On the transverse mode of an atom laser Th. Busch1[†], M. K¨ohl2,3, T. Esslinger2,3, and K. Mølmer1 1Institute of Physics and Astronomy, Aarhus University, Ny Munkegade, DK–8000 ˚Arhus C, Denmark 2Sektion Physik, Ludwig–Maximilians–Universit¨at, Schellingstr. 4/III, D–80799 Mu¨nchen, Germany 3Institute of Quantum Electronics, ETH H¨onggerberg, CH–8093 Zu¨rich, Switzerland (Dated: February 1, 2008) 3 0 ThetransversemodeofanatomlaserbeamthatisoutcoupledfromaBose–Einsteincondensateis 0 investigatedandisfoundtobestronglydeterminedbythemean–fieldinteractionofthelaserbeam 2 withthecondensate. Sinceforrepulsiveinteractionsthegeometryofthecouplingschemeresembles an interferometer in momentum space, the beam is found show filamentation. Observation of this n effect would provethe transverse coherence of an atom laser beam. a J PACSnumbers: 03.75.Fi,32.80.-t 7 2 Research on atom lasers is an active and fascinating we examine its classical, semiclassical and quantum be- ] t areainatomicphysics[1,2]. Severallaboratoriesaround haviour. After we have identified and described the rele- f o the world are now using continuous output couplers to vant processes, we will include the effects of gravity and s produce atom laser beams from Bose-Einstein conden- propagationin more than one dimension. t. sates. It is therefore important to characterize the qual- Atom lasers with radiofrequency output couplers usu- a ities of these beams. Recently, their temporal coherence ally couple different Zeeman substates of the trapped m wasverified[3]andtheir transversedivergencewasmea- atoms. For a Bose-Einstein condensate in a F = - sured [4]. 1,m = 1 state, two output states are possible|. Ei- d F − i n Although atom lasers and optical lasers show strong ther mF = 0 or mF = 1, the first of which has a van- o similarities, the possibility for atoms to scatter off each ishing magnetic moment and the latter experiences a re- c other leads to various effects absent in optical lasers. In pulsive force by the magnetic trap. However, since the [ the following we will show that a strong,inhomogeneous output coupling rate from the mF = 1 state into the 2 repulsive potential, as is represented by the remaining mF = 0 state is usually chosen to be s−mall, subsequent v Bose-Einsteincondensate,canbeasourceofinstabilities transitions into the mF = 1 state can be neglected. We 5 forthebeamand,inparticular,canleadtoitstransverse therefore restrict our considerations to a two–level sys- 5 filamentation. Since most experimental setups involve tem, where the important coupling parameters are the 5 asymmetric traps with two stiff directions, our results Rabi frequency Ω and the detuning ∆ of the rf–field. 0 are immediately applicable to these experiments. Forweakcouplingthe resonanceconditionisdetermined 1 1 Inmagnetictrapsoutputcouplersforatomlaserbeams by the spatialdependence of ∆within the Bose-Einstein 0 are realized by coupling a fraction of the Bose-Einstein condensate. After a transformation into a co–rotating t/ condensate into a magnetically untrapped state. This frame ψmF(t) → e−imFωrftψmF(t) followed by the stan- a process happens inside the trapped sample along a sur- dardrotatingwave–approximation,theequationsforthe m face where the resonance condition for output coupling condensate wave function ψc and the atom laser beam - is fulfilled, subjecting the output coupled atoms to the wave function ψb are given by a set of coupled Gross– d repulsive mean field potential of the condensate. Since Pitaevskii–equations [6] n o gravitydisplacesthe symmetryaxis ofthe Bose-Einstein c condensatewithrespecttothesymmetryaxisofthemag- i¯h∂∂tψi = −2h¯m2 ∇2ψi+Vi(r)ψi−mF¯hωrfψi : netic trapping field, the repulsive potential is however (1) v i nothomogeneousoverthe resonancesurface. In a classi- +U(ψi 2+ ψj 2)ψi+h¯Ωψj , X | | | | cal picture this situation corresponds to particles rolling r off a potential from different heights, leading to a non– with i,j = c,b and U = 4π¯h2a /m. We have as- a s negligible momentum spread or dispersion in the trans- sumed that all triplet scattering lengths have the same versedirections[5]. Moreover,the finite interactiontime value a = a [7], and we will choose a to be pos- ij s s of the falling beam with the remaining condensate leads itive. The external potentials are given by V (r) = c toanon–monotonicincreaseofthetransverseatomicpo- m ω2(x2+z2)+ω2y2 +mgz and V (r)=mgz, where 2 ⊥ y b sition x(t) as a function of the points of resonantoutput g is the gravitational constant, ω and ω are the trap- ⊥ y coupling xi. In the quantum dynamics, this leads to the pin(cid:0)g frequencies of the c(cid:1)ylindrically symmetric magnetic interference of atoms with different transverse momenta field and m denotes the mass of the atoms. within the beam. It has been found, using a separation ansatz for the In the following we will investigate this interference spatial modes, that the atom laser beam in the direc- process. First we consider an idealized, one–dimensional tion of gravity can be almost perfectly described by an model in the direction perpendicular to gravity, and Airy–Function [8, 9, 10]. We therefore first consider the 2 behaviorinthetransversalx–directionasindependentof x 10-6 x 10-3 the other directions. 4 (a) 5 (b) A naive, strictly one–dimensional treatment of the 4 atom laser in the horizontal x–direction would result in 3 3 exactly two resonance points x = ± 2h¯ωrf/mω⊥2 [11]. xi2 t This, however, is not a good approximation to the three 2 p dimensional situation of the experiment. Since gravity 1 1 leads to a displacement of the condensate from the cen- ter of the magnetic field by an amount zg = −g/ω⊥2, 00 0.5 x 1x 10-5 00 0.5 a 1 the resonance shell crosses the condensate with very low curvature [2] and is therefore better approximated by a FIG. 1: (a) Position of atoms starting at xi(t = 0) < xTF plane. This means that in the horizontal directions out- and rolling off an inverted harmonic oscillator potential for put coupling happens along the full Thomas–Fermi dis- a time of t = 2ms. The full line shows the results for a tribution and the initial beam wave function is a scaled potentialthatistruncatedatxTF =4µm(cf.eq.(3))andthe down copy of the condensate wave function [9]. dashedlineshowstheresultsforanuntruncatedpotential. (b) While the atoms fall under gravity,the mean field po- Time needed for the atoms starting at α = xi/xTF to reach tential they experience from the condensate changes. To the condensate boundary at xTF (cf. eq. (5)). The chemical account for this in the one–dimensional approximation potential in both calculations is µ=1700Hz. one has to diminish the mean field potential, U (x,t) = c U ψ (x,z )2 during the evolution according to the free c t fa|ll of the a|toms, z =z +gt2/2. arrival time distribution for the atoms at the Thomas– t g Fermi edge However, the principal physics of the horizontal mode and its instability is most clearly demonstrated by first 1 1+√1 α2 considering the case with g = 0, i.e. taking Uc(x,t) = tTF(α)= ln − , (5) ω α U (x,z ) to be constant in time. The effective one– ⊥ ! c g dimensional Gross–Pitaevskii equation for the beam is with α = x /x , shows a plateau (see Fig. 1b). Once then given by i TF theatomshavepassedtheThomas–Fermiradiusalltheir ∂ψ ¯h2 ∂2 initial potential energy, E = µ(1 α2), has been trans- i¯h b = +U (x)+U ψ 2 ψ (2) − ∂t −2m∂x2 c | b| b formed into kinetic energy and atoms starting closer to (cid:20) (cid:21) the condensate center (x = 0) therefore end up with In the Thomas-Fermi approximation the mean field po- tential is givenby a truncated invertedharmonic oscilla- tor t=1ms t=3ms ) ) Uc(x)= µ 1− xx2T2F for|x|<xTF, (3) (a.u. (a.u. ( 0(cid:16) (cid:17) for|x|>xTF. ( )x ( )x ρ ρ Here µ is the chemical potential and x = 2µ/mω2 TF ⊥ the Thomas–Fermi radius of the condensate. p Let us first analyse eq. (2) classically and neglect the 0 0.5 1 1.5 0 0.5 1 1.5 2 nonlinear term U|ψb|2, because it is normally three or- x x 10-5 x x 10-5 ders of magnitude smaller than U (x). Neglecting for c a moment the truncation of the potential at the con- densate border, i.e., assuming the potential Uc(x) = u.) u.) µ(1 x2/x2 ), x, the classical equation of motion a. a. − TF ∀ ( ( mx¨= dUc(x) (4) ( )p ( )p ρ ρ − dx can be exactly integrated by x(t) = x cosh(ω t), with i ⊥ x the initial position of the atoms at the time of the 0 2 4 6 8 0 2 4 6 8 i p 6 p 6 outcoupling. Sincethecoshisanexponentiallyincreasing x 10 x 10 functionofthetime,theresultingevolutionisaspreading of the initial distribution. This means that atoms with FIG.2: Transversedensitydistributionsofthebeamobtained larger xi will also have larger x at later times t, (see from aclassical (dashed) and a quantum-mechanicalcalcula- dashed line in Fig. 1a). tionfortwodifferentevolutiontimes. Thefiguresinthelower Thedensitydistributionusingthepotentialineq.(3)is row showtheabsolute squareof theFouriertransform of the showninFig.2(dashedlines). Theatomsshowatempo- quantum mechanical wave function. A strong filamentation rary localization at the Thomas–Fermi radius, since the is clearly visible in the quantumresults. 3 higherfinalvelocity. Intheasymptoticlimit,atomsorigi- 160 natingfromthecenterwillhaveovertakenallotheratoms x10-6 and the density distribution that originally had a nega- 2 120 tive slope will have a positive slope. This can be seen x d from the dashed curves in Fig. 2. The full quantum mechanical behaviour can be found ) 1 by solving Eqs.(1) numerically, for which we use a stan- (x 80 1 5 10 r i dardsplit–operator/FFTtechnique. Ascanbeseenfrom min Fig. 2 (solid lines), althoughthe generalfeature of local- 40 izationispreserved,thedensitydistributionismodulated by an interference pattern. The reason is that for finite times, t > t , atoms with different initial positions ar- TF 0 rive at x at the same time t, because the equation 0 1 2 3 4 x x10-5 x(t)=x 1+ω 1 α2(t t (α)) (6) TF TF − − has more thanone(cid:16)solutiopnfor α, andquantum(cid:17)mechan- FIG. 3: Transverse density distribution for t=5ms of the beam from the quantum–mechanical calculation (solid line) ically, atoms coherently outcoupled at different α’s in- andfromtheanalyticalexpressionbasedonFeynmanpathin- terfere. Observation of these fringes would prove the tegralsasexplainedinthetext(dashedline). Theinsetshows transversalcoherence of the atom laser beam. thedistanceδxbetweensuccessiveminimaimin of|ψ|2,start- A mathematically and conceptually very elegant ing at the Thomas–Fermi radius. The units of x is meters. method to calculate the interference pattern is by use The disagreement in amplitude stems from different initial of path integrals [13]. The unitary wave function prop- conditions dueto thedifferent numerical methodsused. agator is obtained by adding phase factor contributions over all paths x(t) along which an argument x in the i initial wave function can evolve into the argument x of Inserting the inverted and truncated harmonic oscillator f the final state potential of eq. (3), V(x) = Uc(x), the action can be calculated to give ψ(x ,t )= D[x(t)]eiS[x(t)]/h¯ψ(x ,t ), (7) f f i i µ 2x x Z S[x ]= α2ω t + f − TF 1 α2 cl ⊥ TF The phase factor is the classical action along the path ω⊥ (cid:20)− xTF − (cid:21) S[x(t)]= tf dt 1mx˙2 V[x(t)] . (8) −µ(tf −ti)(1−α2) p (11) Zti (cid:20)2 − (cid:21) forxf >xTF andthe bracketedexpressionineq.(9)can be calculated to be Path integrals are usually not exactly solvable, but they canbeapproximatedwellwhenforexamplefew classical xf dx −12 1 α gpraatlhtsoxscml(ta)lladreevdioamtioinnasnatr.oWunedcathnetmheSn[xre]s=triSct[xtchle+inδtxe]-. (cid:18)2πikfkiZxi k(x)3(cid:19) =s2πia20√1−α2, (12) This will provide a qualitative understanding of the ob- servedfringesaswellasaverygoodaccountofthequan- where a = ¯h/mω . In Fig. 3 the evolution of the 0 ⊥ titative results of the quantum calculations. wave function according to the semiclassical approxima- p FromFig.1onecanimmediatelymaketheobservation tion to eq. (7) is shown (dashed line). Comparison with that for x > x two classical paths contribute. The the full quantum mechanical evolution of eq. (2) shows TF atoms emerge from two different initial positions, and almost perfect agreement. theywillthereforearriveatxwithtwodifferentmomenta To justify and understand the semiclassical approxi- ¯hk and h¯k . This suggests interference fringes of width mation,letusnotethatrestrictiontoclassicalpathsand 1 2 (k k )−1, i.e., larger fringes for large values of x (see the assumptionofquadraticpotentialsisformallyequiv- 1 2 | − | inset of Fig.3). alentto a WKB approximation[14] andthe conditionof For potentials whose second spatial derivative is con- validity can be written as stant, the propagator can be written as [14] 1 dk x 1 D[x(t)]eh¯iS[x(t)] =eh¯iS[xcl] 2πikfki xf k(dxx)3 −12 k2 (cid:12)(cid:12)dx(cid:12)(cid:12)= a40k(x)3 <1 (13) Z (cid:18) Zxi (cid:19) where we have used(cid:12) the(cid:12) momentum of the atoms given (9) (cid:12) (cid:12) by h¯k(x) = 2m(E V(x)). This condition is well ful- with ki = k(xi), kf = k(xf) and the classical action is − filled as long as V(x) is an inverted harmonic oscillator given by p for all atoms. One may, however, question the validity xf close to the Thomas–Fermi radius, where the slope of S[x ]= E (t t )+h¯ dx k(x). (10) cl − f − i the potential changes strongly within the healing length Zxi 4 leads to a reflection coefficient 2 1 R= <10−2 (15) (cid:12)1−20√1−α2(cid:12) ∼ (cid:12) (cid:12) (cid:12) (cid:12) justifying very wel(cid:12)l the semiclassic(cid:12)al treatment. Let us finally consider a two dimensional situation in- cluding gravity. The beam atoms are subjected to the mean-field potential for a time of the order of 0.5 1ms − assuming a typical condensate radius of 4µm. Most of the atoms therefore do not experience a complete hor- izontal roll–off from the mean–field potential (compare Fig.1b),howeverqualitativelytheabovepictureremains FIG. 4: Two dimensional simulations of the atom laser. (a) unchanged. We havesolvedthe two–dimensionalversion Thebeamafteranevolutiontimeof2.5ms. (b)Acutthrough ofeq.(1)inthex–andz–planenumericallyandinFig.4a the density distribution at z = −15µm, corresponding to the atom laser beam for short evolution times is shown. an evolution time of 1.3ms. (c) The corresponding Fourier Once the beam has left the overlap area with the Bose- transform. Comparison with Fig. 2 shows good agreement. Einstein condensate its evolution within the transverse direction is completely determined by a free evolution and the far–field result can be simply calculated by the of the system, i.e., one might expect the appearance of Fourier transform. A cut through the density distribu- quantum reflection effects at this point. To estimate tion is shownin Fig. 4b and the far field ofthis distribu- the reflectivity of the potential step, we note that the tion is shown in Fig. 4c. Both pictures show good quali- de-Broglie-wavelength of the atoms when reaching the tative agreementwith the results of the one-dimensional Thomas–Fermiradius is largerthan the healing length ξ analysis. In summary we have shown that the transverse mode 2πa2 2πa2 a2 ofanatomlaserisstronglydeterminedbytheinteraction λ = 0 0 > 0 =ξ. (14) dB x2TF −x2i ≥ xTF xTF osaftteh.eDbueeamtotwhiethfinthiteemtimeaeno–ffitehldisoinfttehreacrteisoind,inthgecsoynsdteenm- One can therefopre approximate the edge of the conden- resemblesaninterferometerinmomentumspace andthe sate by an effective step. Since the exact choice of the beam shows filamentation in the transverse directions. position of the effective potential step is not crucial, one This work has been supported by the Danish Natural can from eq. (14) estimate its effective height to be as Science Research Council and the Deutsche Forschungs- large as 10% of the central mean field potential. This gemeinschaft. [†] email: [email protected] School of Physics ’Enrico Fermi’ Course 140, 1999 (IOS [1] M. -O. Mewes et al., Phys. Rev. Lett. 78, 582 (1997); Press Amsterdam). B. P. Anderson and M. A. Kasevich, Science 282, 1686 [8] J. Schneider and A. Schenzle, Appl. Phys. B 69, 353 (1998); E. W. Hagley et. al.,Science 283, 1706 (1999); (1999). 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