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The Roton in a Bose-Einstein Condensate J. Steinhauer1, R. Ozeri2, N. Katz2, and N. Davidson2 1Department of Physics, Technion Israel Institute of Technology, Technion City, Haifa 32000, Israel and 2Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel The roton in a Bose-Einstein condensate is computed, both near and far from a Feshbach resonance. A low-density approximation is made, allowing for an analytic result. A Monte Carlo 4 calculation shows that theroton is larger thanpredicted by thelow-density approximation, for the 0 upper range of densities considered here. The low-density approximation is applied to superfluid 0 4He, roughly reproducingthe results of previous Monte Carlo calculations. 2 PACSnumbers: n a J A roton, first predicted by Landau [1] for superfluid is the relevant length scale is the special case of high- 7 2 4He, is an excitation in a Bose-Einstein condensed fluid, density, for which ro n−1/3. ≈ characterized by a minimum in the excitation spectrum In a low-density BEC,Feynman’s view of the rotonas ] ω(k). By Feynman’s relation[2], this minimum also cor- asingleatommovingthroughthe condensateseemsnat- t f responds to a maximum in the static structure factor ural. Ingeneral,theexcitationspectrumofalow-density o S(k). This peak in S(k) occurs for excitations whose BEC is of the Bogoliubov form [8, 9], which consists of s . wavelength2π/kisequaltothecharacteristicwavelength phononsandsingle-particleexcitations. Wefindthatthe t a ofdensity fluctuations in the ground-statewavefunction roton occurs on the single-particle part of the spectrum, m of the quantum fluid. The peak in S(k) exceeds unity, at k 8/πa, where a is the s-wave scattering length. ≈ - the value for an uncorrelated gas. Althoughthegeneralformofthewavefunction(1)can d n Feynman[2]describedseveralpossiblemicroscopicpic- be applied to both superfluid 4He and BEC, the wave o tures of a roton in superfluid 4He. He found that the function is fundamentally different for these two quan- c most likely description is that the roton is analogous to tum fluids. For both of these fluids, below some temper- [ asingleatommovingthroughthecondensate,withwave ature Ts−wave, the thermal de-Broglie wavelength λdB 2 number k close to 2π/n−1/3, where n−1/3 is the mean becomes longer than the characteristic length scale R of v atomic spacing. the interparticle potential. BelowTs−wave, allscattering 5 Therotoninsuperfluid4HewascalculatedbyaMonte processesexceptfors-wavescatteringbecome negligible. 7 Carlo technique [3, 4] using a many-body Jastrow wave Since a BEC is a dilute gas, n−1/3, which is typically 3 3 function [5], which is of the form about1500˚A,ismuchlargerthanR. Therefore,the crit- 0 ical temperature T for quantum degeneracy, at which c 3 N λ becomes comparable to n−1/3, is much lower than t/0 ψ =j>Yi=1f(|ri−rj|), (1) Theds−rBewafover.aTBhEerCef.oIrne,coonnltyrass-twtaoveBsEcaCt,tesruinpgerpflluaiyds4aHreolies a m wherethepairfunctionf(r)shouldbedeterminedforthe a relatively dense liquid, for which n−1/3 is comparable - quantum fluid under consideration. The wave function to R. Therefore, the temperature Ts−wave also marks d (1) deviates significantly from unity if any two atoms the transitionTc to quantumdegeneracy. EvenbelowTc n become very close to one another. For superfluid 4He, for superfluid 4He, partial waves other than the s-wave o the two-particle correlation function g(r) for this wave contribute to the wave function. c function has fluctuations with a preferential length scale For many properties of a BEC, a is sufficient to de- : v of n−1/3, which results in a peak greater than unity in scribe the interparticle potential, and the form of the Xi S(k) near k 2π/n−1/3, in agreement with Feynman’s potential does not play a role [10, 11]. To quantita- ≈ r picture. tively describe the roton however, the s-wave scattering a Both Feynman’s result and the results of the Jastrow wavefunctionmustbeknowntointerparticleseparations wave function roughly agree with measurements [6] of somewhat smaller than R, where the details of the po- S(k) and ω(k). tential are relevant. The Jastrow wave function (1) has also been consid- Rotons have been predicted in a BEC in the presence ered for a low-density Bose-Einstein condensate (BEC) of optical fields [12, 13], as well as in a dipolar BEC [3],andhasbeenusedtocomputevariouspropertiesofa [14]. In this work, we find that a roton occurs in an high-density BEC [7]. We employ a Jastrow wave func- unperturbed BEC without any dipole interaction. We tion to compute the roton for a low-density BEC. By an consider the enhancement of the roton near a Feshbach analytic calculation, we find that the hard-spheresize r resonance [15, 16], but the physics is qualitatively the o ofthe atomsdetermines the locationofthe roton,rather same as in the unperturbed case. than n−1/3. We see that Feynman’s view that n−1/3 Whenwerefertoaroton,wearereferringtoapeakin 2 S(k), rather than a minimum in ω(k). By the Feynman nator, dr1 N f2(r1 rj ). This can be visualized as relation, the peak in S(k) computed here for a BEC is j>1 | − | not steep enoughto produce a minimum in ω(k), for the the inteRgraloQverthe function shownin Fig. 1 for one di- range of densities considered here. mension. Neglectingthree-bodyinteractionsgreatlysim- We use a low-density approximation [5] to compute plifies this integral. Three-body interactions are rare for g(r) and the roton for a BEC with a positive scattering small values of the gas parameter na3, assuming that length. The low-density approximationis comparedto a the range of three-body interactions is of the same order Monte Carlo calculation. of magnitude as the range of two-body interactions. A For a BEC of alkali atoms, the potential can be taken three-body interaction is represented in Fig. 1 by the 6 as the van der Waals potential C6/r for atomic sepa- points xc and xd, where atoms 1, c, and d interact. − rationsr greaterthana few Angstroms[17, 18]. Forthis Neglecting such interactions, the integral is a function potential, R is givenby (mC6/~2)1/4, which rangesfrom of the volume v indicated by the shaded region in Fig. about 50 to 100˚A[18, 19]. The wave function for s-wave 1. The integral is then given by V (N 1)v, where scattering between two particles in the limit of zero en- v = V drf2(r). This result is i−ndepe−ndent of the ergyis givenby ψ6(r), whichis the solutionofthe radial position−s of the rj. R Schrödinger equation with the van der Walls potential. ψ6(r) can be written in terms of elliptic integrals [17]. For r > R, this wave function is very close in form to 1 a/r [10, 20]. For the case of a R, ψ6(r) therefore 2 − ≫ obtains large negative values for r a [21]. o ≪ We cannot use ψ6(r) as f(r) in (1), because even for r n−1/3, ψ6(r) is significantly less than unity. This is ≫ non-physical, in the sense that no matter how large the 2 gas for fixed density, the value of the many-body wave function (1) depends on the size of the gas. To account formany-bodyeffects, weusethe followingpairfunction which goes to unity for r>n−1/3 [7, 18]. f(r)= ψ6(r)/ψ6(n−1/3) (r ≤n−1/3) (2) xb xc xd xe (1 (r >n−1/3). x 1 To compute (2), R is needed. Throughout this work, we useR=0.05n−1/3. Thisisatypicalexperimentalvalue, FIG.1: Schematicone-dimensionalrepresentationoftheJas- and the results here are rather insensitive to R. trowwavefunction. Thedependenceonthesingledimension Equation(1)with(2)isshownschematicallyinFig. 1. x1 is shown. The overall scaling of the curve is given by the To aid in visualization, Fig. 1 shows the wave function factors in the wave function not involving x1. The labeled squared in one dimension, as a function of the position values of x1 correspond to the positions of atoms other than x1 of atom number 1. The positions of all of the other atom 1. x1 varies overthevolume V. atoms, such as atoms b through e, are fixed. As long Evaluating all of the integrals in (3) similarly to the as atom number 1 is far from the other atoms, the wave function has the constant value ψo. This value is deter- first yields g(|r1 −r2|) ≈ V2f2(|r1 −r2|)[(V −v)V]−1. Since V v, we obtain the result of the low-density mined by the positions of the atoms other than x1. The ≫ approximation[5] rapid oscillations in ψ6(r) for r <R appear in Fig. 1 as dark vertical bands, when atom number 1 is very close 2 g(r) f (r). (4) to another atom. ≈ The correlation function g(r) gives the unconditional The result (4) with (2) is indicated by the solid curve probability of two atoms being at a distance r. g(r) is in Fig. 2 for na3 = 2.2 10−4. This value of na3 is × related to the pair function f(r) by [5] an order of magnitude greater than typical experimen- tal values without a Feshbach resonance. The result for dr3...drN N f2(ri rj ) na3 =0.011 is also shown in the figure. g(|r1−r2|)=V2R j>QNi=1 | − | , (3) moWrehailcecu(4ra)teiscuomsepfuultaftoironobctaaninbinegmaandaelybtyicthreesuMltosn,tae dr1...drNj>i=1f2(|ri−rj|) Carlo technique described in Ref. [3]. This technique ef- R Q fectivelyevaluates(3)byusingaMetropolisalgorithmto where the integrals are over the volume V. (3) can be randomlychooseconfigurationsoftheN atoms(N =100 evaluated starting with the first integral in the denomi- here), according to the probability distribution given by 3 (1)with(2),andcomputingthe distributionofdistances S(k ) and location k of the roton are indicated by the r r between the atoms, with periodic boundary conditions. solidcurvesofFig. 4. S(k)inFig. 3hasseveralmaxima, This distribution, averaged over many likely configura- the tallest of which is taken as the roton. As na3 is var- tions, is proportionalto r2g(r). The result of the Monte ied,thepeakwhichistakenastherotonvaries,resulting CarlocomputationisshowninFig. 2forna3 =2.2 10−4 in the jagged appearance of the solid curves of Fig. 4. and 0.011. These results were obtained with 9 10×8 and For the Monte Carlo computation, S(k) is found by 2 107 iterations, respectively. × inserting g(r) such as is shown in Fig. 2 into (5). The × results are indicated in Fig. 3 by the dash dotted and solidcurves. Forsmallna3,theheightandlocationofthe 3 80 rotonareseeninFig. 4tobethesameforthelow-density approximation and the Monte Carlo result. The low- 60 density approximation is therefore valid for small na3. 40 For the larger values of na3, the height of the roton in 2 ) 20 the Monte Carlo calculation is larger than that of the r ( low-density approximation, as seen in Fig. 4a. g 0 0 1 Byapplyingthelow-densityapproximationtothelimit 1 of a R, we can obtain an analytic expression for S(k) for R≫ a n−1/3. In this range, f(r) can be taken as ≪ ≪ 1 a/r. By (4) and (5), − 3 −1 −2 0 S(k)=1+4πna [π(ka) /2 2(ka) ]. (6) − 0 1 2 3 4 5 6 7 8 Thissmall-RlimitisindicatedbythedottedcurveinFig. 3. The height of the roton in this limit is S(k ) = 1+ r / a r π3na3/8, and the location is given by k =8/πa. These r FIG.2: Thetwo-particlecorrelationfunctionforaBEC.The valuesareindicatedbythedashedcurvesofFig. 4,which solidanddashedcurvesarethelow-densityapproximation(4) as expected, agree with the low-density approximation with (2), with na3 =2.2×10−4 and 0.011, respectively. The (solid curves) for a R, where nR3 is indicated by the dotted curve is the Monte Carlo result for na3 = 0.011. For ≫ dotted lines. As seen in the above expression for k , the na3 =2.2×10−4,theMonteCarloresult isindistinguishable r from the solid curve. The inset shows the curves for na3 = location of the roton is determined by a, rather than by 0.011 only. n−1/3. The small oscillations for small r shown in Fig. 2 are 7 negligible in the computation of the static structure fac- 3 tor. These are the oscillations for r values less than that a n 6 of the first large peak below r/a = 1, in the solid and / dashedcurves. Tosavecomputingtimetheseoscillations ] 5 are not included in f2(r) in the Monte Carlo computa- 1 - 4 tion. ) Fora 1.3R(correspondingtona3 >2.7 10−4inthis k 3 work),T≥hepeakinf2(r)locatedatr <Ris×greaterthan S( unity. The dashed line of the inset of Fig. 2 shows an [ 2 example of sucha peak, for a=4.4R. Because this peak 1 isgreaterthanunity,aclusterofatomsisthemostlikely configuration. Sinceweareinterestedintheun-clustered 0 phase, we do not let the Monte Carlo computation pro- 0 4 8 12 16 20 ceed long enough for the clear transition to clusters to ka occur. The static structure factor S(k) is given by 1 + FIG. 3: The roton in a BEC, computed by (5). The dash n [g(r) 1]eik·rdr, which in general can be written dottedandsolidcurvesaretheMonteCarloresultsforna3 = − 0.011and2.2×10−4respectively. Theresultofthelow-density R ∞ approximation(4)with(2)forna3 =0.011isindicatedbythe sin(kr) S(k)=1+4πn drr2[g(r) 1] . (5) dashed curve. The result of the low-density approximation − kr forna3 =2.2×10−4 isindistinguishablefromthesolidcurve. Z0 The dotted curveis thesmall-R limit, given by(6). We use (5) to compute S(k) for the low-density approximation (4) with (2), as indicated in Fig. 3. The height For na3 = 0.011, the height of the roton given by the 4 Monte Carlo calculation is S(k ) = 1.08, as shown in genttestofthe low-densityapproximation. We findthat r 4 Figs. 3 and4a. Measuringthis 8%effectcouldbe exper- the location of the roton for superfluid He is given by imentally feasible. na3 = 0.011 could be attained by a k = 5.3/b, which differs from the Monte Carlo result r Feshbachresonance[20]. Itshouldbenotedthough,that by 6% [3, 4]. We find that the height of the roton for for this relatively large value of na3, the form of (2) is superfluid4He is S(k ) 1+0.3nb3, whichgivesaroton r ≈ only approximate, so the expressions for the height and ofS(k )=1.1,comparedto S(k )=1.2 fromthe Monte r r location of the roton should be considered as estimates Carlocalculation[3,4],andS(k )=1.5fromexperiment r only. [6]. The order-of-magnitudeagreementbetweenthe low- Due to phonons,the true S(k) is proportionalto k for density approximation and the Monte Carlo result for k .ξ−1, where ξ−1 = a−1√8πna3 is the inverse healing superfluid 4He suggests that the low-density approxima- length. The curves shown in Fig. 3 do not show this tion preserves the essence of (3). linearbehaviorforsmallk becausethe wavefunction(1) In conclusion, we find the height S(kr) and location with (2) does not have the long-range correlations of a kr of a roton in a BEC, for a range of densities. A low- phonon [3, 4]. densityapproximationiscomparedtoaMonteCarlocal- For superfluid 4He the three inverse length scales culation. The values of S(kr) and kr given by the two 2π/n−1/3, k , and ξ−1, are roughly equal. This is also methods agree for the lowest densities. For higher den- r true for a strong rotonin a BEC.As na3 increases,both sities, the Monte Carlo calculation predicts an enhance- kr and ξ−1 approach 2π/n−1/3. More precisely, the ra- mentinS(kr)ofalmostafactorof2overthelow-density tios of ξ−1 and k to 2π/n−1/3 are 2/π(na3)1/6 and approximation. r approximately 4/π2(na3)−1/3, respectively. The latter In contrast to the Monte Carlo calculation for super- ratioimpliesthatanappropriatemeaspurementsystemfor fluid 4He, the small-R limit gives explicit expressions for measuring a roton should be able to probe wavelengths the height and location of the roton. somewhat shorter than n−1/3. 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