the 2D͑final model͒ should give a very good indication of thethe stability properties: in [15], we have rf shield 30 kHz above nrf . The vortices induced in system. comes about in a sequence of very distinct stages as The numberof observedvortices is plotted 120 - Fig. 3. Averagenumber of vorti- the condensate by the optical spoon are then studied using as a functionof stirringfrequency f2 for two ces as a function of the stirring FIG. 1. Transverse absorption images of a Bose-Einstein con- Qf for two different shown in Fig. 1. For this expansion image sequence a not found an indication of energetically unstable Kelvin modesdifferentstirring times of (Fig. skyrmion’s 3). The peak near ring singularities. frequency a time-of-flight analysis. We ramp down the stirring beam densate stirred with a laser beam (after a 27 ms time of flight). = stirringtimes, (0) 100 ms and 60 Hz correspondsto the frequencyf2/2rr 500 ms. Each For all five images, the condensate number is N0 ෇ ͑1.4 6where the in the - (I) point repre- rapidly rotating BEC is first formed by our evaporative slowly (in 8 ms) to avoid inducing additional excitations vr/V, asymmetry trapping 8 80 sents the averageof three mea- Nonetheless we point out that 3D5 studies of the optical tunnel wouldinduceda besurface anexcita- interesting, numerically 0.5K.W.͒ 10 Madisonand the et temperatureal., Phys. Rev. is Lett. below 84 80 (2000) nK. The806. rotationpotential fre- quadrupolar o surements with the error bars in the condensate, and we then switch off the magnetic tion, with angularmomentum 1 = 2, aboutthe given by the standarddeviation. spin-up technique. Then the atom-removal laser is ap- quency V͑͞2p͒ is, respectively, (c) 145 Hz, (d) 152 Hz, (e) L5o FIG. 1. Different stages of the giant vortex formation process. field and allow thechallenging droplet to fall for t extension෇ 27 ms. Because of our present work. axial direction of the condensateP. Engels (the actualet al., Phys.0 Rev. Lett. 90 (2003) 170405. The solid line indicatesthe equi- 169 Hz, (f) 163 Hz, (g) 168 Hz. In (a) and (b) we plot theexcitation vari- of the surfacemode v = libriumnumber of vortices in a plied with a fixed power of 8 fW for a variable amount of frequency E 40 - (a) Starting point: BEC aftercondensate evaporative spin-up. (b)–(h) of the atomic mean field energy, the initial cigar shape of ation of the optical thickness of the cloud along the horizontalr is two times due to the twofold radially symmetric V2v larger z of radius = 29 ptm, at of the The same Rr rotating time, followed by a 10 ms in-trap evolution time and a the atomic cloud transforms into a pancake shape during transverse axis for the images (c) (0 vortex) and (d) (1 vortex).symmetry quadrupolepattern). Laser shoneU onto BECthe for stirring (b)frequency. 14 s,The arrow (c) 15 s, (d) 20 s, (e) 22 s, resonantenhancement in the vortex io production i 0 o0 indicates the radial trapping was observed for a stiff trap, with vr = 298 Hz 0- (f)a 23 s, (g) 40 s, and (h)frequency. 70 s. Pictures are taken after 5.7-fold 45 ms expansion in the antitrapping configuration de- A 3.2. Experimental creation scheme and vz = 26 Hz (aspect ratio 11.5), and has I I I I t recentlybeen studied in great detail for small 20 expansion40 60 of the80 BEC.100 Laser power is 8 fW. (i) Zoomed-in core scribed above. During the atom removal an rf shield is vortex807 arrays(19). StirringFrequency (Hz) Farfrom the resonance,the numberof vor- region of (f). left on to constantly remove uncondensed atoms that tend tices producedincreased with the stirringtime. The method of stabilizing the ring singularities by pinning,By increasing proposedthe stir time up to in 1 s, thevortices presentquadrupoleresonance paper, for 400 ms and should then they must travel to reach their lattice sites. In to decelerate the condensate rotation. Figure 1(a) shows were observedfor frequenciesas low as 23 Hz probedafter different periods of equilibrationin principle,vortex lattices should have already be experimentally realizable with present technology.The procedure(-0.27Vr). Similarly, in a stiffto trap we form observed thethe magnetic ring trap. A flowblurrystructure consistswas al- formed in the rotating,anisotropic trap. We the result of only the evaporative spin-up. This particular vortices down to 85 Hz (-0.29 Vr). From Eq. 1 ready visible at earlyspontaneously times. Regions of low suspect scatteredthat intensityfluctuations photonsof the stirrer blasts atoms out of the one can estimate the equilibriumnumber of column density are probablyvortex filaments or improperbeam alignmentprevented this. condensate contains 180 vortices and has a Thomas- of initially creating a BEC at rest, already in the presencevortices of at the a given optical rotation frequency vortex. to be that were misaligned Thencondensate.with athe axis ring of rotation flow The vortex lattice had lifetimes of several Nv = 2'rrR2fl/K. The observed number was andshowed no ordering(Fig. 4A). As the dwell seconds (Fig. 4, E to G). The observed sta- Fermi radius of 63:5 m when held in the trap. When always smallerthan this estimate,except near time increased,the filamentsUsingbegan to disentan- this newbility of techniquevortex arrays in such large we conden- are able to substantially structure with high vorticity is seeded by means of phaseresonance. imprinting.Therefore, the condensate Indid not [13gle and] align Ruostekoski with the axis of the trap(Fig. 4, B andsates is surprisingbecause in previous work receive sufficientangular momentum to reach and C), and finally formed a completely or- the lifetime of vortices markedly decreased the atom-removal laser is applied for 14 s as in Fig. 1(b), the groundstate in the rotatingframe. In addi- deredAbrikosov lattice increaseafter 500 ms (Fig. the 4D). vorticity with the number in of condensed the BEC atoms (3). and can now routinely Anglin show in detail how the phase structure of the vortextion, becausethe ring, driveincreased similarthe momentof Lattices towith the fewer vortices onecould given be generated byTheoreticalcalculations predict a lifetime in- the number of vortices is increased to 250 and the inertia of the condensate(by weakening the by rotatingthe condensateobtainoff resonance. condensates In versely proportional containingto the numberof vortices more than 250 vortices trappingpotential), we expect the latticeto ro- these cases, it took longerfor regularlattices to (5). Assuming a temperaturekBT - pL,where Thomas-Fermi radius to 71 m. The rotation rate, deter- tate fasterafter the drive is turnedoff. form.Possible explanations [Fig.for 1(b)]. this observation Since kB is the these Boltzmann vortex constant, the predicted cores are too small to be Lookingat time evolutionof a vortexlattice are the weaker interactionbetween vortices at decay time of - 100 ms is much shorterthan mined from side view aspect-ratio images, has increased New Journal of Physics 9 (2007) 85 (http://www.njp.org/) (Fig. 4), the condensatewas driven near the lower vortex densityimaged and the larger when distance observed. the BECAfter 10 s, the is number heldof vortices in trap, the images in from 0:94! to 0:97! . After atom-removal times of 15 Figs. 1–3 and 6 are taken after having let the BEC expand.   www.sciencemag.org SCIENCE VOL292 20 APRIL2001 477 to 20 s, the vortex lattice becomes disordered [Figs. 1(c) For this, a rapid adiabatic passage technique is employed and 1(d)], and the giant vortex core starts to develop in the to change the hyperfine state of the atoms from the center [Figs. 1(e) and 1(f)]. An enlarged view of the core original trapped F 1;m 1 state to the anti- F region in this early stage of giant vortex core formation is trapped F 2;mj ˆ 1 state.ˆÿ Subsequentlyi the mini- F seen in Fig. 1(i), which nicely demonstrates how the mum ofj theˆ magneticˆÿ trappingi field is rapidly moved individual vortices in the center consolidate. For the below the position of the atoms so that the atoms are supported against gravity while at the same time they are radially expelled [17]. After a chosen expansion time the magnetic fields are switched off and the atoms are imaged

FIG. 2. Lattice reforming after giant vortex formation. FIG. 3. Core developing after a 5 ms short, 2.5 pW laser (a) BEC after evaporative spin-up. (b) Effect of a 60 fW, 2.5 s pulse. In-trap evolution time after the end of the pulse is laser pulse. (c),(d) Same as (b), but additional 10 s (c) and 20 s (a) 0.5 ms, (b) 10.5 ms, (c) 20.5 ms, and (d) 30.5 ms. Images (d) in-trap evolution time after the end of the laser pulse. taken after sixfold expansion of BEC. (e) Zoomed-in core Images taken after a sixfold expansion of the BEC. region of (c). 170405-2 170405-2 VOLUME 81, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 12 OCTOBER 1998

Cold Bosonic Atoms in Optical Lattices

D. Jaksch,1,2 C. Bruder,1,3 J. I. Cirac,1,2 C. W. Gardiner,1,4 and P. Zoller1,2 1Institute for Theoretical Physics, University of Santa Barbara, Santa Barbara, California 93106-4030 2Institut für Theoretische Physik, Universität Innsbruck, A-6020 Innsbruck, Austria 3Institut für Theoretische Festkörperphysik, Universität Karlsruhe, D-76128 Karlsruhe, Germany 4School of Chemical and Physical Sciences, Victoria University, Wellington, New Zealand (Received 26 May 1998) The dynamics of an ultracold dilute gas of bosonic atoms in an optical lattice can be described by a Bose-Hubbard model where the system parameters are controlled by laser light. We study the continuous (zero temperature) quantum phase transition from the superfluid to the Mott insulator phase induced by varying the depth of the optical potential, where the Mott insulator phase corresponds to a commensurate filling of the lattice (“optical crystal”). Examples for formation of Mott structures in optical lattices with a superimposed harmonic trap and in optical superlattices are presented. [S0031-9007(98)07267-6]

PACS numbers: 32.80.Pj, 03.75.Fi, 71.35.Lk

Optical lattices—arrays of microscopic potentials in- the lattice, and thus represents an optical crystal with duced by the ac Stark effect of interfering laser beams— diagonal long range order with the periodQuantum imposed by the simulation can be used to confine cold atoms [1–7]. The quantized laser light. The nature of the MI phase is reflected in the motion of such atoms is described by the vibrational mo- existence of a finite gap U in the excitation spectrum. tion within an individual well and the tunneling between Our starting point is the Hamilton operator for bosonic neighboring wells, leading to a spectrum describable as a atoms in an external trapping potential band structure [3]. Near-resonant optical lattices, where quantum2 simulation: new ways to understand complicated (many-body) 3 h¯ 2 dissipation associated with optical pumping produces H ￿ d x cquantumy￿x￿ 2 systems= 1 V0￿x ￿that1 V Tcannot￿x￿ c￿x￿ be fully simulated on a classical computer. cooling, have given filling factors of about one atom per √ 2m ! Z 2 ten lattice sites [1,6]. Higher filling factors will require 1 4pash¯ 3 R. Feynman, Int. J. Theor. Phys. 21 (1982) 467. 1 d x cy￿x￿cy￿x￿c￿x￿c￿x￿ , (1) lower temperatures, and hence will also require mini- 2 m Z mization of the optical dissipation. This can be achieved with c￿x￿ a boson field operator for atoms in a given N in a far-detuned optical lattice (especially with blue detun- Size of Hilbert space of N particle, M mode system: M internal atomic state, V0￿x￿ is the optical lattice poten- ing), where photon scattering times of many minutes have NATURE PHYSICS DOI: 10.1038/NPHYS1614 ARTICLES tial, and VT ￿x￿ describes an additional (slowly varying) been demonstrated [2]. Thus the lattice then behaves as a external trapping potential, e.g., a magnetic trap (see a c conservative potential, which could be loaded with a Bose |R Fig. 1a). In the simplest case, the optical lattice poten- digital quantum simulator: 〉i |r analog quantum simulator: find 〉c condensed atomic vapor [8,9], for which present densities 3 2 tial has the form V0￿x￿ ￿ j￿1 Vj0 sin ￿kxj￿ with wave would correspond to tens of atoms per lattice site. find a system that flexibly Ωc Control atom vectorsa quantumk ￿ 2p￿l systemand l the wavelengthwith the of the laser Spin atom In this Letter we will study the dynamics of ultracold P light, corresponding to a lattice period a ￿ l￿2. V is can evolve according to a Ωr bosonic atoms loaded in an optical lattice. We will show same Hamiltonian but easier 0 proportional to the dynamic atomic polarizability times ∆ that the dynamics of the bosonic atoms on the optical Trotter decomposition theaccess laser intensity. to Theparameters interaction potential and between the Ω Ω lattices realizes a Bose-Hubbard model (BHM) [10–16], p p of the time-evolution |0 describing the hopping of bosonic atoms between the measurements. 〉c |A〉 |B〉 |1〉 lowest vibrational states of the optical lattice sites, the operator. i i c unique feature being the full control of the system’s b d parameters by the laser parameters and configurations. The BHM predicts phase transition from a superfluid (SF) phase to a Mott insulator (MI) at low temperatures and with increasing ratio of the on site interaction U (due to repulsion of atoms) to the tunneling matrix element J [10]. In the case of optical lattices this ratio can be varied by changing the laser intensity: with increasing depth of the optical potential the atomic wave function becomes more and more localized and the on FIG. 1. (a) Realization of the BHM in an optical lattice (see site interaction increases, while at the same time the text). The offset of the bottoms of the wells indicates a trapping potential VT . (b) Plot of the scaled on site interaction U￿ER Figure 1 Set-upH. Weimer of the system. et al., a, TwoNature internal Phys. states A6i and(2010)B i give 382. rise to an effective spin degree of freedom. These states are coupled to a Rydberg | | ￿ | ￿ tunneling matrix element is reduced. In the MI phase the D. Jaksch et al., Phys. Rev. Lett. 81 (1998) 3108. state R i in two-photon resonance, establishing an electromagnetically induced transparency condition. On the other hand, the control atom has two multiplied by a￿as ￿ 1￿ (solid line; axis on left-hand side of | ￿ ¿ internal states 0 c and 1 c. The state 1 c can be coherently excited to a Rydberg state r c with Rabi frequency ￿r, and can be optically pumped into the density (occupation number per site) is pinned at integer graph) and J￿ER (dashed line; axis on right-hand side of graph) | ￿ | ￿ | ￿ | ￿ state 0 c for initializing the control qubit. b, For the toric code, the system atoms are located on the links of a two-dimensional square lattice, with the n 1, 2, . . . , as a function of V E V E (3D lattice). | ￿ ￿ corresponding to a commensurate filling of 0￿ R ￿ x,y,z0￿ R control qubits in the centre of each plaquette for the interaction Ap and on the sites of the lattice for the interaction Bs. c,d, Set-up required for the implementation of the colour code (c) and the U(1) lattice gauge theory (d). 3108 0031-9007￿98￿81(15)￿3108(4)$15.00 © 1998 The American Physical Society computation. It considers a two-dimensional set-up, where the Our choice of the jump operator follows the idea of reservoir spins are located on the edges of a square lattice17. The Hamiltonian engineering of interacting many-body systems, as discussed in

H E0( p Ap s Bs) is a sum of mutually commuting stabilizer refs 18,19. In contrast to alternative schemes for measurement- =− + x z 20 operators Ap i p σi and Bs i s σi , which describe four-body based state preparation , here, the cooling is part of the time ￿ = ∈￿ = ∈ interactions between spins located around plaquettes (Ap) and evolution of the system. These ideas can be readily generalized to ￿ ￿ vertices (Bs) of the square lattice (see Fig. 1b). The ground state more complex stabilizer states and to set-ups in higher dimensions, of the Hamiltonian is the simultaneous eigenstate of all stabilizer as in, for example, the colour code21 (see Fig. 1c). As a final operators Ap and Bs with eigenvalues 1, and gives rise to a remark, we would like to mention that the toric code can also be topological phase: the ground-state degeneracy+ depends on the derived as a perturbative limit of a Hamiltonian with two-body boundary conditions and topology of the set-up, and the elementary interactions on a honeycomb lattice22, of which implementations excitations obey anyonic statistics under braiding. The toric code have been suggested both for cold atoms23 and condensed- shows two types of excitation corresponding to 1 eigenstates of matter systems24. In our approach, the higher-order interactions − each stabilizer Ap (‘magnetic charge’) and Bp (‘electric charge’). arise in a non-perturbative way and the scheme also allows for A dissipative dynamics that ‘cools’ into the ground-state mani- dissipative state preparation. fold is provided by a Liouvillian with quantum jump operators, Implementation of a single time step 1 z 1 x cp σi 1 Ap , cs σj [1 Bs] (1) We now turn to the physical implementation of the digital quantum = 2 − = 2 − simulation. The system and auxiliary atoms are stored in a deep with i p and j s, which￿ act on￿ four spins located around plaquettes optical lattice or magnetic trap arrays with one atom per lattice site, p and∈ vertices∈ s, respectively. The jump operators are readily where the motion of the atoms is frozen and the remaining degree understood as operators that ‘pump’ from 1 into 1 eigenstates of freedom of the system and auxiliary atoms are effective spin-1/2 of the stabilizer operators: the part (1 A −)/2 is a projector+ onto systems described by the two long-lived ground states A and B − p | ￿i | ￿i the eigenspace of Ap with 1 eigenvalue, whereas all states in the and 0 c and 1 c, respectively (see Fig. 1a). We first discuss the − x | ￿ | ￿ 1 eigenspace are dark states. The subsequent spin flip σj transfers elements of the digital quantum simulator for a small local set-up, the+ excitation to the neighbouring plaquette. The jump operators and present the extension to the macroscopic lattice system below. then give rise to a random walk of anyonic excitations on the To be specific, we will focus on a single plaquette in the example of lattice, and whenever two excitations of the same type meet they Kitaev’s toric code outlined above. are annihilated, resulting in a cooling process, see Fig. 2. Similar The implementation of the four-body spin interaction x arguments apply to cs. Efficient cooling is achieved by alternating Ap i σi and the jump operator cp uses an auxiliary qubit the index i of the spin that is flipped. located= at the centre of the plaquette (see Fig. 1b). The general ￿

NATURE PHYSICS VOL 6 MAY 2010 www.nature.com/naturephysics 383 | | | articles

depth of around 13 Er the interference maxima no longer increase in which is of the order of 2 ms for a lattice with a potential depth of 9 strength (see Fig. 2e): instead, an incoherent background of atoms Er. A signi®cant degree of phase coherence is thus already restored gains more and more strength until at a potential depth of 22 Er no on the timescale of a tunnelling time. interference pattern is visible at all. Phase coherence has obviously It is interesting to compare the rapid restoration of coherence been completely lost at this lattice potential depth. A remarkable coming from a Mott insulator state to that of a phase incoherent feature during the evolution from the coherent to the incoherent state, where random phases are present between neighbouring state is that when the interference pattern is still visible no broad- lattice sites and for which the interference pattern also vanishes. ening of the interference peaks can be detected until they completely This is shown in Fig. 3b, where such a phase incoherent state is vanish in the incoherent background. This behaviour can be created during the ramp-up time of the lattice potential (see Fig. 3 explained on the basis of the super¯uid±Mott insulator phase legend) and where an otherwise identicalarticles experimental sequence is diagram. After the system has crossed the quantum critical point used. Such phase incoherent states can be clearly identi®ed by = : U J ˆ z 3 5 8, it will evolveVOLUME in the81, inhomogeneous NUMBER 15 case into P H Y Sadiabatically I C A L R E V mappingI E W L E theT T E population R S of the energy 12 OCTOBER bands1998 onto alternating regions of incoherent Mott insulator phases and coher- the Brillouin zones19,21. When we turn off the lattice potential Quantument super¯uid phases2, phase where the super¯uid fractiontransition continuously adiabatically, from we ®nd that a a statistical mixture of states has been decreases for increasing ratios U/J. Cold Bosoniccreated, Atoms which in homogeneously Optical Lattices populates the ®rst Brillouin zone of

Restoring coherence D. Jaksch,1,2 C. Bruder,1,3 J. I. Cirac,1,2 C. W. Gardiner,1,4 and P. Zoller1,2 super¯uid to a1 Mott insulator in A notable property of the Mott insulatorInstitute state for Theoretical is that phase Physics, University of Santa Barbara, Santa Barbara, California 93106-4030 2Institut für Theoretische Physik, Universität Innsbruck, A-6020 Innsbruck, Austria coherence can be restored very rapidly when3Institut the optical für Theoretische potential Festkörperphysik, Universität Karlsruhe, D-76128 Karlsruhe, Germany is lowered again to a value where the ground state4 of the many-body agasofultracoldatomsSchool of Chemical and Physicala Sciences, Victoria University, Wellington, New Zealand system is completely super¯uid. This is shown in Fig. 3. After only (Received22 26 May 1998) )

4 ms of ramp-down time, the interferenceThe pattern dynamics is fully of an visible ultracold dilute gasr of bosonic atoms in an optical lattice can be described E

Markus Greineragain,*, and Olaf after Mandel 14 ms*, Tilman of ramp-down Esslinger time²,by Theodor a the Bose-Hubbard interference W. HaÈnsch model peaks* & where Immanuel the system ( Bloch parameters* are controlled by laser light. We study the 0 9

continuous (zero temperature) quantum phaseV transition from the superfluid to the Mott insulator phase have narrowed to their steady-state value, proving that phase = 20 ms * Sektion Physik, Ludwig-Maximilians-UniversitaÈt, Schellingstrasseinduced 4/III, by D-80799 varying the Munich, depth Germany, of the optical and Max-Planck-Institut potential, whereτ the MottfuÈr Quantenoptik, insulator phase D-85748 corresponds Garching, to coherence has been restored over the entirea commensurate lattice. The filling timescale of the lattice (“optical crystal”). Examples for formation of Mott structures Germany 0 for the restoration of coherence is comparablein optical to the lattices tunnelling with a time superimposed harmonic trap and in optical superlattices are presented. ² Quantenelektronik, ETH ZuÈrich, 8093 Zurich, Switzerland 80 ms 20 ms t ttunnel ˆ ~=J between two neighbouring lattice[S0031-9007(98)07267-6] sites in the system, Time ...... b ...... PACS numbers: 32.80.Pj, 03.75.Fi, 71.35.Lk 125 For a system at a temperature of absolute zero, all thermal ¯uctuations are frozen out, while quantum ¯uctuations prevail. These Optical lattices—arrays of microscopic potentialsm) in- the lattice, and thus represents an optical crystal with microscopic quantum ¯uctuations canduced induce by a the macroscopic ac Stark effect phase of interfering transition laser in beams— theµ 100 grounddiagonal state of long a many-body range order with system the period when imposed the by the can be used to confine cold atoms [1–7]. The quantized laser light. The nature of the MI phase is reflected in the relative strength of two competing energyz terms is varied across a critical value. Here we observe such a quantum phase transition in a Bose±Einstein condensate with repulsivemotion of suchinteractions, atoms is described held in by a three-dimensional the vibrational mo- opticalexistence lattice of a finite potential. gap U in As the the excitation potential spectrum. depth of the lattice is increased, a transitiontion within is observed an individual from well a super¯uid and the tunneling to a Mott between insulator75 Our phase. starting In pointthe super¯uid is the Hamilton phase, operator each for bosonic neighboring wells, leading to a spectrum describable as a atoms in an external trapping potential atom is spread out over the entire lattice,x withy long-range phase coherence. But in the insulating phase, exact numbers of atoms band structure [3]. Near-resonant optical lattices, where 2 are localized at individual lattice sites, with no phase coherence across the lattice;50 this phase is characterized3 h¯ by2 a gap in the dissipation associated2 ᐜk with optical pumping produces H ￿ d x cy￿x￿ 2 = 1 V0￿x￿ 1 VT ￿x￿ c￿x￿ excitation spectrum. We can induce reversiblecooling, have changes given filling between factors the of abouttwo ground one atom states per of the system. √ 2m ! Z 2 ten lattice sites [1,6]. Higher fillingletters factors to will nature require 1 4pash¯ 3 Width of central peak ( 1 d x cy￿x￿cy￿x￿c￿x￿c￿x￿ , (1) lower temperatures, and hence will also require mini-25 2 m trev=jZn (see refs 1–5). A more detailed picture of the dynamical A physical system that crosses the boundarymization between of the optical two phases dissipation.(here This between can be achieved kinetic andwith interactionc￿x￿ a boson energy)evolution field of isoperatorw can fundamental be seen in for Fig. atoms1, where to the in overlap a given of an arbitrary in a far-detuned optical lattice (especially with blue detun- 4 coherent state bl with the state al (t) is shown for different changes its properties in a fundamental way. It may, for example, quantum phase transitionsinternaland atomic inherently state, differentV0￿x￿ jis fromthe optical normalj lattice poten- 8,9 ing), where photon scattering times of many minutes have0 evolution times up to the first revival time of the many-body state . melt or freeze. This macroscopic change is driven by microscopic phase transitions, which0tial, are and 2usuallyVT ￿x 4￿ drivendescribesIn 6 our by experiment, an 8 the additional competitionwe 10 create coherent (slowly 12 states 14 varying) of the matter wave Condensedbeen demonstrated [2]. Thus matter the lattice then behaves analogues as a field in a potential well, by loading a magnetically trapped Bose– ¯uctuations. When the temperature of the system approaches zero, between inner energy andexternal entropy. trappingEinstein potential, condensatet e.g.,(ms) into a a three-dimensional magnetic trap optical lattice (see poten- conservative potential, which could be loaded with a Bose Fig. 1a). In the simplesttial. For low potentialcase, the depths, optical where thelattice tunnelling poten- energy J is much all thermal ¯uctuations die out. This prohibits phase transitions in The physics of the above-described systemlarger than is the on-site captured repulsive interaction by the energy U in a single well, condensed atomic vapor [8,9], for which present densitiescd1 3 e2 classical systems at zero temperature, as their opportunity(more to changeon quantumBose±Hubbard simulation) model , whichtial has describes the form anVeach0 interacting￿x atom￿ ￿ is spreadj￿1 outbosonVj0 oversin all gas￿ latticekxj in￿ sites.with For wave the case of a would correspond to tens of atoms per lattice site. p lhomogeneousl system with N atoms and M lattice sites, the many- vectors k ￿ 2 ￿ bodyand state can thenthe be wavelength written in second ofquantization the laser as a product has vanished.Figure However, 1 Schematic their three-dimensional quantum mechanical interferenceIn this Letter counterparts pattern we with will measured study can theabsorptionaFigurelattice dynamics 1 Quantum potential. dynamics of ultracold of a coherent The state owing hamiltonian to cold collisions. The images in seconda–g quantizedP form reads: 2 light, correspondingof identical to a single-particle lattice period Bloch wavesa ￿ withl zero￿2. quasi-momentumV0 is show fundamentally different behaviour. In a quantum system, show the overlap kb a t l of an arbitrary coherent state bl with complex amplitude b M † N images taken along two orthogonal directions.bosonic The atoms absorption loaded images in an were optical obtained lattice. after j Wej ð Þ willj show j WlU=J 0 i 1 a^i 0l. It can be approximated by a product with the dynamically evolved quantum state al(t) (seeproportional equation (2)) for an average to thej dynamic¼ / ¼ atomicj polarizability times 2 j 1 over single-site many-body states fil,suchthat WlU=J 0 < ¯uctuations are present even at zero temperature,that the dynamics due to of Heisen- the bosonicnumber atoms of a on3 atoms the at different optical times t. a, t ² 0h=U; b, 0.1 h/U; c, 0.4 h/U; M ÀÁP j j ¼ ballistic expansion from a lattice with a potential depth of V 0 ˆ 10E r and a time of ¯ightj ofj ¼ H ˆ 2 J aütheaà ‡ laser intensity.e nà ‡ Ui The1 filn:à interactionIn nà the2 limit1† of potential large N and1 between†M,theatomnumber the Mott-Insulator / Superfluid quantum phase transitiond, 0.5 h/U; e, 0.6 h/U; f, 0.9 h/U; and g, h/U. Initially, thei phasej of the macroscopici i matter ¼ j i i berg's uncertainty relation. These quantumlattices ¯uctuations realizes a Bose-Hubbard may be model (BHM) [10–16],^ ^ 2 distribution^ of fil in each potential well is poissonian and almost 15 ms. wave field becomes more and more uncertain asi;j time evolves (b), but remarkablyi at trev/2 Q i j h i identical to that of a coherent state. Furthermore, all the matter strong enough to drive a transition fromdescribing one phase the hopping to another, of bosonic(d), when atoms the macroscopic between field has collapsed the such that w < 0, the system has evolved waves in different potential wells are phase coherent, with constant into an exact ‘Schro¨dinger cat’ state of two coherent states. These two states are 1808 out lowest vibrational states of the optical lattice sites,² the relative phases between lattice sites. As the lattice potential depth VA Lattice strength decides type of ground-state of phase, and therefore lead to a vanishing macroscopic field w at these times. More bringing about a macroscopic change. Here aÃi andFigureaÃi 3 Restoringcorrespond coherence. toa the, Experimental bosonicis increased sequence and annihilationJ decreases, usedthe to measure atom and number the distribution restoration in each unique feature being the full controlgenerally, we of can show the that at system’s certain rational fractions of the revival time trev, the system ² 10 A prominent example of such a quantum phase transition is the creation operators of atoms on the ithpotential lattice well site, becomesnà markedlyˆ aà a subpoissonianà is owing to the evolves into other exactof superpositions coherence of coherent after states bringing—for example, the at trev system/4, four into the Mott insulatori phasei i at V ˆ 22E and parameters by the laser parameters and configurations. 2,4 repulsive interactions between the atoms, even0 before enteringr the coherent states, or at trev/3, three coherent states . A full revival of the initial coherent 11–13 change fromab the super¯uid phase to the Mottcd insulator phase in a the atomic number operator counting theMott insulatingnumber state of. After atoms preparing on superposition states fil state is then reached at t h/U. In the graph, red denotes maximum overlap and blue ; j The BHM predicts phase transition fromlowering a¼ superfluid the potential afterwards to V 0 ˆin9 eachE r potentialwhere well, the we system increase is the super¯uid lattice potential again. depth The rapidly vanishing overlap with 10 contour lines in between. system consisting of bosonic particles with(SF) phaserepulsive to a interactionsMott insulator (MI)the atith low lattice temperaturesatoms site, are ®rst and helde ati thedenotes maximum the potentialin energy order depth to create offsetV isolatedfor of20 potential ms, the andi wells.th then The the hamiltonian lattice of equation (1) then0 determines the dynamical evolution of each of hopping through a lattice potential. Thisand system with increasing was ®rst studied ratio of thelattice on site site interaction due toU an external harmonicthese potential con®nement wells. of the 12 potential is decreased to a potential depth of 9 Er in a time t after which the interference macroscopic matter wave field w ka t a^ a t l; which has an The experimental set-up used here to create Bose–Einstein theoretically in the context of super¯uid-to-insulator(due to repulsion transitions of atoms) in toatoms the tunneling. The strength matrix ¼ ofð Þ thej j ð Þ tunnelling term in the hamiltonian 1 2 intriguing dynamicalpattern evolution. of the At atoms first, the is different measured phase by evol- suddenlycondensates releasing in the them three-dimensional from the trapping lattice potential. potential (see liquid helium . Recently, Jaksch et al. haveelement proposedJ [10]. that In such the case a is ofutions characterized optical of the atom lattices number states by this lead the to a collapse hopping of w. However, matrix at Methods) element is similar between to that used in adja-our previous work11,14,15. Briefly, , Width of the central interference peak for different ramp-down times t, based on a integer multiplesb of the revival time3trev h=U all phase factors in2 2we start with a quasi-pure Bose–Einstein condensate of up to ratio can be varied by changing the laser intensity: with ¼ = 5 87 transition might be observable when an ultracold gas of atoms with centthe sum sites of equationi,j J (2)ˆ re-phase2ed moduloxw 2xp, leading2 xi to† a2 perfect~ = 2 210m ‡RbV atomslat x in†† thew xF 22;mxj†,2l state in a harmonic efgh lorentzian ®t. In case of a Mott insulator state£ (®lled circles) coherencej ¼ F ¼ is rapidly increasing depth of the optical potentialrevival of the initial atomic coherent wave state. The collapse time t c depends on magnetic trapping potential with isotropic trapping frequencies of repulsive interactions is trapped in a periodic potential. To illustrate where w x2 2 xi† is a single particle Wannier function localized to the variance j of the atom number distribution, such that tc < q 2p 24Hz: Here F and m denote the total angular momentum function becomes more and more localizedn restored and the already on afterFIG. 4 ms. 1. The (a) Realization solid line¼ is a of£ ®t the using BHM a double inF an exponential optical lattice decay (see this idea, we consider an atomic gas of bosons at low enough the ith0 lattice site (as longtext). as ni The< O offset 1†), ofV thelat( bottomsx) indicates of the wells the optical indicates a trapping site interaction increases, while at the same(t1collapseˆ time0:94 the 7† ms, andt2 ˆ revival10 5† ms). For a phase incoherent state (open circles) using the temperatures that a Bose±Einstein condensate is formed. The lattice potential and m ispotential the massVT . of (b) a single Plot of atom. the scaled The on repulsion site interaction U￿ER tunneling matrix element is reduced. In the MIsame phase experimental the sequence, no interference pattern reappears again, even for ramp- condensate is a super¯uid, and is described by a wavefunction between two atoms on a singlemultiplied lattice by a site￿as is￿ quanti®ed1￿ (solid line; by axis theon-site on left-hand side of density (occupation number per site) is pinned at integer graph) and J E (dashed¿ line; axis on right-hand side of graph) 3 interactiondown matrix times t of element up to 400 ms.U ˆ We￿ ®nd4pR that2a= phasem† j incoherentw x†j4d3 statesx, with are formeda by applying that exhibits long-range phase coherencen .￿ An1, 2, intriguing . . . , corresponding situation to a commensurate filling of as a function of V~0￿ER ￿eVx,y,z0￿ER (3D lattice). appears when the condensate is subjected to a lattice potential in being the scatteringa magnetic ®eld length gradient of an over atom. a time of In 10 our ms duringcase the the ramp-up interaction period, when the which theFigure bosons 2 Absorption can move images from of one multiple lattice matter site wave to the interference next only patterns. by Theseenergy were is verysystem well is still described super¯uid. by This the leads single to a dephasing parameter of theU condensate, due to wavefunction the due to 3108 0031-9007￿98￿81(15)￿3108(4)$15.00 © 1998 The American Physical Society tunnel coupling.obtained after If the suddenly lattice releasing potential the atoms is turned from an on optical smoothly, lattice potential the withshort different rangethe of nonlinear the interactions, interactions in the which system. isc±e much, Absorption smaller images than of the interference potential depths V after a time of ¯ight of 15 ms. Values of V were: a,0E ; b,3E ; c,7E ; patterns coming from a Mott insulator phase after ramp-down times t of 0.1 ms ( ), 4 ms system remains in the super¯uid0 phase as long as the atom±atom0 r latticer r spacing. c , 10 E ; , 13 E ; , 14 E ; , 16 E ; and , 20 E . ( ), and 14 ms ( ). interactionsd arer smalle r comparedf r g tor the tunnelh r coupling. In this In the limitd where thee tunnelling term dominates the hamilto- regime a delocalizedM. Greiner et wavefunctional. Nature 415 (2002) minimizes 39. the dominant kinetic nian, the ground-state energy is minimized if the single-particle energy, andNATURE therefore| VOL 415also| 3JANUARY2002 minimizes the| www.nature.com total energy of the many-© 2002wavefunctions Macmillan Magazines of N atoms Ltd are spread out over the entire lattice with 41 body system. In the opposite limit, when the repulsive atom±atom M lattice sites. The many-body ground state for a homogeneous ˆ : interactions are large compared to the tunnel coupling, the total systemFigure 2 Dynamical (ei evolutionconst of the multiple) is matter then wave interference given pattern by: observed wave field has revived. The atom number statistics in each well, however, remains

after jumping from a potential depth VA 8 Er to a potential depth VB 22 Er and a constant throughout the dynamical evolution time. This is fundamentally different from the energy is minimized when each lattice site is ®lled with the same M. Greiner¼ et al. Nature¼ 419 (2002)N 51. subsequent variable hold time t. After this hold time, all trapping potentials wereM shut off vanishing of the interference pattern with no further dynamical evolution, which is number of atoms. The reduction of ¯uctuations in the atom and absorption images were taken after a time-of-flight period of 16 ms. The hold times²t observed in the quantum phase transition to a Mott insulator, where Fock states are were a,0ms; b, 100 ms; c, 150 ms; d, 250 ms;jeª, 350SFms;Uf,ˆ 4000 m~s; and g, 550 ms.aà Ati formedj0 in each potential well. From the above images 2 the† number of coherent atoms Ncoh number on each site leads to increased ¯uctuations in the phase. first, a distinct interference pattern is visible, showing thati initially the system^ can be is determinedi by first fitting a broad two-dimensional gaussian function to the incoherent iˆ1 ! Thus in the state with a ®xed atom number per site phase coherence described by a macroscopic matter wave with phase coherence between individual background of atoms. The fitting region for the incoherent atoms excludes potential wells. Then after a time of ,250 ms the interference pattern is completely lost. 130 mm 130 mm squares around the interference peaks. Then the number of atoms in £ is lost. In addition, a gap in the excitation spectrum appears. The HereThe vanishing all of the atoms interference pattern occupy is caused by a the collapse identicalof the macroscopic extendedthese squares is counted Bloch by a pixel-sum, state. from which An the number of atoms in the matter wave field in each lattice potential well. But after a total hold time of 550 ms(g) the incoherent gaussian background in these fields is subtracted to yield Ncoh. a.u., arbitrary competition between two terms in the underlying hamiltonian importantinterference pattern is almostfeature perfectly restored, of this showing state that the macroscopic is that matter theunits. probability distribution

52 © 2002 Nature Publishing Group NATURE|VOL 419|5 SEPTEMBER 2002|www.nature.com/nature NATURE | VOL 415 | 3 JANUARY 2002 | www.nature.com © 2002 Macmillan Magazines Ltd 39 week ending PRL 105, 240401 (2010) PHYSICAL REVIEW LETTERS 10 DECEMBER 2010

Realization of a Sonic Black Hole Analog in a Bose-Einstein Condensate week ending PRL 105, 240401 (2010) PHYSICAL REVIEW LETTERS 10 DECEMBER 2010 Oren Lahav, Amir Itah, Alex Blumkin, Carmit Gordon, Shahar Rinott, Alona Zayats, and Jeff Steinhauer Technion—Israel Institute of Technology, Haifa, Israel blocked, so that the boundary between the dark and light shown in Fig. 2(e). The observed radius and length of this (Received 21 June 2009; revised manuscript received 21 October 2010; published 7 December 2010) regions forms the potential step of height V =h 2:0 kHz. condensate, combined with the known trap frequencies, 0 ¼ We have created an analog of a black hole in a Bose-Einstein condensate. In this sonic black hole, sound Initially, the condensate is located to the right of the step, as give the total number of atoms NC in the condensate [26]. shown in Fig. 1. Starting at t 0, the harmonic potential is N , divided by the sum of the pixels of Fig. 2(e), gives the waves, rather than light waves, cannot escape the event horizon. A steplike potential accelerates the flow ¼ C of the condensate to velocities which cross and exceed the speed of sound by an order of magnitude. The accelerated until it reaches the constant velocity of roughly sensitivity of the imaging system. 1 Landau critical velocity is therefore surpassed. The point where the flow velocity equals the speed of 0:3 mm sÀ . The condensate then passes over the potential By comparing the two curves of Fig. 2(f), it is seen that sound is the sonic event horizon. The effective gravity is determined from the profiles of the velocity and step, as shown in Figs. 2(a) and 2(b). We observe no the density in the left region of the figure increases with speed of sound. A simulation finds negative energy excitations, by means of Bragg spectroscopy. increase in the thermal fraction in this process. time, while the density in the right region decreases. We Furthermore, it is seen that the density of the condensate can use this redistribution of density to compute the flow DOI: 10.1103/PhysRevLett.105.240401 PACS numbers: 03.75.Kk, 67.85.De is much smaller to the left of the potential step, whose velocity via the continuity equation, given by ~ nv~ location approximately coincides with the dashed lines in @n=@t [26]. In Figs. 2(a) and 2(b), the flow appearsr Áð toÞ¼ be Figs. 2(a) and 2(b). By conservation of mass, the decrease À The event horizon is a boundaryGeneral around the black hole, condensate relativity [24], and repulsive potential maximaanalogues [5,25]. largely in the x direction. This unidirectional flow is further in density corresponds to an increase in flow velocity. enclosing the region from which even light cannot escape. We achieve the black hole horizon by a steplike potential verified by Figs. 2(c) and 2(d), which show no dynamics in The images of Fig. 2(a) and 2(b) can be converted into It has been suggested that an analog of a black hole could combined with a harmonic potential, as shown in Fig. 1. the y-z plane. The shape in this plane remains similar to the density profiles, averaged over the cross section of the be created in a variety of quantum mechanical [1–6] or (moreWe translate on the quantum harmonic potential simulation) to the left as indicated shape of the condensate in the harmonic trap, shown in classical [7–9] systems. In the case of a quantum fluid such by the horizontal arrow, moving the condensate towardscondensate, as shown in Fig. 2(f). This average density is Fig. 2(e). By assuming that the flow is in the x direction found by the relation n N=xA, where N is the number only, the continuity equation gives as the Bose-Einstein condensate studied here [3], it is the stationary step. While crossing the step, the condensate ¼ sound waves, rather than light waves, which cannot escape. accelerates to supersonic speeds. Thus, the region to theof atoms in a column of pixels of Figs. 2(a) and 2(b), x Analog Gravity: excitations (sound-waves) in a Bose- ¼ x This sonic black hole contains regions of subsonic flow, as left of the step is supersonic, and the region to the right is0:46 "m is the length of a pixel, and A is the cross- 1 @n v dx0: (1) well as regions of supersonic flow. Since a phonon cannot subsonic. There is therefore a black hole horizon at thesectional area of the condensate in the y-z plane, seen in ¼Àn 0 @t Einstein condensate behave in some regime like particles Z propagate against the supersonic flow, the boundary be- location of the step. Figs. 2(c) and 2(d). To calibrate N, the sensitivity of the tween the subsonic and supersonic regions marks the event The condensate consists of 1 105 87Rb atoms in theC. imagingBarcelo system et al., Living is required. Rev. ThisRelativity sensitivity 14, (2011), is found 3 by The integrand of Eq. (1) is given by the ratio Án=Át, where propagating in general relativistic space-times. Án is the difference between the profiles in Fig. 2(f) (after horizon of the sonic black hole. The analogy was later F 2, mF 2 state and is initially prepared in the har-studying the condensate in the harmonic potential only, extended to include excitations with a nonlinear dispersion monic¼ part of¼ the potential, a magnetic trap with oscillation normalizing them to the same total atom number) and Át is relation, in addition to phonons [10–12]. frequencies of 26 and 10 Hz in the radial and axial (y) the time difference between the profiles. This integrand is 3 discretized by the pixels of the imaging system. The origin The experimental challenge is to create a steady flow directions, respectively. The x coordinate of the minimum 30 (a) (c) (e) (g) which exceeds the speed of sound [1,3,13,14]. Consider a of the harmonic trap is controlled by adjusting the trap ) x 0 is the left edge of Fig. 2(f). Figure 2(g) shows the

analogue Hawking radiation in 1

Hawking radiation − ¼ phonon with momentum k. In the reference frame of the frequencies, which adjusts the sag due to gravity. The 2 resulting velocity profile. This calculation is repeated for m)

moving fluid, the phonon@ has energy E kc(fluids), where c is Bosesteplike potentialEinstein is created condensates by a large diameter, red- µ 0 various times, giving the velocity profiles of Fig. 3(a). The the speed of sound. In the laboratory frame,¼ @ by a Galilean detuned laser beam with a Gaussian profile (1=e2 radius y ( 1 error bar indicates the standard error of the mean. We v (mm s transformation [15], this energy becomes E E W.k G.v, Unruhof 56, Phys.m, wavelengthRev. Lett. 46 812 (1981) nm). Half 1351. of this beam is − attribute this variability to fluctuations in the initial posi- 0 ¼ þ Á −30 where v is the flow velocity. For the case of supersonic@ flow 0 tion of the harmonic magnetic trap. The experimental 12 x (µm) z (µm) z (µm) 0 10 2030 (v>c), E0 can be zero, resulting in the unstable produc- 2 2 results agree well with a 3D simulation of the Gross- (b) (d) (f) (h) tion of phonons. This instability is thought to prevent the 10 30 ) )

3 Pitaevskii equation shown in Fig. 4(b). 3 −

supersonic flow required to realize a sonic black hole, a 8 − For comparison with v, the profile of the speed of sound m) phenomenon referred to as the Landau critical velocity cm cm 6 µ 0 1 1 can be computed from the density profiles of Figs. 2(f) and 13

[3,13,15]. By momentum conservation, however, the pro- 13 y ( 2(h), via the relation c gn =m [27], where g is the 4 av duction of such phonons requires an additional body such V/h (kHz) ¼ n (10 interaction parameter, m is the atomic mass, and nav is the as an impurity particle [16] or a container with a rough wall 2 −30 n (10 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 average of the several density profiles shown in the figure. [15]. This body provides momentum in the opposite direc- 0 102030 0 102030 0 10 2030 0 10 2030 V/h a 0 In contrast to v, the computation of c relies on the absolute tion to the flow. Thus, we have arranged an experimental 0 b x (µm) z (µm) x (µm) x (µm) −2 calibration of n discussed above. The profiles of c are apparatus which does not supply much momentum in this 0 20 40 60 80 100 direction, allowing for supersonic flow during the time x (µm) indicated by the solid black curves in Figs. 2(g) and 3(a). JAIN et al. FIG. 2O. (color). Lahav, Inet situal. Phys.absorption Rev. imagesLett. 105 of the (2010) sonic black 240401. PHYSICAL REVIEW A 76, 033616 ͑2007͒ scale of the experiment [3]. The free flow required to hole, with the potentials on. Gravity is in the x direction. As seen in Fig. 3(a), the peak flow velocity vm exceeds FIG. 1. The profile of the potential, shown at a time before the À À overcome the Landau critical velocity also helps prevent (a) Condensate in the presence of the steplike potential, at c by an order of magnitude. The black hole horizon xH is the production of quantized vortices, which usually limit harmonic potential (a) has reached the steplike potential (b). The200 ms. The average of 2 images is shown. The dashed (dash- indicated by a filled circle. The flow decreases below the horizontal arrow indicates the motion of the harmonic potentialdotted) line indicates the position-dependent profile of the black speed of sound again at the white hole horizon, indicated the flow to speeds much lower than the speed of sound [17]. relative to the stationary steplike potential. The dashed line hole (white hole) horizon. (b) Like (a), at 205 ms. (c),(d) Side Suggested schemes for forming a sonic blackanalogue hole in a indicates the cosmological chemical potential of the condensate. Gravity10 is in 10 by a ‘‘ ’’ [3]. In order to find the position-dependent views (y-z plane) of (a) and (b), imaged simultaneously with þ condensate include a Laval nozzle [18,19], flow along a the x direction. The profile of the potential step is derived from profiles of the horizons, v and c are computed for 3 (a) and (b). (e) Side view of the condensate in the harmonic ring or a long, thin condensate [3,20], a gradientparticle in the ancreation imageÀ of the laser radiation beam. The harmonic potential is derived horizontal slices in Figs. 2(a) and 2(b), rather than for the coupling constant [21,22], a soliton [2,23], an expanding from the measured frequency of the harmonic magnetic trap. potential only. (f) Density profiles, derived from (a) and (b) in entire image. The dashed and dash-dotted lines indicate the k k 5 blue and green, respectively. The density is averaged over5 the y-z resulting profiles of the black hole horizon and white hole N N plane (see text). (g) v (red curve) and c (black curve). These horizon, respectively. 0031-9007=10=105(24)=240401(4) 240401-1 Ó 2010 The American Physical Societycurves are derived fromÀ (f), as described in the text. The filled circle and plus indicate the black hole and white hole horizons, The black hole analogy requires that v and c be sta- tionary (independent of time). We find that the motion of 0 respectively. (h) Time evolution of the density profile.0 The red, the black hole horizon itself5 is a reasonable overall check 50 green, cyan, blue, magenta, and5 black curves indicate equal50 100 −5 100 −4 intervals from 200 to 225 ms. Each curvex 10 is derived from an of stationarity.150 Specifically, thex speed 10 vH of the black hole S. Hawking, Nature 248 (1974) 30. 150 0 0 average of 2 images. horizon should be much less than c. In Fig. 3(a), the total (a) k t (c) k t | | 240401-2| |

10 10 k k 5 5 N N

0 0 5 50 2 −4 50 −3 100 x 10 100 x 10 150 0 150 0 (b) k t (d) k t | | | | 5 FIG. 4. ͑Color online͒ de Sitter expansion: time dependence of Bogoliubov mode populations. Parameters are CNL ͑¯t=0͒=1ϫ10 , N0 =107, and X=2ϫ103. The ͑blue͒ points on each curve show where each mode crosses over from phonon to free-particle behavior ͑as defined ¯2 by kc /2=CNL͒. The ͑green͒ dashed curve at the final time shows the analytic prediction in the acoustic approximation based on the Bogoliubov theory and Hamiltonian diagonalization as outlined in Sec. V A. The ͑red͒ solid curve at the final time shows the analytic prediction ͑including quantum pressure͒ for a sudden transition from Eq. ͑125͒.

−4 cycles ͑mϾ1͒, which correspond to m expansion and con- and ͑ii͒ m=10, X=2, and ts =1ϫ10 . The corresponding tractions. The peak wave-vector kres for parametric resonance simulation results are given by Figs. 7͑c͒ and 7͑d͒. Clearly from Eq. ͑128͒ is shown by the position of the red solid line the mode population is peaked near the resonance condition. in each case.

1. Single cycle „m=1… IX. DISCUSSION We further consider two specific sets of parameters: ͑i͒ A. Inflationary universe models −5 m=1, X=2000, and ts =1ϫ10 and ͑ii͒ m=1, X=2000, and −4 The results for the de Sitter and tanh cases can be inter- ts =1ϫ10 . The corresponding simulation results are given pretted as follows. The healing length increases as the expan- by Figs. 7͑a͒ and 7͑b͒. In both cases there is a transient phase sion proceeds; a mode k that starts as phononic where there is dramatic quasiparticle production in the time- ͓kϽ1/␰͑0͔͒ will at a later time t crossover to a particlelike dependent Bogoliubov basis; however, the net quasiparticle c regime ͓kϳ1/␰͑t ͔͒. According to the discussion of Sec. 3 production at the end of the cycle t=t is small in both c ͑ f͒ we expect particle production to be dominant before this cases, and in particular is very close to zero for the faster time, the mode populations becoming fixed after this time. cycle in Fig. 7 a . Note that for t =1ϫ10−5 the resonance ͑ ͒ s This prediction is borne out in the results shown in Figs. 4 condition occurs for a wave-vector larger than the cutoff for and 5. the projector i.e., k Ͼk . ͑ res cutoff͒ For fast expansions, the analytic prediction in the acoustic approximation overestimates the quasiparticle production for 2. Multiple cycles „mϾ1… modes that crossover from phononlike to particlelike. The The excitation of the field due to parametric resonance is role of the crossover is further illustrated in Fig. 8. Here we much greater when there are multiple cycles of the scaling have plotted the quasiparticle numbers Nk at four different factor, and we can therefore consider a smaller driving am- times, corresponding to the results from Fig. 4͑c͒ for −4 plitude X. In particular, we consider two specific sets of pa- ts =1ϫ10 . For each time, the blue points represent the cal- −4 rameters for this subcase: ͑i͒ m=5, X=100 and ts =1ϫ10 culated Nk from the classical field simulations, the green

033616-18 (part 2) Rydberg atoms

• atoms in states with large principal quantum number n ~ 40-100. • Large size ~ n2 (85nm for n=40) • Large polarizability ~ n7 • Long life times ~ n3 (40µs for n=40) long range interactions • 4 3 V ~ n /d dipole-dipole 11 6 Johannes Rydberg V ~ n /d Van-der-Waals

Rydberg 1880: Wavelengths of alkali spectral lines behave according to: 1 1 1 = R λ n2 − n2 ￿ 1 2 ￿ Almost classical orbits

semi-classical description: de-Broglie wavelength compared to orbital radius becomes smaller and smaller => Understanding that can be gained from classical pictures increases

PHYSICAL REVIEW LETTERS week ending PRL 100, 243004 (2008) circular wave packet, Bohr-atom20 JUNE 2008 classical and quantum dynamics

FIG. 2 (color online). Snapshots showing the evolution of the FIG. 1 (color online). Time evolution of a classical Rydberg J. J. Mestayer et al., Phys. Rev. Lett. 100 (2010) 243004. W. Wintgen et al., Phys. Rev. A. 36 (1987) 131. pump 1 simulated wave packet following application of a pump field electron trajectory ni 306 in a field Fz 20 mV cmÿ pump 1 ˆ † ˆÿ Fz 20 mV cmÿ for 22 ns to quasi-1D ni 306 atoms. directed along the z axis. The electron orbit is initially oriented ˆÿ ˆ pump These are taken at the times shown after turn-off of Fz and along the x axis, and the motion, projected into the xz plane, is also can 2study quantum chaos represent projections on the xz plane. Scaled units x0 x=ni followed for one Stark precession period TS 43 ns. and z z=n2 are used. ˆ Inset: Calculated evolution of the distribution in the y component 0 ˆ i of angular momentum Ly for the wave packet. Scaled units x0 2 2 ˆ x=n and z0 z=n are used. snapshot in Fig. 2 is taken 7 ns after the field is turned i ˆ i off.) The coordinate of the wave packet moves in an approximately circular orbit with angular frequency !n of points, is followed by solving Hamilton’s equation of 3 ˆ nÿ given by motion for each initial point. In the presence of the pump pump field Fz , the distribution in Ly remains narrow and oscillates sinusoidally, periodically becoming strongly peaked at values of L 270 (or 270) 35, which y ÿ ‡  corresponds to an electron traveling anticlockwise (or clockwise) around a nearly circular orbit. Switching off the pump field at such times ‘‘freezes’’ the Ly distribution, leaving the atom in a nearly circular state. This operation is similar to the ‘‘=2 pulse’’ used to manipulate nuclear spins [23] and to a theoretical proposal to produce sta- tionary circular states [9,24]. Figure 2 shows snapshots of the simulated wave packet pump resulting from application of a field Fz that is turned off after 22 ns ( T =2), i.e., when L reaches its largest  S y negative value. A localized wave packet is evident that moves in a nearly circular orbit. As seen in Fig. 3, which shows the time evolution of the electron distribution in both angle and radius, the wave packet remains well local- ized for several orbits. It circulates with near constant radius ( n2) and, since its angular position depends line- arly on time, with constant angular velocity and momen- 1 FIG. 3 (color online). Time dependence of (a) the angular and tum ( nÿ ). The localization of the wave packet in (b) the radial distributions of the wave packet following turn-off  1 azimuth initially improves with time becoming optimum of a pump field of 20 mV cmÿ applied for 22 ns. The 7–12 ns after turn-off of the pump field, when its full azimuthal angles are measuredÿ from the x axis. Scaled units width half maximum amounts to 1 radian. (The first r r=n2 are used. ‡  0 ˆ i 243004-2 Rydberg-dipole blockade LETTERS PUBLISHED ONLINE: 11 JANUARY 2009 DOI: 10.1038/NPHYS1178 Interactions between Rydberg atoms at micrometer distances is 12 OOM larger than between ground-state atoms. Micrometer Observation of Rydberg blockade between distances allow individualLETTERS laser addressing. => interesting for two atoms PUBLISHEDquantum ONLINE: 11 JANUARY computation 2009 DOI: 10.1038/NPHYS1183

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker and M. Saffman* 2314The blockade effectSaffman, Walker, and Mølmer: QuantumApplication information with Rydbergto quantum atoms gates

Blockade interactions whereby a single particle prevents the aby1 flowObservation or excitation of other particles provide a mechanism of collective for Control Target excitation of two control of quantum states, including entanglement of two or |r〉 11 1–3 B 10 Coulomb~1/R more particles. Blockade has been observed for electrons , w = 10 m a) b)

m µ photons4 and cold atoms5. Furthermore, dipolar interactions µ 780 nm z 480 nm 3 Rydberg between highly excited atoms have been proposed as a π π 7 d-d~1/R = 10 10 B 6,7 Z individual atoms in the Rydberg2π y2 blockade100s regime mechanism for ‘Rydberg blockade’ , which might provide a 8–11 |r> novel approach to a number of quantum protocols . Dipolar |1〉 interactions between Rydberg atoms were observed several |0〉 3 123123 12 1 1 1 10 2 2 decades ago and have been studied recently in a many-body vdW~1/R6 regimeAlpha using Gaëtan cold atoms13–,18 Yevhen. However, to Miroshnychenko harness Rydberg c, Tatjana Wilk ,y Amodsen Chotia , Matthieu Viteau , mag d-d blockade for controlled quantum2 dynamics, it is necessary2 to z 1 0.1 . 1 achieveDaniel strong Comparat interactions between, Pierre single pairs Pillet of atoms., Antoine Browaeys * and Philippe~1/R Grangier3 π π 2π π π 2π Here, we demonstrate that a single Rydberg-excited rubidium 12 orders of magnitude atom blocks excitation of a second atom located more than -5 10 µm away. The observed probability of double excitation is 10 When two quantum systems interact strongly with each other, ab6 less than 20%, consistent with a theoretical model of the 3vdW~1/R C3/R |1> Rydbergtheir simultaneous interaction augmented excitation by Monte by Carlo the simulations same driving pulse may be 10 -9 ω µ 10 |r,r〉 that account for experimental imperfections. m 2 5 10 20 50 100 10 forbidden. The phenomenon is known as blockade of excitation. 1 r,r |0> The mechanism of Rydberg blockade is shown in Fig. 1a. Two ∆E | 〉 ∆E atoms,Recently, one labelled extensive ‘control’ and studies the other have ‘target’, been are placed devoted in to the so-called 2E control target control target proximityRydberg with blockade each other. The between ground state neutral1 and Rydberg atoms, state which appears when √2Ω r of each atom form a two-level system that| ￿ is coupled by laser Figure 1 Rydberg blockade mechanism. a, Conceptual operation of a |the￿ atoms are in highly excited electronic states,| owing to 3 Ω ∆φ =π ∆φ =0 beams with Rabi frequency Ω. Application of a 2π pulse (Ωt 2π Rydberg blockade phase gate betweenFIG. control 1. and͑Color target¬C atoms3/R eachonline with ͒ Two-body interaction strength for t t withthet being interaction the pulse duration) induced on the by target the atom accompanying results= in internal qubit large states dipole0 , 1 and Rydberg state r . b, Experimental geometry | ￿ | ￿ ground-state| ￿ Rb atoms,|g,r〉 Rb, |r,g〉 atoms| excitedΨ 〉 to the 100s|Ψ level,〉 excitation and de-excitation of the target atom giving a phase shift with two trapping regions separated by Z 10 µm. c, Experimental + ¬ moments. Rydberg blockade has been proposed as a basic = E FIG. 2. ͑Color online͒ Rydberg blockade controlled phase gate of π on the quantum state, 1 t 1 t. If the control atom is fluorescence image of the1 atomic–5 and density ions. created by averaging 146 excitedtool toin the quantum-information Rydberg state before| ￿ →− application| ￿ processing of the 2π pulse, the withexposures neutral each atoms with one atom, in the control and target sites.

Two-atom energy Two-atom operating on input states ͑a͒ |01͘ and ͑b͒ |11͘. Quantum infor- dipole–dipole interaction r r shifts the Rydberg level by an and can be used toc deterministicallyt generate entanglement Ω amount B that detunes the| excitation￿ ↔ | ￿ of the target atom so that it is for internal-state manipulation. The ten-to-one ratio we achieve √2Ω D.mation Jaksch, iset al. stored Phys. inRev. the Lett. basis 85 (2000) states 2208. |0͘, |1͘ and state |1͘ is blockedof several and 1 atoms.1 . Thus, the Here, excitation we dynamics demonstrate and phase of Rydbergbetween the interaction blockade length and the wavelength of the control | ￿t → | ￿t |g,g〉 coupled to a Rydberg level ͉r͘ with excitation Rabi frequency thebetween target atom two depend atoms on the state individually of the control atom. trapped Combining in opticallight is a tweezers significant step at towardsand havedemonstration0 been of a used universal to demonstrate high fidelity gates 6 thisa distanceRydberg-blockade-mediated of 4 m. controlled-phase Moreover, operationwe showwith experimentallyquantum gate between that neutral atoms in a scalable architecture. |g,g〉 M.⍀ .D. The Lukin controlled, et al. Phys. phaseRev. Lett. gate 87 is(2001) implemented 037901. with a three π/2 single-atom rotationsµ between states 0 t and 1 t of the To carry out the blockadeand operation small of Fig. algorithms1a, the atoms must ͑BlattR and Wineland, 2008͒. Neu- targetcollective will implement two-atom the CNOT behaviour, gate between| ￿ associated two| atoms.￿ We withbe close the enough excitation to have a strong interaction, yet far enough apart pulse sequence: ͑1͒ ␲ pulse on control atom ͉1͘ ͉r͘, ͑2͒ 2␲ have previously demonstrated the ability to carry out ground-state that they can be individuallytral controlled atom (Fig. 1 qubitsb). To satisfy represent these another promising approach → of an entangled state19 between the ground and Rydberg E. UrbanFigure 1et Rydbergal., Nature excitation Phys. 5 of (2009) two atoms 110. in the blockade regime. M.pulse Saffman on,target et al. Rev. atom Mod.͉1͘ Phys.͉r͘ 82͉ 1(2010)͘, and 2312.͑3͒ ␲ pulse on control rotations at individual trapping sites , as well as coherent excitation conflicting requirements, we͑Bloch, first localize single2008| atoms͒. Theyto regions share many features in common → → fromlevels, ground enhances to Rydberg states the at a allowedsingle site20. Here, single-atom we describe that excitation. are formed by These tightly focuseda, beams Principle froma of far-detuned the Rydberg laser. blockade between two atoms separated by a atom ͉r͘ ͉1͘. ͑a͒ The case where the control atom starts in |0͘ experiments that demonstrate the Rydberg blockade effect between The lasers that control the coupling between internal states are → observations should be a crucial step towards the deterministicwithA. Gaëtan,distance trapped Ret. Twoal., ion statesNature systemsg Phys.and r5are including(2009) coupled 115. with long-lived Rabi frequency encodingΩ. When and is not Rydberg excited so there is no blockade. ͑b͒ The two neutral atoms separated by more than 10 µm, which is an focused to a small waist w 10 µm so the atoms can be separately| ￿ | ￿ enablingmanipulation step towards creation of entanglement of entangled atomic ofstates. two Previous or moremanipulated atoms, by displacing with∼of thethe quantumcontrol two lasers, atoms even information are though in state the r,r , they in atomic interact strongly, hyperfine which leads states to and case where the control atom is in |1͘ which is Rydberg excited demonstrations of neutral-atom entanglement have relied on short- trapping sites are close together. In addition, we excite high-lying| ￿ 3 possible implications for quantum-information science, as wellthesymmetrical possibility energy of shifts manipulating￿E (C3/R and). When measuring this shift becomes the largerqubit range collisions at length scales characterized by a low-energy Rydberg levels with n 79 and 90. The strength of the long-range =± leading to blockade of the target atom excitation. scatteringas for length quantum of aboutmetrology, 10 nm (refs 21,22). the Our study results, using of stronglyinteraction betweencorrelated two= Rydbergstatethan atoms usinghΩ scales, the as resonant lasern11, with isn outbeing of laser resonance pulses. with the transition coupling the 87 23 ¯ laser-cooledsystems and in optically many-body trapped Rb, physics, extend the distance and for fundamentalthe principal quantum studies number .singly As will be and discussed doubly in more excited detail states, and only one atom at a time can be strong two-atom interactions by three orders of magnitude, and below, the 79d(90d) Rydberg levelsNeutral provide B atoms/2π > 3(9.5) distinguish MHz themselves from ions when placein quantum us in a regime where physics. the interaction distance is large compared of blockade shift at Z 10.2 µm,transferred which is sufficient to the for Rydberg a strong state. b, When the atoms are in the blockade = pear more difficult than it may be for an array of weakly withA 1 µm, large which is experimental the characteristic wavelength effort of is light nowadays needed two-atom devoted blockade to effect. theweregime, consider the state theirΨ state, described dependent in the text, interaction is only coupled to properties, the ground | +￿ interacting neutral atoms. Several approaches to scal- production of entanglement, that is quantum correlations, betweenwhichstate areg,g essentialwith a strength for√2 implementingΩ, whereas the state two-qubitΨ is not coupledquantum by | ￿ | −￿ individual quantum objects such as atoms, ions, superconductinggates.the laser Figure to the1 statesshowsg,g theand dependencer,r . The atoms are of therefore the two-particle described by ability in trapped ion systems are being explored includ- | ￿ | ￿ Departmentcircuits, of Physics, spins University or photons. of Wisconsin, There1150 University are Avenue, several Madison, ways Wisconsin to 53706, engineer USA. *e-mail:interactionan [email protected]. effective two-level strength system. on separation R for singly charged ing the development of complex multizone trap tech- 110entanglement in a quantum system. Here, we focusNATURE on PHYSICS a methodVOL 5 FEBRUARY 2009 www.nature.com/naturephysics nologies ͑Seidelin et al., 2006͒ and anharmonic traps ͑Lin | | ions, ground-state| neutral atoms, and Rydberg atoms. that relies on a blockade mechanism where the strong interaction between two atoms 10 µm apart, by showing that when one atom The interaction of ground-state atoms is dominated by et al., 2009͒. We note that some of the attractive features between different parts of a system prevents their simultaneous is excited to a Rydberg state, the excitation of the second one is of Rydberg-mediated interactions may also be appli- excitation by the same driving pulse. Single excitation is still1/Rgreatly6 van suppressed der Waals18. However, forces at the short enhancement range and of the 1/ excitationR3 mag- cable to trapped ion systems ͑Müller et al., 2008͒. possible but is delocalized over the whole system, and results in theneticrate dipole-dipole of one atom when forces twobeyond atoms are about present, 30 explained nm. At by spac- the production of an entangled state. This approach to entanglementingsexcitation greater of than an entangled 1 ␮m state the in interaction the blockade is regime, weak, has less not beenthan is deterministic and can be used to realize quantum gates1 or observed until now. to entangle mesoscopic ensembles, provided that the blockade1 HzHere, in frequency we study two units, individual which atoms, implies held at that a few an micrometres array of A. Rydberg-mediated quantum gates is effective over the whole sample2. Blockade effects have beenneutraldistance atom by two qubits optical tweezers. can be The structurally ground state stable.g and a RydbergOn the observed in systems where interactions are strong such as systemsotherstate hand,r of excitation an atom are of separated Rb atoms by an to energy the| 100￿E (sees RydbergFig. 1a) The idea of using dipolar Rydberg interactions for 6 | ￿ of electrons using the Coulomb force or the Pauli effectiveleveland results can be coupled in a very by a strong laser. For interaction non-interacting that atoms, hasa reso-and neutral atom quantum gates was introduced in 2000 7 interaction , as well as with photons and atoms coupled to an b, the two-atom spectrum exhibits two transitions3 at the same 8 nant dipole-dipole character, scaling as 1/R , at short ͑Jaksch et al., 2000͒ and quickly extended to a meso- optical cavity . Recently, atoms held in the ground state of the frequency E/￿, connecting states g,g to r,g or g,r , and then6 wells of an optical lattice have been shown to exhibit interactiondistancesto r,r . This and enables van der the simultaneous Waals| character,￿ excitation| ￿ scaling of| the￿ two as atoms 1/R , scopic regime of many-atom ensemble qubits ͑Lukin et blockade, due to s-wave collisions9. An alternative approach usesat longto| state distances.￿ r,r . However, As will if the be two discussed atoms interact in Sec. stronglyII the whenchar- al., 2001͒. The basic idea of the Rydberg blockade two- the comparatively strong interaction between two atoms excited to in state |r,r￿, this energyR level is shifted by an amount ￿E and qubit gate is shown in Fig. 2. When the initial two-atom acteristic| length￿ scale c where the Rydberg interaction Rydberg states. This strong interaction gives rise to the so-calledchangesthe laser character excitation cannot depends bring on the the two atoms principal to state quantumr,r .A state is |01͓͘Fig. 2͑a͔͒ the control atom is not coupled to Rydberg blockade, which has been observed in clouds of cold fundamental consequence of this blockade is that the atoms| ￿ are 10–15 16 number n. For the 100s state the crossoverik r lengthik r is the Rydberg level and the target atom picks up a ␲ atoms as well as in a Bose condensate . A collective behaviour excited in the entangled state Ψ (1/√2)(e · a r,g e · b g,r ), | +￿= | ￿+ | ￿ phase shift. When the initial state is |11͘ both atoms are associated with the blockade has been reported in an ultracoldclosewhere to Rrac =9.5and rb␮m,are and the positionsat this length of the scale two atoms the ratio and ofk atomic cloud17. Recently, an experiment demonstrated the blockadetheis Rydberg related to interaction the wave vectors to the of ground-state the exciting lasers. interaction More coupled to the Rydberg level. In the ideal case when the is approximately 1012. two-atom “blockade” shift B due to the Rydberg inter- The applicability of Rydberg atoms for quantum infor- action is large compared to the excitation Rabi fre- 1Laboratoire Charles Fabry, Institut d’Optique, CNRS, Univ Paris-Sud, Campusmation Polytechnique, processing, RD 128, 91127 which Palaiseau is the cedex, central France, 2 topicLaboratoire of Aiméthis re- quency ⍀, excitation of the target atom is blocked and it Cotton, CNRS, Univ Paris-Sud, Bâtiment 505, Campus d’Orsay, 91405 Orsayview, cedex, canFrance. be *e-mail: traced [email protected]. to the fact that the two-atom inter- picks up no phase shift. The evolution matrix expressed action can be turned on and off with a contrast of 12 in the computational basis ͕|00͘,|01͘,|10͘,|11͖͘ is NATURE PHYSICS VOL 5 FEBRUARY 2009 www.nature.com/naturephysics orders of magnitude. The ability to control the interac-115 | | | 10 0 0 tion strength over such a wide range appears unique to the Rydberg system. We may compare this with trapped 0 − 10 0 U = , ͑1͒ ions whose Coulomb interaction is much stronger but is 00− 10 always present. The strong Coulomb interaction is ben- ΂00 0− 1 ΃ eficial for implementing high fidelity gates ͑Benhelm et al., 2008b͒ but the always on character of the interaction which is a controlled-Z ͑CZ͒ gate. As is well known makes the task of establishing a many-qubit register ap- ͑Nielsen and Chuang, 2000͒ the CZ gate can be readily

Rev. Mod. Phys., Vol. 82, No. 3, July–September 2010 week ending PRL 107, 153001 (2011) PHYSICAL REVIEW LETTERS 7 OCTOBER 2011

week ending PRL 104, 195302 (2010) PHYSICAL REVIEW LETTERS Nonlocal Nonlinear14 MAY Optics2010 in Cold Rydberg Gases

1 1 2 1 Three-Dimensional Roton Excitations and Supersolid FormationS. Sevinc¸li, N. Henkel, C. Ates, and T. Pohl 1 in Rydberg-Excited Bose-Einstein CondensatesMax Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany 2School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom N. Henkel, R. Nath, and T. Pohl (Received 10 June 2011; published 3 October 2011) Max Planck Institute for the Physics of Complex Systems, No¨thnitzerWe Straße present 38, an 01187 analytical Dresden, theory Germany for the nonlinear optical response of a strongly interacting Rydberg gas (Received 19 January 2010; revised manuscript received 30under March conditions 2010; published of electromagnetically 11 May 2010) induced transparency. Simple formulas for the third-order optical We study the behavior of a Bose-Einstein condensate in whichsusceptibility atoms are weakly are derived coupled and to a shown highly to be in excellent agreement with recent experiments. The obtained excited Rydberg state. Since the latter have very strong van derexpressions Waals interactions, reveal this strong coupling nonlinearities, induces which in addition are of highly nonlocal character. This property effective, nonlocal interactions between the dressed groundtogether state atoms, with which, the enormous opposed to strength dipolar of the Rydberg-induced nonlinearities is shown to yield a unique interactions, are isotropically repulsive. Yet, one finds partiallaboratory attraction platformin momentum for nonlinear space, giving wave phenomena, such as collapse-arrested modulational instabilities in rise to a roton-maxon excitation spectrum and a transition to a supersolid state in three-dimensional a self-defocusing medium. condensates. A detailed analysis of decoherence and loss mechanisms suggests that these phenomena are observable with current experimental capabilities. DOI: 10.1103/PhysRevLett.107.153001 PACS numbers: 32.80.Ee, 42.50.Gy, 42.65. k À DOI: 10.1103/PhysRevLett.104.195302 PACS numbers: 67.80.K , 03.75.Kk, 32.80.Ee, 32.80.Qk À Advances in designing materials with highly intensity- an ideal nonlinear medium to study nonlocal wave phe- Since being introduced by Landau in a series of seminal i dependentresponding refraction transition [1–3 and] have projection ushered operators in numerous^ ð Þ nomena, in which the strength, the range, and even the sign articles [1], the notion of a roton minimum in the disper- ¼ studies i of nonlocali ( ; nonlineare; g), the resulting wave phenomenaN-particle interaction [4–8]. In of the nonlocal interaction kernel can be widely tuned with sion of a quantum liquid has been pivotal to understanding jcanih bej written¼ as superfluidity in helium. This later led to the prediction ofnonlocal a systems, such as nematicRydberg liquid crystals [1dressing,2] or high accuracy. and To demonstrate EIT this potential, we present i j i peculiar solid state upon softening of the roton excitationthermal mediaH^ [3], theV nonlinearr ^ eeð Þ^ eeð responseÞ Á depends^ eeð Þ H^ not; only numerical results for the propagation of cw laser light I ¼ eeð ijÞ À þ L energy [2], simultaneously possessing crystalline and su-on the local intensityXi Rydberg0 denotes state: the van Very der Waalssmall transparency. Quantum interference time provide for superfluid nonviscous flow. Forty yearstransparencyeeð (EIT)ijÞ¼P in6 ultracoldij multilevel atoms [9,10] capabilities. after its conjecture, this apparent contradiction continues to (vdW)Rydberg interaction admixture between two makes Rydberg ground-state atoms at a dis- effects leads to zero population in provides an elegant mechanism to suppress photon11 loss Consider first the propagation of a beam with wave attract theoretical interest and has ushered in an intense tance rij ri rj. Because of the strong C6 n scaling and simultaneously¼ À increase light-matter interaction times number k and amplitude E that couples to the atomic search for experimental evidence in solid 4He, whose ofinherit the vdW coefficient, some of such the Rydberg-Rydberg extreme atom Rydberg inter- middle level. Extremelyp sensitive to to enhance nonlinear effects. Combined with sufficiently medium with a Rabi frequency } E = [see interpretations are currently under active debate [4,5]. actionsproperties. are orders of magnitude larger than those of ground interactions acting on |3>. p ¼ 12 p Here, we demonstrate how three-dimensional roton ex-largestate nonlinearities, atoms. We this are interested holds great in the potential potential for surface few- Fig. 1(a)], as described by the paraxial wave equation@ citations can be realized in atomic Bose-Einstein conden-photon nonlinear optics [11,12] and may enable applica- week ending sates (BECs),PRL thereby104, 195302 introducing (2010) an alternativePHYSICAL system REVIEWtions to LETTERS in communication and quantum14 MAY 2010 information science. study supersolidity. The supersolid phase transition is Recently, it was recognized that EIT schemes involving shown to arise from effective interactions, realized throughhighly excited atomic Rydberg levels provide promising off-resonant optical coupling [6–8] to highly excitedperspectives for such applications [13–23]. In particular, Rydberg states. Owing to the strong increase of atomicthe huge polarizability of Rydberg states gives rise to giant interactions with their principal quantum number n, reso- nantly excited Rydberg gases have proved to be an idealKerr coefficients [16] but also entails strong long-range platform to study strong interactions in many-body sys-interactions, which render Rydberg-EIT media intrinsi- tems [9] on short microsecond time scales. The presentcally nonlinear. Indeed, a recent theory for two-photon approach—based on off-resonant two-photon excitationpulses revealed the emergence of strong effective photon- [see Fig. 1(a)] of Bose-condensed alkaline atoms—permitsphoton interactions [24], while experiments [21] and nu- us to utilize the strong Rydberg interactions over muchmerical calculations [25] demonstrated greatly enhanced FIG. 1 (color online). (a) Schematics of the considered three- longer timesFIG. of 3 (color100 online). ms. In (a) particular, Roton gap  as we a function consider of the cou-nonlinear absorption coefficients in the opposite limit of 2 2 interaction parameter . (b) Energy density " 0 É level atom, illustrating the laser coupling between the atomic pling to nS Rydberg states with vanishing¼ orbital2N h j À2@Mr angularþ H^ É for different crystal symmetries relative to the energy largeground photon state numbers.n S and the Rydberg state nS . For Á , the momentum, which,j i as2 opposed to dipole-dipole interac- 0 1 1 FIG. 1 (color online). (a) Three-level scheme for isolated density "hom  =3 of a homogeneous BEC. Panel (c) pro- In this Letter,j wei develop an analyticalj i theory for the tions, givesvides rise an to enlarged isotropically¼ view around the repulsive transition point. interaction The effective po- system reduces to an effective two-level atom, with the states 87 atoms, where the atomic ground state 1 , an intermediate state Rydberg state lifetime for excitation of Rb to n 60 (C6 nonlinearg opticaln0S and responsee nS coupled of a with strongly a two-photon interacting Rabi j i tentials for9 the:7 10 ground20 a:u: [26]) state with Á atoms,50 MHz and and, thus,10¼14 cm ensures3 ¼is j i  j i j i  j i 0 À frequency and detuning Á. (b) Effective potential resulting 2 , and a highly excited Rydberg state 3 are mutually driven by stability of theshown condensate. in (d). ¼ ¼ Rydberg-EIT medium to monochromatic multiphoton light j i j i FIG. 4 (colorfrom online). the Snapshotsoff-resonant of the BEC coupling dynamics to for a the strongly interacting a strong control and a weak probe field with Rabi frequencies c The system is described as a gas of N atoms with masssources.time-varying interaction Based parameter on the t approach,shown in (a). Panels we (b)– give a simple formula sequence, the system generally approaches a glassy state L.Rydberg Santos, states et al. for , ðPhys.nÞ 60 Rev.and Lett.Á 5085 MHz, 1791. Panels (2000). (c) and and S. Sevincli,p, respectively. et al. , (b)Phys. In aRev. gas Lett. of atoms, 107, the 153001 strong (2011). van der for(e) show the the nonlinear density along orthogonal absorption slices¼ through coefficient the simu-¼ that provides an M at positionswithr short-rangei, each possessing ordered density a modulations, ground state when start-gi andlation box(d) at times provide indicated an in enlarged (a). The upper view and right of axes the potential showing the Waals interaction between atoms in state 3 inhibits multiple ing from a homogenous initial state (see below). We, thus,j i 87 an excited nS Rydberg state, denoted by ei . The twoexcellentin (a) showN. theHenkel, actual description time et and al. Rabi , of frequencyPhys. recent forRev. a measurementsRb Lett.BEC 104, 195302 on cold (2010). ru- j i contributions from both ground14 3 state-Rydberg atom and ground Rydberg excitations within a blockade radius R , giving rise to a used variational calculations, based on periodicallyj i ar- with n 60, Á 50 MHz, and 0 2 10 cmÀ . c states are opticallyranged Gaussians coupled with varying with a width two-photon and lattice constant, Rabi fre-bidium¼state-ground gases¼ [21 state]. atom¼ For interactions large single-photon (solid line) as detunings, well as the strongly nonlinear optical response of the medium. The resulting to provide the proper initial wave function for a subsequent sole contribution from the latter (dashed line). quency and detuning Á [see Fig. 1(a)]. Defining cor-absorptionwhich extended superfluid is shown fractions to of be nearly greatly constant den- suppressed—yet main- imaginary time evolution according to Eq. (2). nonlinear beam propagation, for example, leads to modulation sity coexist with density-modulated domains [Fig. 4(c)]. Some of the obtained energies are shown in Fig. 3(b). tains huge refractive nonlinearities that exceed previous The latter increase in time [Fig. 4(d)], and ultimately instabilities, as shown in (c) for a rubidium 70S1=2 Rydberg gas For small values of the BEC ground state is a homoge- merge to form sizable ‘‘crystallites’’ of regular density 13 3 0031-9007=nous10=104(19) superfluid.= At195302(4) a critical value of 30:1, one 195302-1records in ultracoldÓ Kerr2010 media The [ American10] by several Physical orders Society of with a density of 8 10 cmÀ and p=2 0:35 MHz, suso modulations [Fig. 4(e)].  ¼ finds a transition to a stable supersolid state. This first- magnitude.Turning to a discussion Combined of the experimental with their feasibility, long range, this makes for c=2 80 MHz, and Á=2 1:2 GHz. order transition precedes the roton-instability [21] and ¼ ¼ 2k2 we consider a particular example of coupling to 60S takes place at a finite roton gap of  0:66 rot . The 87 suso  @2M Rydberg states in a Rb condensate, for which Rydberg existence of several competing states with similar energies excitation has recently been demonstrated [23]. Figure 5 but different crystal symmetries [see Fig. 3(b)] may gen- 0031-9007shows the corresponding=11=107(15) ‘‘phase diagram’’=153001(5) for a typical 153001-1 Ó 2011 American Physical Society 14 3 erally complicate the experimental preparation of ordered density of 10 cmÀ , also including a finite s-wave scat- states. In this respect, the dynamical tunability of the tering length a>0. The latter only leads to some increase ~ interaction strength C6 via changing the laser intensity of the critical Rabi frequencies for inducing the roton can serve as a useful tool to steer the BEC evolution. instability. Importantly, the transition to a supersolid can In order to demonstrate this point, we also studied the be realized with Rabi frequencies of a few hundred kHz, time evolution, starting from a homogenous BEC. As a and the condition Á can be well fulfilled deep in the specific example, we discuss the BEC dynamics for a roton-instability regime.j j  Yet, the detuning is sufficiently simple time dependence of , shown in Fig. 4(a). The small to avoid excitation of nearby Rydberg states and calculation starts from a homogenous condensate with near-resonant dipole coupling to adjacent pair states for small random phase noise and uses a complex time inte- distances * Rc. For typical values of Rc, the number of gration with a small imaginary contribution [22]. The atoms within a single density peak is on the order of 103, instantaneous increase of at time t 0 from 0–60 which justifies the applied mean-field description in terms induces the roton instability. This¼ sudden parameter¼ of Eq. (2). quench, however, causes relaxation towards a short-range Major limitations for the stability of Rydberg gases ordered, ‘‘glassy’’ [5] state [Fig. 4(b)], as discussed above. generally stem from the finite lifetime of the involved As is decreased close to the phase transition some of the excited states and from autoionization of close Rydberg structures vanish entirely, leading to a mixed phase in atom pairs, initiated by near-resonant dipole-dipole cou- 195302-3 Excitation transport week ending PRL 105, 053004 (2010) PHYSICAL REVIEW LETTERS 30 JULY 2010 sical) nuclear positions, we now(more move to aon quantum quantum nu- and simulation) atom 3 to accelerate. In this way the initial momentum clear wave function. is transferred through the chain to atom 7, realizing a week ending The spatial wave function of each atom is assumed microscopic version of Newton’s cradle. PHYSICAL REVIEW LETTERS Gaussian with aresonant standard deviation 0 dipole-dipole. Hence, we takePRLTo lay 105, interactions:the basis053004 for the treatment (2010) of entanglement dy- 30 JULY 2010 the complete initialFluctuations wave function (i.e., between containing nuclear dipolesnamics, of we different next discuss the atoms excitation transfer, shown in and excitonic degrees of freedom) as Fig. 3(c), which is strongly coupled to the atomic motion as 0.6 can be seen in Fig. 3(b). The excitation gets transferred, N (c) (molecules) can lead to excitation(a)always remaining transport. localized on the two instantaneously The diabatic representation of the wave function allows É t 0 ’rep R G Rn ; 0.5 j ð ¼ Þi ¼ j ð Þi ð Þ nearest atoms, in accordance with the structure of exciton n 1 (6) Y¼ eigenstates outlined in [17]. After 5:5 s the momentum  R N exp R R 2=22 ; a straightforward propagation for short chains. For longer Photosynthetic light-Gð nÞ¼ ðÀ½ n À 0nŠ 0Þ transferred through the chain kicks out the last atom, and a 0.4 n=1 well-defined close proximity pair no longer exists. The n=2 where R0n is the center of mass of the nthChain Gaussian andofN Rydberg atomsa x x x chains and for the interpretation of the results, the adiabatic (b) 0 0 0 n harvesting iscomplex a normalization factor. exciton state then assumes the shape for an equidistant p 0.3 n=3 N To confirm the applicability of the quantum-classical chain,p delocalized over the entire chain (consisting of the n=4 ~ remaining N 2 atoms), which subsequently slowly n=5 representation É R n 1 0n R ’n R is helpful. numerical treatment, we consider the smallest nontrivial À 0.2 chain N 3. Figure 2 shows the quantum mechanical spreadss out. From the occupation of the initially populated j ð Þi ¼ ¼ ð Þj ð Þi ¼ repulsive adiabatic state, which is also shown in Fig. 3(b), Here the adiabatic basis ’ is defined via probability to find an atom at a certain position, in perfect 0.1 n agreement with the corresponding graph obtained with the el P j i quantum-classical hybrid approach. H R ’n R Un R ’n R . For each R there are 0 The excellent agreement between the two disparate 0.2 0.4 0.6 0.8 1 ð Þj ð Þi ¼ ð Þj ð Þi methods for N 3 gives confidence thatAdiabatic Tully’s surface g N excitonic eigenstates ’ R labeled by the index n. ¼ a/x n hopping produces reliable results also for longer chains, 0 j ð Þi such as N 7, which we consider now. Theexcitation corresponding The corresponding eigenenergies Un R define the adia- atomic motion¼ and excitation transfer, whentransport starting in the ð Þ exciton state with highest energy, are shown in Fig. 3. Let batic potential surfaces. The two representations are re- us first consider the atomic motion. As expected, initiallyFIG. 1 (color online). (a) Sketch of the initial total density the two excited atoms strongly repel each other. When ~ atom 2 has approached atom 3, the main repulsion isdistribution of N 5 Rydberg atoms. (b) Visualization of the lated by 0n R m ’n R %m 0m R . now between those two, causing atom 2 to slow down ¼ ð Þ¼ h ð Þj i ð Þ electronic state %1 . (c) Trapping of the electronic excitation in For long chains, we solve the time-dependent j i P the repulsive exciton state by a perturbation of the regular chain. Schro¨dinger equation with Hamiltonian (1) using a mixed 2 G. McDermott et al. Shown are the populations pn %n ’rep R as a function of quantum-classical method, Tully’s surface hopping algo- Nature 374 (1995) 517. ¼ jh j ð Þij a=x0. rithm [18,19]. In this approach an ensemble of trajectories S. Wüster et al. , Phys. Rev. Lett. 105, 053004 (2010). is propagated, and each trajectory moves classically on a single adiabatic surface U R , except for the possibility so large that the overlap of their electronic wave functions mð Þ can be neglected. The total Hamiltonian of the system is of instantaneous switches among the adiabatic states. The equations of motion read FIG. 2 (color online). Nuclear dynamics in the case N 3. The time evolution of the total atomic density n x; t (a) is shown¼ FIG. 3 (color online). Dynamics of atomic motion andN excita- 2 ð Þ 5 together with a comparison of Tully’s surface hopping calcula- tion transfer. (a) Total atomic density averaged over 10 realiza- 2 el N tions (black solid line) with the full quantum evolution (red tions. We actually plotHpn.R (b) Mean trajectories of the R H R ; (1) @ dashed line) in the other panels. (b) Spatial slice n x; t0 , with t0 individual atoms (white) and electronic excitation probabilities@ n ð Þ 2 ffiffiffi T 2M _ as indicated by the first vertical white lines in (a). Arbitrary units. (diabatic populations) cm ,ðcm Þ¼ÀmOnmc~m. Then latter1 are r þ ð Þ i c~ U R c~ i R d c~ ; (4) (c) Relative population n dR ~ R 2 (n c~ 2 in encoded as the width ofj thej copper¼ shading surrounding each k k k kq q 2 2 2 2 P ¼ j ð Þj ¼ j j trajectory. (c) Population on the adiabatic surface ‘‘rep’’X¼ (black @t ¼ ð Þ À Á Tully’s algorithm) on the adiabaticR surface (index 2) that is q 1 energetically nearest to the initial repulsive one. This is a line) and individual diabatic populations. (d) Purity P Tr ^ 2 ¼T ½ Š X¼ measure of the propensity of nonadiabatic transitions. Thewhereof the reducedR electronic densityR1 matrix; ...^ (solid;R greenN line) andis the vector of nuclear posi- deviation of curves in (b) and (c) is shown magnified in the bipartite entanglement En;n 1 for neighboring atoms as defined ¼ðþ Þ insets. (d) Initial repulsive state. tions.in the text. The The dotted electronic line indicates 1. Hamiltonian € el 053004-3 MR R ’m R H R ’m R : (5) el ¼Àr h ð Þj ð Þj ð Þi H R Vnm Rnm %n %m (2) ð Þ¼ nm ð Þj ih j The N complex amplitudes c~k define the electronic state X via É R;t N c~ t ’ R;t and the d are non- j ð Þi ¼ n 1 nð Þj nð Þi kq contains the dipole-dipole coupling between atoms n and adiabatic coupling¼ vectors. Besides their appearance in m. We consider the case with all atoms in m 0 azimu- P l ¼ Eq. (4), they also control the likelihood of stochastic jumps thal angular momentum states, such that Vnm Rnm from the current surface m to another surface m0 in Eq. (5), "2=R3 without angular dependence [16]. ð Þ¼ _ nm which is proportional to R dm m . Further details about À Our numerical calculations use an atomic mass M j Á 0 j ¼ this scheme can be found in Refs. [17,19,20]. The under- 11 000 a:u: (which is roughly the mass of lithium) and a lying quantum-classical correspondence is discussed in transition dipole moment " 1000 a:u:, corresponding to Refs. [21,22]. We randomize the initial classical positions transitions between s and p ¼states with n 30; ...; 40. % and velocities for the trajectories according to the Wigner The full many-body problem posed by the Hamiltonian distribution of the initial state, described in Eq. (6). This is (1) becomes quickly intractable as the number of atoms N essential for a correct description. is increased. For small N, however, it is no problem to Initially we assume that the Rydberg atoms are arranged directly solve the equation of motion. Expanding the full in a straight line. The distance between the first two atoms wave function in electronic (diabatic) states according to is denoted by a and assumed shorter than the equal dis- É R N 0 R % , we arrive at the Schro¨dinger n 1 n n tances (x0) between the other atoms, as sketched in Fig. 1. jequationð Þi ¼ (in atomic¼ ð units)Þj i P For fixed classical positions, one of the N eigenstates of N 2 the electronic Hamiltonian (2) leads to a situation where i0_ R rRm 0 R V R 0 R : (3) initially all atoms repel each other [17]. In the following we n 2M n nm nm m ð Þ¼m 1À ð Þþ ð Þ ð Þ will focus on this state, which we label with ‘‘rep.’’ Since X¼ the dipole-dipole interaction between the first two atoms is In order to validate the semiclassical method pre- much stronger than between all others, the excitation in sented below, which in turn will be faithfully used this repulsive state is mainly localized on these two atoms. for longer chains, we solve Eq. (3) for N 3. In our For a x0 this initial state can be approximately written figures we will not show the full N-dimensional¼ ( as %1 %2 =p2. In Fig. 1(c) the electronic population nuclear wave function but focus on the more intuitive total ðj i À j i on the various atoms is shown as a function of a=x0. For atomic density, which is given by n R ffiffiffi our simulations, we have taken x0 5 "m and a N N N 1 2 N 1 ð Þ¼ ¼ ¼ j 1 m 1 d À R j 0m R . Here d À R j de- 2 "m, i.e., a=x0 0:4. Then the dipole-dipole interaction ¼ ¼ f gj ð Þj f g ¼ notesP P integrationR over all but the jth nuclearR coordinate. between the first two atoms is about 5 times larger than for The density n R gives the probability to find an atom at the rest of the chain and more than 90% of the excitation is position R. ð Þ localized on the first two atoms. From the pointlike (clas-

053004-2 1.2. Revision

Let’s move to blackboard Appendix NATURE|Vol 443|28 September 2006 ARTICLES

Michelson interferometer to study phase spatial correlations. Exper- at the nonlinear threshold29,30, down to half of the polariton line- iments discussed inARTICLES this work were performed for a slightly positive width in the linear regime (Fig. 3a). The line broadening observedNATURE at|Vol 443|28 September 2006 cavity–exciton detuning (3 to 8 meV). higher excitation is due to decoherence induced by polariton self- interaction31. We studied the coherence time more directly using a Thermalization and condensation Michelson interferometer (not shown). This measurement gives a Under non-resonantphysics and high soon excitation, exits the the regime polariton of emission weakly interacting in coherence bosons time that of 1.5 psdisplaying below the a vacuum nonlinear field threshold, Rabi splitting and 6 ps of 26 meV (ref. 26). The CdTe-basedARTICLES microcavitiesdescribes becomes ultracold highly atoms; nonlinear second,18,27–30 the. We lifetime first isabove short enoughthreshold, that consistentmicrocavity with thewas spectralNATURE excited narrowing|Vol by 443 a|28 observedcontinuous-wave September at 2006 Ti:sapphire laser, analyse the spectralwe and must angular confront distribution the role of the of non-equilibriumemission as a threshold. physics25. Never- combined with an acousto-optic modulator (1-ms pulse, 1% duty function of the excitationtheless, the power. principal Figure experimental 2a displays pseudo-3Dcharacteristics expectedSignatures for of BEC polaritoncycle) coherence to reduce in sample CdTe-based heating. microcavities The pulse duration is sufficiently images ofphysics the angular soonare exits clearly distribution the reported regime of here:of the weakly spectrallycondensation interacting integrated into bosons the ground thathave been statedisplaying previously arising a vacuum reportedlong (by field28,29 four Rabi. Macroscopic orders splitting of magnitude) of coherence 26 meV (ref. in in comparison the 26). The with the charac- emission.describes Below threshold ultracoldout of aatoms; (left), population the second, emission at the thermal lifetime exhibits equilibrium;is short a smooth enough the thatmomentum developmentmicrocavity plane of was wasteristic observed excited times by above a of continuous-wave the the system nonlinear to guarantee threshold Ti:sapphire28 a. steady-state laser, regime. The 25 distributionwe aroundmust confrontquantumv x v y the coherence,08 role, that of is, non-equilibriumindicated around k by long-range0. When physics the. spatial Never-However, coherence,combined the use and with of alaser ansmall acousto-optic beam excitation was carefullyspot modulator (3 mm shaped (1-diameter)ms pulse,into pre- a 1% ‘top duty hat’ intensity profile ¼ ¼ k ¼ excitationtheless, intensity the issharpening principal increased, experimental ofthe the emission temporal characteristics from coherence the zero expected momen- of the emission. for BECvented relaxationcycle) to reduce intoproviding the sample lowest heating. apolariton uniform The energyexcitation pulse durationstates: spot polariton is of sufficiently about 35 mm in diameter on BECtum stateare becomes clearlyin reported predominant other here: condensationat threshold (centre) intophysical the andground a sharp state arisingstimulationlong occurredsystems (by four orders in excited of magnitude) states and in was comparison thus only with remotely the charac- the sample surface, as shown in Fig. 4i. The excitation energy was peak formsout at ofk a populationExperimental0 above threshold at thermal procedure (right). equilibrium; Figure 2b the shows development the connected of teristic with times BEC. of An1.768 the experiment system eV, well to under guarantee above conditions the a polariton steady-state more ground favour- regime. state The (1.671 eV at cavity k ¼ energy andquantum angle-resolved coherence,The sample emission indicated(some we studiedintensities. by long-range consistsexamples) The spatial width of a coherence, ofCdTe/CdMgTe the able and to microcavitylaser BEC beam (25-m wasm-diameterexciton carefully resonance), spot), shaped in into which at a the ‘top polaritons first hat’ reflectivity intensity could profile minimum of the Bragg momentumsharpening distributiongrown of the shrinks temporal by molecular with coherence increasing beam of excitationepitaxy. the emission. It intensity, contains 16condense quantumproviding into wells, the a lowest uniformmirrors, energy excitation allowing state, spot indirectly proper of about showed coupling 35 mm the in build-to diameter the intra-cavity on field. This 29 and above threshold, the emission mainly comes from the lowest up of macroscopicthe sample surface,coherenceensures asin shown that real polaritons space in Fig. above 4i. initially The threshold excitation injected.How- energy in the was system are incoherent, Experimental procedure energy state at k 0. The polariton occupancy has been extracted ever, that1.768 measurement eV, well abovewhich was obtained the is polariton a necessary under ground pulsed condition excitationstate (1.671 for (150-fsdemonstrating eV at cavity BEC. In atomic The samplek ¼ we studied consists of a CdTe/CdMgTe microcavity exciton resonance), at the first reflectivity minimum of the Bragg from such emission patterns by taking into account the radiative pulses), thus precludingBEC steady or superfluid state in the helium, system and the temperaturemixing high is the parameter driving lifetime ofgrown polaritons. by molecular beam epitaxy. It contains 16 quantum wells,polaritonmirrors, densities allowing at short proper times and coupling low densities to the intra-cavityat long times field. on This the phase transition. Here the excitation power, and thus the injected exciton polaritonsFigure 3a shows the occupancy of the ground state, as well as its the sameensures spectra. that polaritons initially injected in the system are incoherent, polariton density, is an easily tunable parameter, and so we chose it as emission energy and linewidth as a function of excitation power. Figurewhich 3b displays is a necessary the occupancy condition of for polaritons demonstrating as a function BEC. In of atomic in semiconductorWith increasing excitation power, the occupancy first increases their energy.BEC or The superfluid occupancythe helium, experimental is computed the temperature by control measuring is para the the parametermeter. intensity The driving large exciton binding linearly, then exponentially, with sharp threshold-like behaviour. It of the signal,the phase taking transition. intoenergy account Here in the CdTe excitation polariton quantum power,radiative wells and recombina- (25 thus meV), the injected combined with the large microcavity should be noted that the occupancy at threshold is close to unity, tion ratepolariton and thedensity, efficiencynumber is an of easilythe of collection quantum tunable parameter, set-up. wells in The the and uncertainty microcavity, so we chose it is as crucial in maintain- consistent with a polariton relaxation process stimulated by the may bethe estimated experimental to being roughly control the strong a factor para couplingmeter. of two. The regime The large estimation of exciton polaritons has binding at high carrier density. ground-state population: a specific feature of bosons. The emission been performedenergy in for CdTe differentThe quantum far-field detunings wells polariton (25 and meV), the emission threshold combined pattern is with always thewas large measured to probe the blue shift was measured to be less than a tenth of the Rabi splitting at observednumber for occupancies of quantumpopulation of wells the order in thedistribution of microcavity, one, in agreement along is crucial the with inlower maintain- all polariton branch. The a pumping level ten times above the nonlinear threshold, confirming previousing measurements. the strong couplingspatially For the regime resolved sake of of simplicity, emissionpolaritons and we at high have its coherence carrier arbitra- density. properties are accessible that the microcavity is still in the strong coupling regime. We rily adjustedThe far-field the ground-state polaritonin a real-space emissionoccupancy imaging pattern to be wasone set-upmeasured at threshold. combined to probe At with the an actively stabilized measured the ordinary lasing excitation threshold and found it to very lowpopulation excitation distributionpower, the polariton along the occupancy lower polariton is not therma- branch. The be 50 times higherquantum than the condensation well threshold inside (not shown). resonatorlized29,32,33spatially. Close resolved to threshold, emission the and occupancy its coherence can properties be fitted withare accessible a The linewidth of the k 0 emission shows significant narrowing Maxwell–Boltzmannin a real-space distributionimagingdispersion set-up,indicatingapolaritongasin combined withrelation an actively stabilized k ¼

Figure 3 | Polariton occupancy measured at 5 K. a, Occupancy of the k 0 excitation powers. For each excitation power, the zero of the energy scale ground state (solid black diamonds), its energy blue shift (solid greenk ¼ corresponds to the energyFigure of the 2k | Far-field0 ground emission state. The measured occupancy at 5 is K for three excitation k ¼ circles) and linewidth (open red triangles) versus the excitation power. The deducedFigure from far-field 2 | Far-field emissionintensities. emission data measured (seeLeft Fig. panels, 2b), at 5 taking0.55 K forP threethr into; centre account excitation panels, the P thr; and right panels, condensation 22 blue shift is plotted in units of the Rabi splitting Q 26 meV. At low radiativeintensities. lifetime of polaritons.Left panels,1.14 P Atthr 0.55 the; whereP excitationthr; centreP thr threshold, panels,1.67 kWP thr the cm; and polaritonis right the panels, gasthreshold power of ¼ 22 ¼ excitation densities, the ground-state occupancy increases linearly with the is fully thermalized,1.14 P thr; where as indicatedPcondensation.thr 1.67 by the kW Boltzmann-like cma, Pseudo-3Dis the threshold exponential images power of the decay of far-field of emission within the excitation and then, immediatelyFigure 1 | Microcavity after threshold, diagram increases and energy exponentially dispersion. thea, A distribution microcavitycondensation. function. is a a, Pseudo-3Dangular Above¼ threshold, cone images of ^23the of8, the withground far-field the state emission emission becomes intensity within displayedthe on the vertical axis before becomingFigure 1 linear| Microcavityplanar again. Fabry–Perot This diagram sharp and transition resonator energy dispersion. is with accompanied two Bragga, A microcavityby mirrors a atmassively is resonance a angular occupied, with cone whereas of ^23(in8, the arbitrary with excited the emission units). states Withare intensity saturated, increasing displayed which excitation on is the typical vertical power, axis a sharp and intense peak decrease ofplanar the linewidth Fabry–Perotexcitons by about resonator in quantum a factor with of wells two two, Bragg (QW). corresponding mirrors The exciton at resonance to an is an opticallywithof BEC. active(in The arbitrary polariton dipole units). thermalis With formed cloud increasing in is the found centreexcitation to beof the power,at 19 emission K a without sharp distribution and intense peakv v 08 ; ð x ¼ y ¼ Þ increase ofexcitons the polariton in quantumthat coherence. results wells from (QW). Further the The Coulomb increase exciton in interaction is linewidth an optically is between due active to dipoleansignificant electronis in formed changes the in when the centre increasing of the the emission excitation distribution to twicev thex thresholdvy 08 ; b a corresponding to the lowest momentumð ¼ ¼ stateÞ k 0. , Same data as in interactionthat between results polaritons fromconduction the Coulomb in the band condensate. and interaction a hole The in between the polariton valence an electron ground band. in In the microcavitiespower. Thecorresponding operating low-energy to partbut the of lowestresolved the polariton momentum in energy. occupancy state Fork such cannot0. a measurement,b, usually Same data be ask ¼ a in slicea of the far-field conduction band and a hole in the valence band. In microcavities operating but resolved in energy. For such a measurement,k ¼ a slice of the far-field state slightly blue shifts,in by the less strong than 7%coupling of the regimeRabi splitting of the for light–matter densities up interaction,properly 2D fitted excitons by a Bose distributionemission corresponding function, as expected to vx 0 for8 is BEC dispersed of by a spectrometer and to seven timesin the the strong thresholdand coupling 2D density, optical regime staying modes of the well give light–matter below rise to the new interaction, uncoupled eigenmodes, 2D excitons calledinteracting microcavityemission particles. corresponding Theimaged error bars to onvx indicate a charge-coupled08 is standarddispersed deviation by device¼ a spectrometer (CCD) for each camera. and The horizontal axes and 2D optical modes give rise to new eigenmodes, called microcavity imaged on a charge-coupled device¼ (CCD) camera. The horizontal axes exciton (E X) and photonpolaritons. mode (Ebph,) Energy energies. levels This as provides a function clear of evidence the in-planepoint; wavevector and thek absolutein a uncertaintydisplay the in emission occupation angle factor (top and axis) polariton and the in-plane momentum (bottom polaritons. b, Energy levelsb as a function of the in-plane wavevector k in a displayk the emission angle (top axis) and the in-plane momentum (bottom of the strong couplingCdTe-based regime. , Polariton microcavity. occupancy Interaction in ground- between and excitonk andenergy photon is given modes, as the blackaxis); scale the bars. vertical axis displays the emission energy in a false-colour scale CdTe-based microcavity. Interaction between exciton and photon modes, axis); the vertical axis displays the emission energy in a false-colour scale excited-state levels is plottedwith parabolic in a semi-logarithmic dispersions (dashed scale for curves), various gives rise to lower and upper with parabolic dispersions (dashed curves), gives rise to lower and upper (different for each(different panel; the for units each for panel; the colour the units scale are for number the colour of counts scale are number of counts polariton branchespolariton (solid branches curves) (solid with dispersions curves) with featuring dispersions an anticrossing featuring anon anticrossing the CCD camera,on thenormalized CCD camera, to the integration normalized time to the and integration optical411 density time and optical density typical of thetypical strong of coupling the strong regime. coupling The excitation regime. Thelaser excitation is at high energy laser is atfilters, high energy divided byfilters, 1,000 so divided that 1 bycorresponds 1,000 so that to the 1 levelcorresponds of dark counts: to the level of dark counts: and excites freeand carrier excites states free carrierof the quantum states of well. the quantum Relaxation well. towards Relaxation the towards1,000). Below the threshold1,000). (left Below panel), threshold the emission (left panel), is broadly the emissiondistributed is in broadly distributed in exciton levelexciton and the level bottom and of the the bottom lower polariton of the lower branch polariton occurs by branch occursmomentum by and energy.momentum Above and threshold, energy. the Above emission threshold, comes thealmost emission comes almost acoustic andacoustic optical phonon and optical interaction phonon and interaction polariton scattering. and polariton The scattering.exclusively The from theexclusivelyk 0 lowest from energy the k state0 lowest (right panel). energy A state small (right blue panel). A small blue radiative recombinationradiative recombination of polaritons results of polaritons in the emission results inof photonsthe emission that of photonsshift of about that 0.5 meV,orshiftk of¼ 2%about of the 0.5 Rabi meV,or splitting,k ¼ 2% of is the observed Rabi splitting, for the ground is observed for the ground can be used tocan probe be used their to properties. probe their Photons properties. emitted Photons at angle emittedv correspond at angle vstate,correspond which indicatesstate, that which the microcavity indicates that is still the in microcavity the strong couplingis still in the strong coupling to polaritonsto of polaritons energy E and of energy in-planeE wavevectorand in-planek wavevectorE="c sinvk: E="c sinregime.v: regime. k ¼ð Þ k ¼ð Þ 410 410 78 R. M. Godun et al.

Figure 1. A Mach ± Zehnder light interferometer where the beam splitters and mirrors separate and recombine the paths of the light waves.

In the situation pictured above, the two paths are the same length and so the interfering waves will add together in phase with each other, that is U 0. The detector will ˆ record a spot of bright light resulting from this constructive interference. If, however, the path length diŒerence is changed, the phase diŒerence between the superposing waves, U, will vary. The detector will then record the intensity of the light as varying between bright and dark, giving rise to a set of interference fringes as shown by the solid line in ®gure 2. To look at just one of many examples of how this device can be useful, we consider how the ! Mach ± Zehnder interferometer in ®gure 1 can be used as an ! instrument for measurin78 g rotations. The principlR. M. Godun et al. e is the same as that of a Sagnac interferometer which is described in [4]. Suppose that the device is subjected to a rotation Figure 2. The solid line shows the interference fringes as the about an axis perpendicular to the page. When stationary, Atom interferometry phase diŒerence between the two interfering waves is scanned in the light waves from the two interferometer arms add a stationary interferometer. The dotted line shows the same scan together in phase, as described above. When the instrument for a rotating interferometer. The phase oŒset from the interferometer on an atom-chip is rotating, however, the two paths to the detector acquire stationary interferometer’s fringes is DU. ARTICLES an additionaARTICLESl phasatome diŒerence interferometer, DU, proportiona vs.l to the rate optical interferometer of rotation. The resultinFigure 1.gA Macinterferench ± Zehnder light interferometee patterr where the beanm splittersees annd mirrorats separattheand recombine the paths of the light exploring the intrinsic phase dynamics in complex interacting a a b waves. 80 Atom chip quantum systems (for example, Josephson oscillations20,21)or detector is the78 n R. M. Godun et al. where k and c are the wavelength and speed of the light and the influence of the coupling to an external ‘environment’ I d.c. 45° RF In the situation pictured above, the two paths are the (decoherence22). Technologically, chip-based atom interferometers wire wire same length and so the interfering waves will add together A is the area enclosed by the two paths of the inter-

23 m) 60 in phas2e with each other, that is U 0. The detector will promise to be very useful as inertial sensors on a microscale . µ ˆ Itotal 2recorEd a spo1t of brighcost light resultinUg from DthisUconstructiv, e ferometer. A measurement of the phase shift DU and a For all of these applications it is imperative that the deterministic 50 µm interference0 . If, however, the path length diŒerence is y Gravity / ‰ ‡ … ‡ †Š coherent quantum evolution of the matter waves is not RF field changed, the phase diŒerence between the superposing waves, U, will vary. The detector will then record the knowledge of k and A allow the rate of rotation X to be 40 perturbed by the splitting process itself. Although several Prospects fointensitr atoy ofmthe lighinterferometrt as varying between brighyt and dark, x d.c. 79 atom-chip beam-splitter configurations have been proposed and Trap separation ( and will be oŒset fromgivinthg risee tostationara set of interferenceyfringeinterferometes as shown by the r fringes determined. The device can therefore be used as a magnetic experimentally demonstrated12,24–27,noneofthemhasfulfilledthis b solid line in ®gure 2. To look at just one of many examples trapping by DU, as shown by thof hoewdottethis device dcan linbe usefule , inwe conside®gurr howethe2. gyroscope with a sensitivity which depends on how crucial requirement. 20 when defield Broglie postulated that any particle witMach h ± Zehnder interferometephaser in ®shifgure 1 catn beisused asgivean n by equation (1). If, however, an We present an easily implementable scheme for a phase- instrument for measuring rotations. The principle is the The phase shift duesamtoe as tharotatiot of a Sagnac interferometen at rangulawhich is describerd velocity X is precisely the resulting interference fringes allow DU to be preserving matter-wave beam splitter. We demonstrate momentum, p, should have an associated wavelengthin ,[4]. Suppose thainterferometet the device is subjected to ra rotatioofn equivalent area is built using matter 123 Figure 1. A Mach ± Zehnder light interferometer where the beam splitters and mirrors separate and recombine the pathFigurs ofe th2.e lighThte solid line shows the interference fringes as the RF current RF d.c. current d.c. about an axis perpendicular to the page. When stationary, experimentally, for the first time, coherent spatial splitting and z RF frequency (MHz) Magnetic given by waves. phase diŒerence between the two interfering waves is scanned in the light waves from the two interferometer arms add measured. kd B , relatebias fieldd through his renowned equation waves then the measurea stationardy interferometerphase. Thshife dottedtlinewilshows lthe besame scan subsequent stable interference of matter waves on an atom together in phase, as described above. When the instrument for a rotating interferometer. The phase oŒset from the 40 In the situation pictured above, the two paths are the c d is rotating, however, the two paths to the detector acquire stationary interferometer’s fringes is DU. chip. Our scheme is exclusively based on a combination of x Imaging same length and so the interfering waves will add together Until the 1920s, no one had ever considered making such Data points in phase with each other, thaant isadditionaU 0. Thel detectophase rdiwilŒerencel , DU, proportional to the rate static and radio-frequency (RF) magnetic fields forming an ˆ h record a spot of bright light resultinof rotationg from thi. Ths constructive resultin4pe g interference pattern seen at the 4p 28 35 interferometers with matter sources because the wave-like adiabatic potential . The atomic system in the combined magnetic Non-interacting kdB , interference. If, howeverD, thUdetectoe patlighth rlengtis theh dinŒerence is X A, DUwhereatomk and c are the1wavelengtXh and speeAd of the light and 2 c point sources changed, the phase diŒerence between the superposing A is the area enclosed by the two paths of the inter- fields can be described by a hamiltonian with uncoupled ˆ p waves, U, will vary. The detector will thenˆrecordkthce2 ˆ k…dB†v … † Itotal 2E 1 cos U DU , ferometer. A measurement of the phasnature shift DU ande aof matter was not known. All this changed in 1924 m) 30 Interacting intensity of the light as varying between bright and/dark,0 ‰ ‡ … ‡ †Š adiabatic eigenstates, so-called dressed states. For sufficiently strong µ giving rise to a set of interference fringes as shown by the knowledge of k and A allow the rate of rotation X to be wave packets 100 µm amplitudes of the RF field, transitions between the adiabatic levels solid line in ®gure 2. To look anat jusd wilt onlebeof oŒmanseytexamplefrom ths e stationary interferometer fringes determined. The devickce can therefore be used as a 25 Vbar of how this device can be usefulby D,Uwe, asconsideshowrnhobyw ththee dotted line in ®gure 2. gyroscope with a sensitivitDy whicUh depends on how are inhibited as the strong coupling induces a large repulsion d Mach ± Zehnder interferometer in ®gure 1 can be used as an light RF (Sagnac)The phase shift due to rotation at angular velocity X is precisely the resulting interference fringes allow DU to be instrument for measuring rotations. The principle is the ˆ kdBv Energy where h is Planck’s constant. Typical de Broglie wavegiven- by of these levels. This property of the combined static and RF same as that of a Sagnac interferometer which is described measured.

Fringe spacing ( 20 in [4]. Suppose that the device is subjected to a rotation Until the 1920s, no one2had ever considered making such fields could be exploited to form trapping geometries using Figure 2. The solid line shows the interference fringes as the about an axis perpendicular to the page. When stationary, 4p mc 29 lengths for atoms at room temperature are about 1000 DU phasXe diŒerencA, e between the two interferin1 g waveinterferometers is scanned in s with matter sources because the wave-like the effective potential acting on the dressed eigenstates , and the light waves from the two interferometer arms addlight ˆ kc … † a stationary interferometer. The dotted line shownaturs the esamofe scamatten r was not knownDU. Allightl this change, d in 1924 15 together in phase, as described above. When the instrument for a rotating interferometer. The phase oŒset from the ± related demonstration experiments with thermal atoms have times smaller than wavelengths of visible lightis rotating, makin, however, thegtwo pathits to the detector acquire stationary interferometer’s fringes is DU. ˆ hx been performed30. Position Position an additional phase diŒerence, DU, proportional to the rate di￿cult to detect eŒects related to this wave-likof rotatione. Thnaturee resulting interferenc. e pattern seen at the In a more general case than considered in ref. 29, the 10 detector is then where k and c are the wavelength and speed of the light and 62 64 66 68 70 72 atoms are 11 orders of A is the area enclosed by the two paths of the inter- orientation, in addition to the intensity and frequency, of the RF amplitude (mA) I 2E2 1 cos U DU , ferometer. A measurement of the phase shift DU and a Nevertheless, experimental evidence of matter wavetotals /cam0 ‰ ‡ e… ‡ †Š where k is the de Broglie wavelength and v is the atomic knowledgdeBof k and A allow the rate of rotation X to be RF field determine the effective adiabatic potential Veff at the Figure 1 Operation principle of the beam splitter. a,Astraightwirecarryinga and will be oŒset from the stationary interferometer fringes determined. Thmagnitudee device can therefore be used as a more sensitive position r: static (d.c.) current ( 1A)isusedtotrapaBEConanatomchipdirectlybelowajust three years later in 1927 when Davisson andby DGermeU, as shown by therdotte[5d lin]e in ®gure 2. speedgyroscop. The witeh ratia sensitivitoy whicofh dependthes onmeasurehow d phase shifts in the matter ∼ The phase shift due to rotation at angular velocity X is precisely the resulting interference fringes allow DU to be Figuresecond 2 The splitting wire carrying of BECs a RF is controlledcurrent ( over60 mA a wide at 500 spatial kHz). range. The d.c.By wire adjusting has a amplitude width and frequency of the RF field, we have been able to reach splittinggiven by measured. 2 ± demonstrated electron diŒraction from a crystal lattice. The Until the 1920s, no one had ever considered making such x 2 2 distances of up to 80 µm. a,Acomparisonofthemeasuredsplittingdistances(redcircles)tothetheoreticalexpectation(blacklines)yieldsgoodagreementforthree∼ and light interferometerthan photonss is therefore mc /h , where m is V (r) m µ g B (r) hω µ g B (r)/2 . of 50 µm, it is separated by 80 µmfromtheRFwire(width10µm). Placing the trap 4p interferometers with matter sources because the wave-like eff F B F d.c. RF B F RF 1 DUlight X A, 1 = [ − ¯ ] +[ ⊥ ] different80 strengthsµmfromthechipsurfaceattheindicatedpositionallowsforsymmetric of transverse confinement (gradients 1.1,1.9wav and 2.e4kGcmnatur− ,toptobottom).e of b,Theexperimentaldataarederivedfromatoms was ®rst observein situ absorptiond imagesin (Roper1930 ˆbykc th…e† particlnature of matteer wasmasnot knowns. Allinthis changea dmattein 1924 r interferometer and x is the Scientific MicroMAX: 1024BFT). c,ThefringespacingisplottedasafunctionofRFamplitude(redcircles).Asimpleapproximationoftheexpectedfringespacingbasedonan ! horizontal splitting. b,Topviewontotheatomchip(mountedupsidedowninthe Here mF is the magnetic quantum number of the state, gF is theexpansion of a non-interacting gas from two points located at the two double-well minima agrees well with the data for sufficiently large splittings (solid line). For small experiment): an elongated BEC is transversely split. AllEsterman images are takenn alongan thed Stern who diŒracted helium atoms from light frequency in an optical interferometer. This ratio Lande´ factor, µB is the Bohr magneton, h is the reduced Planck’ssplitting distances (large fringe spacings), inter-atomicT. interactionsSchumm affect et the al., expansion Nature of the Phys. cloud. A numerical1 (2005) integration 57. of the time-dependent Gross–Pitaevskii e.g. R.M. Godun et al., Contemp. Phys. 42 (2001) 77. ¯ indicated direction. c,Left:theRFmagneticfieldcouplesdifferentatomicspin constant, Bd.c. is the magnitude of the static trapping field andequation using our experimental parameters takes this effect intosingl accounte (dottedcrysta line). dl,Interferencepatternsobtainedafter14mspotential-freetime-of-flightexpansionsurfaces of NaCl [6]. Neutron matter waves suggests that atom interferometers could make measure- states (only two shown for simplicity). Right: the initial d.c. trapping potential is ωRF is the frequency of the RF field. BRF is the amplitude of theof the two BECs. For splittings below our imaging resolution (d < 6 µm), the splitting distances can be derived from these interference patterns. 1 1 component of the RF field perpendicular⊥ to the local direction of deformed to an effective adiabatic potential under thewer influencee late of the RFr fieldobserve with a d in the 1940s through re¯ection [7], ments with 10 times more accuracy than light inter- frequency below the Larmor frequency at the trap minimum ( 1G).Inthevertical the static trapping field. The directional dependence of this term ∼ (y)direction,thespatiallyhomogeneousRFcouplingstrengthleadstoaslightdiŒraction [8] and interference [9] of neutron beams. ferometers. These huge gains have yet to be realized as implies the relevance of the vector properties of the RF field, whichsinusoidally alternating current through this wire provides the RF well by fine-tuning the position of the original trap relative to enables the formation of a true double-well potential. field thatrelaxation splits of the the static trap. trap For (dashed smallsplitting green line). distances Along theWit horizontal(<6 µhm)th (x we)direction,thee abilitthe RFy wire.to manipulate electron, neutron or atom atom interferometers have not achieved areas or particle Figure 1 illustrates the operation principle of the beam splitter.rampadditional the amplitude effect of oflocal the variations RF current of the fromRF coupling zerobreaks to its finalthe rotational value symmetryTo characterize the splitting, the split cloud is detected in situ. A standard magnetic microtrap5 is formed by the combined fields(typicallyof the 60–70 trap and mA) allows at for a constant the formation RF frequency of a double-well (wave500 potential kHz).s withit This ashoul well We ared ablethe ton splitbe BECspossibl over distancese to of upbuil to 80d µmattemwithoutr wave ¯uxes comparable with light interferometers. However, the ∼ of a current-carrying trapping wire and an external bias field;frequencyseparation is slightlyd and potentialbelow the barrier Larmor height frequencyVbar (solid blue ofanalogue the line). atoms at thes ofsignificantlight lossinterferometers or heating (determined, in time-of-flightbut how imaging).do atoms ®eld of atom interferometry is still developing and the a static magnetic field minimum forms where atoms in low-minimum of the static trap ( 1Gcorrespondingto 700 kHz). By The measured splitting distances are in very good agreement with ∼ ∼ field-seeking states can be trapped. An RF field generated by anapplying the ramp, we smoothly split a BEC confinedcompar in the single-e wittheh theoreticalelectron expectationss and forneutron differents configurationsfor use ofin the inter- techniques are rapidly improving. independent wire carrying an alternating current couples internalwell trapNote into that two. in our The configuration splitting is performed the magnetic transverselyferometry near-field to the part? Atom ofinitial the singles hav welle (Fig.large 2a). r masses and hence smaller de atomic states with different magnetic moments. Owing to thelongRF axis field of the completely trap, as shown dominates. in Fig. 1b. The distance between To study the coherence of the splitting process we recombine strong confinement in a microtrap, the angle between the RF fieldthe two wellsWe complete can be further the interferometer increased by raising sequenceBrogli the frequency ande measurewavelengththe the split cloudss (fo in time-of-flightr a give expansionn velocity after a non-adiabatically) than electrons and the local static magnetic field varies significantly over shortof therelative RF field phase (up to between 4 MHz the in our split experiment). BECs by recombining The atoms are the cloudsfast (<50 µs) extinction of the double-well potential. Typical 3. Designing an atom interferometer distances, resulting in a corresponding local variation of the RFdetectedin time-of-flightby resonant absorption expansion. imaging In (see our Fig. experimentsan 2) alongd neutron the weweak foundsmatter-wavean an d thi interferences make patternss ato obtainedm interferometr by taking absorptiony techni- coupling strength. By slowly changing the parameters of the RFtrappinginterference direction, pattern that is, integrating with a fixed over phase the long as axis long of asthe the one- twoimages wells 14 ms after releasing the clouds are depicted in Fig. 2d. dimensional clouds. The images are either takencallin situy mucor afterh morThe transversee di￿cult density. profileHowever derived, fromit theseis advantageou images contains s to Just as with light interferometers, the key components of current we smoothly change the adiabatic potentials and transform are not completely separated. The phase distribution remains non- time-of-flight expansion. information on both the distance d of the BECs in the double- random and its centre starts to evolve deterministicallyuse atoms oncerathe the r than electrons or neutrons for several atom interferometers are the source and the elements to a tight magnetic trap into a steep double well, thereby dynamically Unbalanced splitting can occur owing to the spatial well potential and the relative phase φ of the two condensates. We splitting a BEC without exciting it. We accurately control theinhomogeneitywells are entirely of the separated RF field, soowing that to tunnelling asymmetriesreasons is fully in. inhibited theFirstlydetermine on, atomi the fringec source spacing "zsandar thee phasereadilφ byy fittingavailabl a cosine e and manipulate the waves such as the mirrors and beam splitting distance over a wide range. The potential barrier betweenstaticall magnetic experimental trap and timescales. owing to gravity. Although the splitting function with a gaussian envelope to the measured profiles (Fig. 3). the two wells can be raised gradually with high precision, thusprocess itselfThe is experiments very robust, imbalancesare performed lead to inatom a the rapid following evolutions can way.beForprecisel We large splittingsy controlle (d > 5 µm ford ourby experimentalelectromagneti parameters),c ®elds splitters. Let us begin by considering the source. 20 5 enabling access to the tunnelling regime as well as to the regimeof theroutinely relative prepare phase of BECs the of two up condensates to 10 rubidium-87 once they atoms are inthe the fringe spacing is given by "z ht/md, where h is Planck’s ∼ = of entirely isolated wells. separated.F m TheF influence2 hyperfine of gravity state in can microtraps be eliminatedactin near byg the splittingon surfacetheiconstant, ofr aninternat is thel states expansion. timeThi ands controm is the atomicl is essentia mass. Thisl to be In light optics, lasers are the ideal source for many = = 31 32 The beam splitter is fully integrated on the atom chip, asthe trapatom horizontally. chip . Our In smooth the experiment microwires we balanceablenablee the usto double to createguidapproximatione purethe matte of a non-interactingr waves gasaroun expandingd fromprecis two pointe paths. interferometry experiments because they have large coher- the manipulating potentials are provided by current-carrying one-dimensional condensates (aspect ratio 400) with chemical microfabricated wires. The use of chip-wire structures allows one potential µ hω in a trap with highAlthoug transverse∼ h confinementelectrons are also readily available and can be ence lengths, are well collimated and have a high photon to create sufficiently strong RF fields with only moderate currentsnature physics(ω VOL2π 1 OCTOBER2∼.1 kHz;¯ ⊥ 2005 refswww.nature.com/naturephysics 33,34). By tilting the external bias field, we 59 and permits precise control over the orientation of the RF field. position⊥ = the× BECs directly below an auxiliarymanipulate wire© 200 (Fig.5 Nat 1).ured P Aub smalleasilylishing Gro,uptheir electric charge results in interac- ¯ux. The coherence length is the maximum distance along tions both with other electrons and with stray external the wave over which the phase at all points has a well Untitled-1 3 9/22/05, 8:06:52 PM 58 nature physics VOL 1 OCTOBER©! 2005!""#®elds!www.nature.com/naturephysicsNature. PubliThishisng leadGroup! s to unwanted phase shifts in an electron de®ned relationship. Two points separated by more than © 2005 Nature Publishing Group interferometer. Neutrons have the advantage over electrons this distance will not have a ®xed phase relationship

Untitled-1 2 that they9/22/05,do 8:06:51no PMt interact electromagnetically with each between them and interference cannot be observed with a ©!!""#!Nature Publishing Group! other; however, they are not as useful as atoms because detector which has a response time longer than the time they do not interact with applied electric ®elds. Addition- scale of the phase ¯uctuations. Larger coherence lengths ally, neutron sources could not be incorporated into a therefore allow larger path diŒerences between the inter- table-top experiment. ferometer arms before the interference fringes start to Atom interferometers also have several advantages over disappear. The fringes in ®gure 2, for example, would wash light interferometers. The range of physical phenomena out after fewer periods if the coherence length of the source which can be probed by matter interferometers goes were smaller. For accurate measurements of the phase shift, beyond the possibilities available to light interferometers. DU, it is desirable to take readings over as many fringes as For example, properties of the particles themselves such as possible and therefore a large coherence length is required. electric polarizabilitie s or collision cross-sections can be High photon ¯ux also improves the sensitivity of an probed. Gravitational interactions can also be explored interferometer as it increases the signal-to-nois e ratio in the since the interfering waves have mass. Additionally, atom interference fringes. The lower the noise in the results, the interferometers have the potential to measure phase shifts more accurately the phase of the interference fringes is much more accurately than light interferometers. Consider de®ned and the more precisely DU can be measured. once again the example of a rotating Mach ± Zehnder In atom optics, there are two main choices of atomic interferometer. If the interfering waves are light, then the source: atom beams and cold atom clouds. Atom beams are REPORTS

40 ns. As tI is progressively increased, Gilbert devices with functionalities that are not possible 12. M. Hayashi et al., Phys. Rev. Lett. 97, 207205 (2006). damping causes the amplitude of the oscillations with charge-based devices. 13. E. Saitoh, H. Miyajima, T. Yamaoka, G. Tatara, Nature 432, 203 (2004). to decrease, so that depinning is observed only in 14. L. Thomas et al., Nature 443, 197 (2006). an increasingly narrower range of tI and tp. References and Notes 15. S. S. P. Parkin, U.S. Patent 6,834,005 (2004). In good agreement with the 1D model, the 1. Z. Li, S. Zhang, Phys. Rev. B 70, 024417 (2004). 16. A. P. Malozemoff, J. C. Slonczewski, Magnetic Domain Walls in Bubble Material (Academic Press, New York, 1979). probability of DW depinning on tI occurs out of 2. G. Tatara, H. Kohno, Phys. Rev. Lett. 92, 086601 (2004). phase for unipolar and bipolar pulses. For uni- 3. S. E. Barnes, S. Maekawa, Phys. Rev. Lett. 95, 107204 17. More details are available as supporting material on (2005). Science Online. polar pulses, the second pulse leads to DW 4. S. Zhang, Z. Li, Phys. Rev. Lett. 93, 127204 (2004). 18. The second resonance occurs for pulse lengths of 3/2 depinning when the interval is a multiple of the 5. A. Thiaville, Y. Nakatani, J. Miltat, Y. Suzuki, Europhys. oscillation periods, which is three times longer than that precession period, whereas for bipolar pulses, Lett. 69, 990 (2005). of the first resonance (1/2 oscillation period). depinning is observed for odd multiples of half 6. N. Vernier, D. A. Allwood, D. Atkinson, M. D. Cooke, 19. We acknowledge financial support from Defense Microelectronics Activity. the precession period. R. P. Cowburn, Europhys. Lett. 65, 526 (2004). 7. A. Yamaguchi et al., Phys. Rev. Lett. 92, 077205 Supporting Online Material By using the concept of resonant amplifi- (2004). www.sciencemag.org/cgi/content/full/315/5818/1553/DC1 cation, DWs can be excited and moved with 8. M. Klaui et al., Phys. Rev. Lett. 95, 026601 (2005). Materials and Methods much reduced power, and, moreover, by tailoring 9. M. Klaui et al., Phys. Rev. Lett. 94, 106601 (2005). Fig. S1 pinning potentials, individual DWs in neighbor- 10. M. Yamanouchi, D. Chiba, F. Matsukura, H. Ohno, Nature Reference 428, 539 (2004). ing sites can be addressed. These results thus 11. D. Ravelosona, D. Lacour, J. A. Katine, B. D. Terris, 16 November 2006; accepted 16 January 2007 facilitate magnetoelectronic memory and logic C. Chappert, Phys. Rev. Lett. 95, 117203 (2005). 10.1126/science.1137662

PHYSICAL REVIEW LETTERS week ending sation in three dimensions,PRL 96, for130404 example, (2006) is in the 7 APRIL 2006 Critical Behavior of a Trapped same universality class as a three-dimensional XY model for magnets. Moreover,Noise the physics Thermometry with Two Weakly Coupled Bose-Einstein Condensates Interacting Bose Gas of quantum phase transitions occurring at zero temperature can often be mappedRudolf onto Gati, ther- Bo¨rge Hemmerling, Jonas Fo¨lling, Michael Albiez, and Markus K. Oberthaler mally driven phase transitions in higher spatial on March 20, 2012 T. Donner,1 S. Ritter,1 T. Bourdel,1 A. Öttl,1 M. Köhl,1,2* T. Esslinger1 Kirchhoff-Institut fu¨r Physik, Universita¨t Heidelberg, Im Neuenheimer Feld 227, D-69120 Heidelberg, Germany* dimensions. (Received 17 January 2006; published 6 April 2006) The phase transition scenario of Bose- Here we report on the experimental investigation of thermally induced fluctuations of the relative phase The phase transition of Bose-Einstein condensation was studied in the critical regime, where Einstein condensation in a weaklybetween interact- two Bose-Einstein condensates which are coupled via tunneling. The experimental control over fluctuations extend far beyond the length scale of thermal de Broglie waves. We used matter-wave ing atomic gas is unique, asthe it coupling is free strength of and the temperature of the thermal background allows for the quantitative analysis of interference to measure the correlation length of these critical fluctuations as a function of impurities and the two-body interactionsthe phase fluctuations. are Furthermore, we demonstrate the application of these measurements for thermom- temperature. Observations of the diverging behavior of the correlation length above the critical precisely known. As the gasetry condenses, in a regime where standard methods fail. With this we confirm that the heat capacity of an ideal Bose temperature enabled us to determine the critical exponent of the correlation length for a trapped, trapped bosonic atoms of agas macroscopic deviates from that of a classical gas as predicted by the third law of thermodynamics. weakly interacting Bose gas to be n = 0.67 ± 0.13. This measurement has direct implications for number accumulate in a singleDOI: quantum10.1103/PhysRevLett.96.130404 state PACS numbers: 05.30.Jp, 03.75. b, 05.40. a, 74.40.+k the understanding of second-order phase transitions. and can be described by the condensate wave www.sciencemag.org ÿ ÿ function, the order parameter of the tran- REPORTS The generation of two independent matter-wave packets parameters: the temperature of the system randomizing the sition. However, it has proven to be exper- description of the experimental situation, a tion can be approximatedUltra-coldby splitting by 〈C†(r a)C single(0)〉 º Bose-Einsteinwhere n is condensate the criticalphase (BEC) exponent is a of thephase cor- and thetransition tunneling coupling of the two matter-wave hase transitions are among the most Near a second-order phase transition, imentally2 2 difficult to access the physics of Bose gas in a harmonic trap, has remained exp(−pr /ldB), where well-establishedr is the separation technique of relation [1–3] lengthin the andfieldT ofc is atom the critical op- temper-packets stabilizing the phase. The results depicted in striking phenomena in nature. At a macroscopicelusive. quantities show a universal thethe two phase probed transition locationstics. itself. ( New9) (Fig. In phenomena particular, 1). Non- arise theature. if the The two value separated of the parts critical can exponentFig. 1(b) show that the phase fluctuations become more phase transition, minute variations in scaling behavior that is characterized by critical regime has escaped observation be- P We report on a measurement of the trivial fluctuations ofstill the order coherently parameter interactC independs analogy only to Josephson on the universality junctions classpronounced of the as the temperature is increased since the fluc- the conditions controlling a system can trig- criticalcorrelation exponents length (1) of that a trapped depend Bose only gas on closecause to the it critical requires temperaturein an condensed extremely become matter visible close physics andsystem. [4] and superfluid helium tuations outweigh the stabilizing effects. From this point of ger a fundamental change of its properties. generalwithin properties the critical of regime the system, just above such the as tran- its whencontrolled their length approach scaleJosephson becomes to the weak larger critical links than [5]. tem- AnAlthough advantage for of noninteracting the realization systemsview the it is expected—and also experimentally observed For example, lowering the temperature below dimensionality,sition temperature. symmetry The visibility of the of order a matter- pa- theperature. thermal Meanwhile, de Broglieof weakly a wavelength.dvanced coupled theoretical BECThe incritical a double-well exponentsDownloaded from potential can be [6] calculated is the exactly[Fig. 1(c)]—thatthermometry keeping the temperature constant and a critical value creates a finite magnetization rameter,wave or interference range of interaction.statistical pattern gave Accordingly, us direct mechanicsdensitymethods matrix have of increased a homogeneouspossibility and our to understanding observeBose phase gas the phase(1, 12 difference), the presence between of the interactions two increasing adds the tunneling coupling leads to a reduction of of ferromagnetic materials or, similarly, al- phaseaccess transitions to the first-order are classified correlation in termsfunction. of forofr the> ldB criticalcan be expressed regimemacroscopic in by athe gas correlation wave of weaklyfunctionsrichness directly. to the Our physics experimental of the system.the De- fluctuations. A measure for the fluctuations is the lows for the generation of superfluid cur- universalityExploiting classes. our experimentaltransition Bose-Einstein temperature conden- res- offunctioninteracting interacting bosonsinvestigation (2–5). Yetsystem ofa theoretical this relativetermining phase the reveals value that of the it is critical not exponent rents. Generally, a transition takes place olution of 0.3 nK (0.002 times the critical locked to zero but exhibitsthrough fluctuations. Landau Two’s theory fundamental of phase transitions(a) (b) temperature), we observed the divergence of results in a value of n = ½ for the ho- between a disordered phase and a phase ex- types1 of fluctuations are discussed in the literature—quan- the correlation length and determined its † mogeneous system. This value is the result of hibiting off-diagonal long-range order, which Fig. 1. Schematics of the 〈C r C 0 〉 ºtum fluctuationsexp −r=x [7] and1 thermally induced fluctuations [8]. critical exponent n. This direct measurement ð Þ ð Þ r ð ÞðÞ both a classical theory and a mean-field ap- is the magnetization or the superfluid den- correlation function and the In this Letter we report on the experimental investigation correlationof n through length the close single to -particle density (10, 11), where x denotesof thermal the fluctuations correlation ofproximation the relative to phase quantum arising systems. from However, sity in the above cases. Near a second-order matrix complements the measurements of length of the order parameter. The correlation calculations by Onsager (13) and the more phase transition point, the fluctuations of the the phase transition temper- the interaction of the BEC with its thermal environment, atureother of Bose-Einstein critical exponents conden- in liquid He (6–8), length x is a function ofwhich absolute is always temperature present. recent techniques of the renormalization order parameter are so dominant that they which is believed to be in the same univer- T and diverges as the system approaches the group method (1) showed that mean-field sation. Above the critical The essential prerequisite for the investigation of these completely govern the behavior of the system sality class as the weakly interacting Bose phase transition (Fig. 1). This results in the theory fails to describe the physics at a temperature T the condensate thermally induced phase fluctuations is the ability to pre- (c) gas. c algebraic decay of the correlation function phase transition. Very close to the critical on all length scales (1). In fact, the large-scale pare a BEC adiabatically in a symmetric double-well fractionIn is zero,a Bose and gas, for theT >> physicsTc of fluctuations with distance †(r) (0) º 1/r at the phase temperature—in the critical regime—the fluctuations in the vicinity of a transition 〈C Cpotential〉 and to adjust its temperature. In our experiments the correlationof the order function parameter decays is governed by transition. The theory of critical phenomena fluctuations become strongly correlated and already indicate the onset of the phase on the approximately as a Gaussian this is achieved by splitting a single 87Rb BEC produced other side of the transition. different length scales. Far above the phase predicts a divergence of x according to a aperturbativeormean-fieldtreatmentbe- on atransition length scale temperature, set by the classical thermal fluc- power law, and trapped in an optical dipolecomes trap impossible, by slowly making ramping this up regime very a barrier in the center. The tunneling coupling is adjusted 1 thermaltuations de Broglie dominate. wavelength Their characteristic length challenging. Institute of Quantum Electronics, Eidgenössische Tech- by the barrier height and its strength can be deduced from nische Hochschule (ETH) Zürich, CH-8093 Zürich, Switzer- ldB.Asthetemperatureap-scale is determined by the thermal de Broglie Consider a weakly interacting Bose gas −n 2 numerical simulations of the BEC in the trap using the on March 20, 2012 land. Cavendish Laboratory, University of Cambridge, proacheswavelength the criticalldB,andthecorrelationfunc- temper- x º T − Tc =Tc 2 with density n and the interaction strength jð modelÞ describedj inð [9].Þ The temperature of the BEC is Cambridge CB3 0HE, UK. ature, long-range fluctuations parameterized by the s-wave scatteringFIG. length 1. Observation of thermal phase fluctuations. The experi- week ending 1/3 PHYSICAL REVIEW LETTERS *To whom correspondence should be addressed. E-mail: start to govern the system and the correlation length x increases markedly.adjusted Exactly by holding at the criticalthe clouda = 5.3 in nm the in trap, the dilute where limit duen to a < ldB (Eq. of the 1). trap parametersthe critical energy regime, is transferred mean-field to theorysate fails (solid line) is prepared in a double-well potential (dashed of a trapped Bose gas close to the the atoms. Once the final temperaturebecause the is fluctuations reached a standing of C becomeline) more by adiabatically ramping up the barrier. The relative phase critical temperature. Shown is the light wave is ramped up generatingdominant thana barrier its mean in the value. center, Thiscan can be be measured after a time-of-flight expansion by analyzing have confirmed that the heat capacity of a degenerate 1556 16 MARCH 2007visibility VOL of a 315matter-waveSCIENCE interfer- www.sciencemag.org leading to an effectivedetermined double-well by trapping the Ginzburg potential criterionthe resultingx > double-slit interference patterns (black line). (b) Po- Bose gas vanishes in the zero temperature limit as pre- lar plots of the relative phase obtained by repeating the experi- ence pattern originating from two [upper part of Fig. 1(a)]. l2 /(p128p2a) ≈ 0.4 mm(14, 15). Similarly, dB ment up to 60 times. The graphs show measurements for four dicted by the third law of thermodynamics [20]. regions separated by r in an atomic When the potential isthese switched enhanced off, the fluctuations matter-wave are responsi- cloud just above the transition different temperatures T at constant tunneling coupling energy We thank T. Bergeman very much for the numerical packets start to expand,ble overlap, forffiffiffiffiffiffiffiffi a andnontrivial form ashift double-slit of the critical tem- www.sciencemag.org temperature. The gray line is a Ej, i.e., constant barrier height. The phase fluctuations increase perature of Bose-Einstein condensation calculation of the relevant parameters and the valuable Gaussian with a width given by interference pattern which depends on their relative phase with increasing temperature. (c) Polar plots of the relative phase as indicated in the lower( part2–4, of16 Fig.). The 1(a). critical Repeating regime the of a weakly theoretical support. We would also like to thank B. ldB, which changes only marginally for a constant temperature at four different tunneling coupling for the temperature range consid- interference measurementsinteracting reveals that Bose this gas relative offers an phase intriguingenergies, pos- i.e., different barrier heights. Here the fluctuations are Eiermann and T. B. Ottenstein for discussions and ered here. The experimental data is not constant but fluctuatessibility around to study zero. physics The general beyond thereduced usual with increasing coherent tunneling coupling showing M. Cristiani, Th. Anker, and S. Hunsmann for their con- show phase correlations extending behavior of these phase fluctuationsmean-field approximation is connected to (17 two), whichthe until stabilization. tributions to the experimental setup. This work was funded far beyond the scale set by ldB. The now has been observed in cold atomic gases solid line is a fit proportional to only in reduced dimensionality (18–21). by Deutsche Forschungsgemeinschaft Schwerpunkts- 0031-9007=06=96(13)=130404(4)$23.00In our experiment, we let 130404-1 two atomic Downloaded from  2006 The American Physical Society (1/r) exp(−r/x) for r > ldB. Each programm SPP1116 and by Landesstiftung Baden- data point is the mean of 12 mea- beams, which originate from two different surements on average; error bars areT. ±SD. Donner et al., Science 315 (2007) 1556.locations spaced by a distance r inside the Wu¨rttemberg–Atomoptik. R. G. thanks the Landes- trapped atom cloud, interfere. From the visi- graduiertenfo¨rderung Baden-Wu¨rttemberg for financial bility of the interference pattern, the first-order support. correlation function (22) of the Bose gas aboveFIG. 4. Heating up of a Bose gas. The filled circles correspond Fig. 3. Divergence of the correla- the critical temperature and the correlation tion length x as a function of to measurements employing the phase fluctuation method and ∆Tc 1/3 length x can be determined. temperature. The red line is a fit 687the open circles to the results obtained with the standard time-of- = C0an We prepared a sample of 4 × 10 Rb of Eq. 2 to the data, with n and Tc C0 = 1.3 flight method. The gray shaded region shows the critical tem- atoms in the |F =1,mF = −1〉 hyperfine *Electronic address: [email protected] as free parameters. Plotted is one Tc perature expected from the experimental parameters and their data set for a specific temporal ground state in a magnetic trap (23). The [1] M. R. Andrews et al., Science 275, 637 (1997). trapping frequencies were (wx, wy, wz)=2p × offset t0. The error bars are ±SD, uncertainties. The solid line is a fitting function assuming a V. A. Kashurnikov et al., Phys. Rev. Lett. 87 (39,(2001) 7, 29) 120402. Hz, where z denotes the verticalR. Gati et al., Phys. Rev. Lett. 96 (2006)d 130404.. [2] J. E. Simsarian et al., Phys. Rev. Lett. 85, 2040 (2000). according to fits to Eq. 1. They also power law for the heat capacity C T=Tc of the Bose gas reflect the scattering between dif- axis. Evaporatively cooled to just below the / † [3] T. Schumm et al., Nature Physics 1, 57 (2005). below the critical temperature Tc 59 4 nK, a constant heat ferent data sets. Inset: Double critical temperature, the sample reached a ˆ † [4] A. Barone and G. Paterno, Physics and Applications of the 13 −3 capacity above the critical temperature, and a temperature inde- logarithmic plot of the same data. density of n =2.3× 10 cm ,givingan Josephson Effect (Wiley, New York, 1982). −1 pendent transfer rate of energy. From this fit we deduce d elastic collision rate of 90 s .Thetem- [5] J. C. Davis and R. E. Packard, Rev. Mod. Phys. 74, 741 perature was controlled by holding the atoms2:7 6 , which is consistent with the theoretical prediction of d ˆ (2002), and references therein. in the trap for a defined period of time,3 for † an ideal Bose gas in a three-dimensional harmonic trap.ˆ [6] M. Albiez et al., Phys. Rev. Lett. 95, 010402 (2005). during which energy was transferred to theThe dashed line represents the expected behavior of an ideal atoms as a result of resonant stray light, [7] A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001). classical gas for increasing temperature which makes the differ- fluctuations of the trap potential, or background [8] L. Pitaevskii and S. Stringari, Phys. Rev. Lett. 87, 180402 ence arising from quantum statistics evident. (2001). www.sciencemag.org SCIENCE VOL 315 16 MARCH 2007 1557 [9] D. Ananikian and T. Bergeman, Phys. Rev. A 73, 013604 (2006). can be applied and confirms the consistency of the two [10] J. Javanainen, Phys. Rev. Lett. 57, 3164 (1986). approaches in the overlap region. The solid line corre- [11] M. W. Jack, M. J. Collett, and D. F. Walls, Phys. Rev. A 54, sponds to a fitting function for the temperature where we R4625 (1996). assume a mean critical temperature of Tc 59 nK (de- [12] A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, duced from independent measurements), aˆ temperature Phys. Rev. Lett. 79, 4950 (1997). independent transfer rate of energy per particle and a [13] F. Sols, Physica B (Amsterdam) 194–196, 1389 (1994). power law for the temperature dependent heat capacity [14] A. Imamoglu, M. Lewenstein, and L. You, Phys. Rev. Lett. d 78, 2511 (1997). C d 1 Cth T=Tc where Cth is the heat capacity of a classicalˆ ‡ gas.† The shown† excellent agreement is obtained [15] G.-S. Paraoanu, S. Kohler, F. Sols, and A. J. Leggett, for a heating rate of 2:3 2 nK=s for a classical gas and d J. Phys. B 34, 4689 (2001). [16] R. Gati et al., Appl. Phys. B 82, 207 (2006). 2:7 6 . Thus the expected † exponent d 3 for an ideal Boseˆ † ˆ [17] The atom-atom interaction energy can be estimated by gas in a three-dimensional harmonic trap [19] is experi- V =k 27 nK E =k 51 nK 1 exp ÿ 0 Bÿ =N and the tun- mentally confirmed. The expected increase of temperature c B  ÿ 41 nK †† neling coupling energy by Ej=kB 196 nK of a classical gas is indicated by the dotted line and shows V =k 27 nK   N exp ÿ 0 Bÿ for N 2500 and 24 nK

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