RAPID COMMUNICATIONS

PHYSICAL REVIEW A, VOLUME 63, 061601~R!

Method for collective excitation of a Bose-Einstein condensate

Zbyszek P. Karkuszewski, Krzysztof Sacha, and Jakub ZakrzewskiInstytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagiellon´ski, ulica Reymonta 4, PL-30-059 Krako´w, Poland

~Received 2 February 2001; published 8 May 2001!

It is shown that by an appropriate modification of the trapping potential, one may create collective excitationin a cold atom Bose-Einstein condensate. The proposed method is complementary to earlier suggestions. Itseems to be feasible experimentally—it requires only a proper change in time of the potential in atomic traps,as realized in laboratories already.

DOI: 10.1103/PhysRevA.63.061601 PACS number~s!: 03.75.Fi, 05.30.Jp, 32.80.Pj

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Spectacular experimental realizations of the BoEinstein condensate~BEC! in cooled and trapped atomigases@1–3# stimulated intensive investigations of possibmodifications, control, and manipulations of this new statematter. Here a macroscopic sample of atoms is in a wdefined quantum state. Thus several typically quantumechanical phenomena may now be investigated on a mroscopic level.

As an example of manipulation of the condensate, omay consider the splitting of the condensate into two pa@4#, well separated in space and yet coherent with each oThe latter property may be tested by superimposing, at slater time, the two parts and an observation of the interence fringes@4–6#. Another example is the leakage of atomfrom the condensate that may be used to prepare an ‘‘alaser’’ @7,8#.

Other fascinating possibilities are revealed when one csiders possible collective excitations of the condensate. Seral schemes have been proposed to create either sowaves or vortices in the condensate. Both these typeexcitations are the solutions of the time-dependent GroPitayevsky equation~GPE! as appropriate for the mean-fieldeffective single-particle description of the gas of weaklyteracting bosons in the limit of vanishing temperature~forreviews see@9,10#!. In analogy to nonlinear optics@11#, onemay consider bright solitons~bell shaped structures propagating without dispersion!, dark solitons~with a node in themiddle—an analog of the first excited state in the nonintacting particles picture!, or the intermediate gray solitons.

The early propositions for creation of solitions in the BEutilized collisions between spacially separated condens@12,13#. Soon it was realized that less violent approachesalso possible. Typical for atomic laser control—resonant Rman excitation scheme—the excitation of vortex statesbeen proposed@14#. This approach relies on the resonancondition, which is, however, modified during the transprocess due to the nonlinearity of the GPE. Another pobility that takes the nonlinearity fully into account is thadiabatic scheme of Ref.@15#. It effectively utilizes internalatomic transitions combined with appropriate states ofcondensate for a controlled laser induced adiabatic tranpopulating solitonic or vortex solutions of GPE, dependion the details of the process. The latter approach seemsrobust against typical experimental uncertainties. A yet dferent approach produces a phase shift between two parthe condensate—such a phase imprinting method, origin

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proposed in Ref.@16#, has been actually utilized to creadark solitons both in a cigar shaped BEC@17# and in thespherically symmetric condensate@18#. The same methodhas been successfully applied to create vortices@19#. Thelatter have also been demonstrated experimentally usinlaser stirring approach@20#.

The aim of this Rapid Communication is to propose yanother scheme for effective collective excitation of tBEC. The method is in some sense similar, in another seopposite, to the adiabatic passage of Ref.@15#. In the ap-proach of Ref.@15#, one slowly tunes the laser frequencadiabatically following the levels. The transfer of populatiobetween two internal atomic states is accompanied byappropriate change of the condensate wave function indark soliton, two-soliton, or vortex solution of the GPE.our proposition, discussed below, we consider asingle inter-nal state and sweep the laser across the trap, modifyinthis way the trapping potential.

For an explanation of the effect, we first assume thatcondensate consists of noninteracting particles. While sucondensate is not realized in nature, it may provide a gstarting point for an analysis of the weakly interacting Bogas. We later show that the picture remains valid for intacting particles by considering the numerical example wattractive atom-atom interactions@3#.

To collectively excite a condensate, we are goingmodify the trapping harmonic potential along one of thedependent directions only. Therefore, as the noninteracparticles system is separable, it is enough to consider atomotion restricted to one dimension~the generalization to athree-dimensional case is simple!. Originally, the condensateoccupies the ground state of the trap@9,10,21#. Since weconsider noninteracting particles, it is sufficient to considesingle-particle picture. By imposing a laser beam, beingpropriately tuned off~but close to! the resonance with respect to an internal atomic transition, we may modify ttrapping potential by adding a Gaussian-shaped local we

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In the following, as above, we use the trapping harmonoscillator units, i.e.,vt for time and A\/mv for length,wherev is the harmonic-oscillator frequency, whilem standsfor atomic mass. A similar modification of the potential h

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RAPID COMMUNICATIONS

KARKUSZEWSKI, SACHA, AND ZAKRZEWSKI PHYSICAL REVIEW A 63 061601~R!

been used to split the condensate into two parts@4#—thereinstead of a local well, a potential barrier has been creaWe suggest here to produce such a well on the very edgthe harmonic potential~thus not affecting the condensate!.Then we slowly sweep the well across the potential~by mov-ing the laser beam!, simultaneously decreasing the depththe well ~by adjusting the intensity of the beam!—it corre-sponds, forU0.0, to a change ofx0 from some negativevalue to zero, see Fig. 1.

Assume that a particle is originally in the ground statethe harmonic potential. For a sufficiently slow sweep,levels in the ‘‘time-dependent’’ potential may be followeadiabaticallyexceptin the vicinity of avoided crossings. Byappropriately choosingU0 ands in Eq. ~1!, we may arrangethe situation in which a narrow~with respect to a mean levespacing! avoided crossing between the ground and the fiexcited state of the potential occurs when the local wsweeps the trap; see Fig. 2. If the avoided crossing is narenough, it may be passeddiabatically, and when the loca

FIG. 1. Plots of the potential~1! for U056.4, s50.5, andx0

525.0 ~a!, x0524.0 ~b!, x0522.0 ~c!, andx0520.3 ~d!. Dashedlines give the unperturbed harmonic trapping well.

FIG. 2. Energy levels for a single particle in the potential~1! forU056.4 ands50.5 as a function ofx0. Note the narrow avoidedcrossing between the ground and first excited states arounx0

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potential well disappears, the particle is left with a high proability in the excited state. This is nothing other than tLandau-Zener transition. The Landau-Zener effect has bexplored in the BEC but for the transition of internal~notexternal! atomic degrees of freedom@8#.

Assume that a single-particle probability for the transitifrom the ground to the lowest excited state isp. Then forNnoninteracting particles originally being in the ground staof the potential, the transition probability ofk particles is( k

N)pk(12p)N2k. Thus the mean energy of the condensatethe final state is (p11/2)N\v with variance Np(12p)\2v2. For N sufficiently large~say of the order of thou-sands! andp close to unity, we get at the end of the potentsweep a macroscopic part of the condensate in the excmotional state. The final state of the noninteracting condsate is in fact describable as a time-dependent wave pashowing beats with the frequency of the trapping potentThis is seen from the final single-particle reduced probabidensity, which independently onN, reads as

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wherec0(x) andc1(x) are harmonic-oscillator ground anexcited states~in the real representation!, respectively.

To check whether it is possible to realize an efficietransfer using the method proposed, we have simulatedsituation numerically. Choosing, without any special optimzation attempt, the parameters of the potential~1! as U056.4, s50.5 and changingx0 from 25 to 0 with the ve-locity 0.1, we getp50.97. The final single-particle reduceprobability density of the condensate is then depicted in F3 at various times of its periodic behavior.

As a specific example, we propose in this Rapid Commnication to excite the condensate by sweeping the trapppotential using the local potential well. However, the excition may be realized in different ways—the key point isarrange, in the level dynamics, a narrow isolated avoidcrossing between the ground and excited states.

FIG. 3. Single-particle reduced probability density of the codensate~corresponding to the noninteracting particles model!, Eq.~2!, for p50.97 atvt5p/2 ~solid line! andvt5p ~dashed line!.

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RAPID COMMUNICATIONS

METHOD FOR COLLECTIVE EXCITATION OF A BOSE- . . . PHYSICAL REVIEW A 63 061601~R!

One may argue that the proposed model of noninteracparticles is very simple. The role of the interactions maysubtle. They clearly modify the energy levels of the systeSuch a modification will be felt mostly close to avoidecrossings that may be shifted and broadened. Will this scompletely the proposed scheme? It is our belief that it wnot, although an adjustment of the laser beam intensityother parameters may be necessary to optimize the tranof population. As a test of this assumption, we considerexcitation of the BEC with attractive atom-atom interactias realized for Li atoms@3#. While the one-dimensional approach for interacting atoms is not exact~nonlinearitycouples different degrees of freedom!, a one-dimensional approach based on the GPE is often used and may be justfor asymmetric traps@22–26#.

We integrate the time-dependent GPE,

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with g525 @27#, starting with the condensate in the groustate of the harmonic trap. By adjusting the parameters ofpotential~to compensate the influence of the interaction! tak-ing U0510, s50.3 and changingx0 from 25 to 0 with thevelocity 0.05, we were able to get a 97.5% transfer of polation into a collective state corresponding to the first excistate in the independent particle model. The final wave fution obtained via integration of the time-dependent GPEdepicted in Fig. 4.

It is interesting to compare the present ‘‘diabatic’’ aproach with the adiabatic scheme considered in Ref.@15# ~asthe closest in spirit among the techniques proposed!. Wemust admit that the method of Ref.@15# may be more robusand flexible; in particular, it is adaptable to the excitationvortices and multiple solitons. The latter may also be reized by our method, one needs simply to apply the sweeppotential two or more times~this as well as application torepulsive atom-atom interactions will be considered infuture!. While the method of Ref.@15# uses two internalstates~two-component condensate!, which is the common

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trend also in other treatments of collective excitations, oapproach considers a single internal state. This may bevantageous in some applications. Importantly, also the abatic scheme@15# takes necessarily much longer time for aeffective transfer~of the order of 200 or more periods of thharmonic trap! than our diabatic approach~here a typicaltransfer time is 20 periods!. While such comparisons may bquite encouraging, the best way of verifying our schewould be a laboratory test. Experimental setup requires oslight modifications of the present atomic traps; thus, suchexperiment can be realized immediately.

To summarize, we have proposed a simple schemeenables us to create a collective excitation of the BoEinstein condensate. The proposed scheme may servehope, as an alternative to other proposed and already exmentally used methods.

We are grateful to Maciek Lewenstein and KazRzazewski for the reading of the manuscript and several uful suggestions. J. Z. acknowledges support of KBN unProject No. 2P03B 00915, while research of K.S. is a para KBN proposal.

FIG. 4. The wave packet obtained as a solution of the timdependent GPE, Eq.~3!, at the end of the potential sweeping~solidline! and at the momentvt5p later ~dashed line!.

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