Fast Sweeping into Feshbach Molecules as a Probe of Fermionic Condensates

a r X i v :0805.4295v 1 [c o n d -m a t .o t h e r ] 28 M a y 2008Fast Sweeping into Feshbach Molecules as a Probe of Fermionic CondensatesS.Matyja´s kiewicz,1,2M.H.Szyma´n ska,3,4and K.G´o ral 51Department of Physics,King’s College London,London WC2R 2LS,UK2Department of Applied Mathematics,Warsaw University of Life Sciences,Nowoursynowska 159,02-776Warsaw,Poland3Department of Physics,University of Warwick,Coventry CV47AL,UK 4Cavendish Laboratory,University of Cambridge,Cambridge CB30HE,UK5Clarendon Laboratory,Department of Physics,University of Oxford,Parks Road,Oxford OX13PU,UK(Dated:May 28,2008)Fast sweep projection onto Feshbach molecules has been widely used as a probe to explore the properties of fermionic condensates,in particular the condensate fraction.By determining the exact dynamics of a pair of atoms in time varying magnetic fields across a Feshbach resonance,we calculate the number of condensed and noncondensed molecules created after fast magnetic field sweeps from the BCS to the BEC side of the resonance.We analyse the dependence of the molecular conversion efficiency for condensed and noncondensed molecules on the sweep rate,and on the initial and final magnetic fields,for 40K and 6Li resonances;and discuss the relation between the initial fermionic condensate fraction and the molecular condensate fraction measured after the sweep.PACS numbers:03.75.Ss,03.75.Kk,05.30.FkEver since the first realisations of fermionic conden-sation in two-component gases of 40K [1]and 6Li [2]the question how these fragile states can be accurately probed has been of great experimental importance.Since current techniques rely on the time-of-flight expansion,during which the Cooper pairs dissociate,indirect meth-ods have been proposed.One of the most commonly used is the so called fast-sweep-projection technique [1],which consists in a rapid magnetic field ramp from the BCS to the BEC side of the resonance.The key point behind this technique is to make the field variation fast with re-spect to the many-body time scales thus preventing pair collisions from changing the pairs’momentum distribu-tion and re-thermalisation.When this condition is satis-fied,the condensed molecules measured after the sweep originate only from the Cooper pairs,already present at the initial time of the dynamics,whereas the noncon-densed molecules are converted only from the noncon-densed atoms.The expectation is that this allows the information about the initial state to be easily unravelled from the measurements of molecules created during the sweep.This technique has been used in a wide range of experiments on cold fermions [3].The aim of this Letter is to describe the fast sweep regime of fermion dynam-ics and to determine whether and how the information about the initial fermionic condensate can be extracted from the momentum distributions and the production ef-ficiency of molecules after the sweep.Despite its significant experimental relevance and fun-damental interest,the theoretical understanding of the dynamics of cold Fermi gases following variations in the interaction strength is still relatively limited.The early analyses of the molecular production were based on cal-culating the overlap of the initial pair state with the final molecular wave function [4].This corresponds to an in-finitely fast jump of the magnetic field strength and thuscannot account for the sweep rate dependence.Theoret-ical analysis of time-dependent precesses has so far con-centrated on the mean-field dynamics [5],which by def-inition ignores the noncondensed molecules.Thus,such methods are inadequate for studying the sweeps where large numbers of noncondensed molecules were reported to be created during the ramp [3].Finally,an inter-esting approach was proposed in [6],in which the de-pendence of the molecular production on the sweep rate was estimated analytically without determining the ac-tual dynamics.This was done by overlapping the initial state with a sweep-rate-dependent “final state”molecular wave function.Clearly,more rigorous calculations,which would determine the dynamics of the gas in time-varying magnetic fields and allow us to capture both condensed and noncondensed molecules created during this process,are still missing.Making use of a systematic cumulant expansion [7],one can obtain the dynamic equations for the density matrix and the pair function of the gas.On this basis,the number of molecules beyond the mean-field level,i.e.accounting for noncondensed pairs,can be calculated.In this Letter we numerically solve these dynamical equa-tions for a limited case of fast sweeps,during which pair collisions are insignificant.In such a case,the dynamical aspects are captured by the two-body time evolution of a single atom pair in the spirit of the original rationale be-hind the fast sweep projection technique.We determine the two-body time evolution exactly by numerically solv-ing the time-dependent Schr¨o dinger equation for a pair of atoms.We start from the single-channel HamiltonianˆH = k sǫk a †k s a k s + kk ′qV kk ′a †k +q ↑a †−k ↓a −k ′↓a k ′+q ↑,(1)involving Fermi operators a †k s and a k s (hereafter k ,k ′,and q denote wave vectors).In the following,to ac-2 count for the larger region of the crossover extendingbeyond the universal regime,we use afinite-range sepa-rable potential of the form V kk′=V0(B)χ(k)χ(k′).Hereχ(k)=exp[−(kσbg)2/2]and the parameters V0(B)andσbg are chosen to recover the correct magneticfield de-pendence of the scattering length a(B)=a bg(1−∆B∂t′[ˆΓ(k,t′)ˆΓ(k−2k′,t′)]}.The dynamic equations for the density matrix ˆΓ(k,t)=1∂tU2B(t,t0)=ˆH2B(t)U2B(t,t0),whereˆH2B is equal to(1)for a pair of atoms.We are then in a position to evaluate n c m and n nc m from(3)and(4),respectively.Since our main focus is on establishing the relation between the final state after a fast sweep and the initial state,we take the simple mean-field thermodynamic initial conditions. Wefirst analyse the dependence of n c m,n nc m and n m= n c m+n nc m on thefinal magneticfield for different linear1FIG.1:(color online).Upper panels:The bound state energy, E b(B),of the40K(left panel)and the6Li(right panel)Fesh-bach moleculeφb(B)in the vicinity of the202G(40K)and the834G(6Li)resonances from two-(solid lines)and single-channel approaches(dashed lines).The scattering length is shown by dashed-doted lines.Lower panels:Total(top curves,blue),condensed(middle curves,red)and noncon-densed(lower curves,black)molecular densities versus the finalfield,Bfin,for the initialfield B ini−B0=0.12G for40K (left panel)and B ini−B0=4.62G for6Li(right panel)and slew rates(1/˙B):1µs/G(40K),0.003µs/G(6Li)(dashed lines)and10µs/G(40K),0.01µs/G(6Li)(solid lines).The case of an infinitely fast jump is shown by dotted lines.The initial atomic densities are1.5(40K)and2.9(6Li)1013cm−3. For40K the parameters are from[1]and for6Li are motivated by[3,12].(unlike for the infinitely fast jump)the number of pro-duced molecules depends on thefinalfield only if it is in the region close to the resonance–it saturates forfields further away from the resonance.The exact value of the3field at which the molecular production saturates is pro-portional to the sweep rate–the faster the sweep the fur-ther from the resonance the saturation takes place.Thissupports the physical picture used in[6]that sufficiently far from the resonance E b is so large and interactions sosmall that the molecular state adiabatically follows theramp(no more molecules are created or dissociated dur-ing this part of the dynamics)[3].Forfinalfields nearthe resonance the molecular production is close to whatone would obtain from an infinitely fast jump,i.e it is independent of the sweep rate.There is,however,anintermediate region offields for which the molecular pro-duction is not yet saturated,but it depends on the rate(i.e it differs from the infinitely fast jump).The presenceof this region and also the exact position of the crossover to adiabatic dynamics is expected to be the main sourceof quantitative discrepancy between the exact dynamicspresented here and analytical estimates discussed in[6] for condensed pairs(additional differences arise for non-condensed molecules–these are discussed later). Note,that there is a qualitative difference between the202G resonance in40K and the834G resonance in6Li [10,13]which might affect the theoretical descriptionneeded for quantitative predictions.In the case of6Liand slew rate,1/˙B,of0.003µs/G used in the experi-ment of[12],which is close to being fast on the many-body time scales[17],the saturation of the molecular pro-duction takes place around250G below the resonance,where the single and two-channel predictions already dif-fer(there is around15%difference in E b between the two approaches).This suggests that if we were to cal-culate the molecular production for sweeps even fasterthan those used in[12]an extension to the two-channel description would be necessary.In contrast,for40K thevalues of E b(B)from the single-and two-channel ap-proaches coincide in the relevant region B−B0>−10Gand are close for anyfield(see Fig1).Thus,the singlechannel description is sufficient for quantitative predic-tions of molecular production in40K for any sweep rate. The dependence of n c m on the initialfield(Fig.2)is re-lated to the dependence of the number of Cooper pairs,n c(B)= d k|ˆΦ(k,B)|2,on the interaction strength at thatfield,but it is different for different sweep rates.Asexpected,the number of condensed molecules,n c m,cre-ated during the linear sweep is always smaller than the number of initial Cooper pairs but larger than what onewould obtain after an infinitely fast jump.The number of noncondensed molecules,n nc m,increases as the initialfield is shifted deeper to the BCS side.This can be un-derstood by noting that,in the fast sweep regime,the noncondensed pairs are created only from atoms whichwere initially out of the condensate and the number ofthese atoms increases towards the BCS side.Finally,the dependence on the sweep rate is shown in Fig.3and compared with[6]and,for n m,with the Landau-Zenner(LZ)theory from[3].Since Cooper pairsFIG.2:(color online).Lower panel:Molecular condensate,2n c m/n,(solid lines),non-condensate,2n nc m/n,(dashed lines) and total,2n m/n,(doted lines)conversion efficiencies versusthe initialfield B ini for thefinalfield Bfin−B0=−10G and slew rates:1µs/G(lower curves,blue)and10µs/G(uppercurves,red).The condensed part,2n c m/n,for an infinitely fastjump,which produces virtually no noncondensed molecules,is also shown(lowest solid curve,green).For comparison,top solid curve(black)shows the ratio2n c/n of atoms initially(at B ini)paired into Cooper pairs to the total number of atoms. Upper panel:Molecular condensate fraction,n c m/n m(solid curves with closed circles)for the same parameters.Low-est solid curve(black)shows the initial condensate fraction 2n c/n.Calculations are done for40K202G resonance and atomic density of1.51013cm−3.k f in the upper axis of the lower panel indicates the Fermi wave vector.convert to molecules more efficiently than free atoms(as pointed out in[6]),LZ theory,which does not distin-guish between the two,is unlikely to give an accurate account of molecular production from a partially con-densed Fermi gas.Indeed,as can be seen from Fig.3, LZ results differ substantially from the results of our dy-namical calculations.The differences with[6]are also pronounced,especially for the production efficiency of noncondensed molecules,which in[6]has a linear depen-dence on the slew rate with a coefficient being dependent on the density but independent of the initialfield.Here, it is instructive to compare our dynamics with an ap-proach based on the asymptotic dissociation spectrum given by(B12)in[8,15].It can be shown[15]that in the asymptotic limit of t0going to(minus)infinity and t to infinity the transition probability is given by|T(k)|2= |a bg∆B|3a bg∆B4FIG.3:(color online).Upper panel:Conversion efficiencyfor condensed,2n c m/n,noncondensed,2n nc m/n,and total,2n m/n,molecules versus the dimensionless inverse sweep raten∆Ba bgfor the40K202G resonance and atomic density of˙B1.51013cm−3for B ini=B0and Bfin−B0=-10G.For com-parison analytical formulas from[6]are indicated by dottedlines(total,condensed and noncondensed from top to bot-tom);and for the total molecules the Landau-Zener predic-tion from[3](dashed-dotted line)is shown.x symbols showthe asymptotic approximation as discussed in the text.Lowerpanels:Condensate fraction after sweeps as in upper panel asa function of the initial condensate fraction(left panel,lineswith circles,the line without circles is a guide for the eye)andversus the inverse rate(right panel,solid line).The analyticalprediction from[6]is shown by the dotted line,and the initialcondensate fraction,2n c/n,by theflat dashed line.the exact transition probability,|T(k′,t,t0)|2,turns outto be a half of n nc m obtained using the asymptotic spec-trum.On the basis of numerical evidence we can thusprovide a simple formula for the number of noncondensedmolecules for B ini=B0as a function of the slew ratea bg∆Bn nc m≈|a bg∆B|35ali,P.Pieri,and G.C.Strinati,Phys.Rev.Lett.95, 010407(2005).[5]R.A.Barankov and L.S.Levitov,Phys.Rev.Lett.93,130403(2004);R.A.Barankov,L.S.Levitov,and B.Z.Spivak,Phys.Rev.Lett.93,160401(2004);A.V.An-dreev,V.Gurarie,and L.Radzihovsky,Phys.Rev.Lett.93,130402(2004);M.H.Szyma´n ska,B.D.Simons,and K.Burnett,Phys.Rev.Lett.94,170402(2005);E.A.Yuzbashyan,V.B.Kuznetsov,and B.L.Altshuler,Phys.Rev.B72,144524(2005);E.A.Yuzbashyan,O.Tsyply-atyev,and B.L.Altshuler,Phys.Rev.Lett.96,097005 (2006);E.A.Yuzbashyan and M.Dzero,Phys.Rev.Lett.96,230404(2006).[6]E.Altman and A.Vishwanath,Phys.Rev.Lett.95,110404(2005).[7]T.K¨o hler and K.Burnett,Phys.Rev.A65,033601(2002).[8]K.G´o ral et al.,J.Phys.B37,3457(2004).[9]T.K¨o hler,K.G´o ral,,and P.S.Julienne,Rev.Mod.Phys.78,1311(2006).[10]M.H.Szyma´n ska,K.G´o ral,T.K¨o hler,and K.Burnett,Phys.Rev.A72,013610(2005).[11]T.K¨o hler,T.Gasenzer,and K.Burnett,Phys.Rev.A67,013601(2003).[12]L.Tarruell et al.,arXiv:cond-mat/0701181(2005).[13]W.Yi and L.-M.Duan,Phys.Rev.A73,063607(2006).[14]J.von Stecher and C.H.Greene,Phys.Rev.Lett.99,090402(2007).[15]T.M.Hanna et al.,Phys.Rev.A74,023618(2006).[16]M.W.Zwierlein,Private communication(2007).[17]Note that the relevant dimensionless slew rate whichcharacterises the ramp speed is roughly n∆Ba bg˙B[14]and thus0.003µs/G for6Li corresponds to around 12µs/G for40K).。