We have analyzed a single vortex at T=0 in a 3D superfluid atomic Fermi gas
across a Feshbach resonance. On the BCS side, the order parameter varies on two
scales: $k_{F}^{-1}$ and the coherence length $\xi$, while only variation on
the scale of $\xi$ is seen away from the BCS limit. The circulating current has
a peak value $j_{max}$ which is a non-monotonic function of $1/k_F a_s$
implying a maximum critical velocity $\sim v_F$ at unitarity. The number of
fermionic bound states in the core decreases as we move from the BCS to BEC
regime. Remarkably, a bound state branch persists even on the BEC side
reflecting the composite nature of bosonic molecules.Comment: 4 Pages, 4 Figure

A. J. Leggett

M. Randeria

Mohit Randeria

Rajdeep Sensarma

Tin-Lun Ho

English

arXiv

We have analyzed a single vortex at T=0 in a 3D superfluid atomic Fermi gas across a Feshbach resonance. On the BCS side, the order parameter varies on two scales: k_F^-1 and the coherence length ξ, while only variation on the scale of ξ is seen away from the BCS limit. The circulating current has a peak value j_max which is a nonmonotonic function of 1/k_Fa_s implying a maximum critical velocity ∼vF at unitarity. The number of fermionic bound states in the core decreases as we move from the BCS to the BEC regime. Remarkably, a bound state branch persists even on the BEC side reflecting the composite nature of bosonic molecules

Sensarma, Rajdeep

Randeria, Mohit

Ho, Tin-Lun

KnowledgeBank at OSU

PRL 96, 090403 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending10 MARCH 2006Vortices in Superfluid Fermi Gases through the BEC to BCS CrossoverRajdeep Sensarma, Mohit Randeria, and Tin-Lun HoDepartment of Physics, The Ohio State University, Columbus, Ohio 43210, USA(Received 27 October 2005; published 9 March 2006)0031-9007=We have analyzed a single vortex at T  0 in a 3D superfluid atomic Fermi gas across a Feshbachresonance. On the BCS side, the order parameter varies on two scales: k1F and the coherence length ,while only variation on the scale of  is seen away from the BCS limit. The circulating current has a peakvalue jmax which is a nonmonotonic function of 1=kFas implying a maximum critical velocity vF atunitarity. The number of fermionic bound states in the core decreases as we move from the BCS to theBEC regime. Remarkably, a bound state branch persists even on the BEC side reflecting the compositenature of bosonic molecules.DOI: 10.1103/PhysRevLett.96.090403 PACS numbers: 03.75.Hh, 03.75.Kk, 03.75.SsThe recent discovery of vortices in 6Li is a milestone inthe study of superfluidity in atomic Fermi gases [1]. For thefirst time, the phase coherence of a superfluid atomic Fermigas is demonstrated unambiguously. Fermi gases of alkali-metal atoms, however, are very unusual many-body sys-tems. Their interactions, which are related to s-wave scat-tering length, are highly tunable using Feshbach resonance.By tuning the inverse scattering length 1=as continuouslyfrom 1 to 1, the ground state of these systemschanges from a weak coupling BCS superfluid to a mo-lecular Bose-Einstein condensate (BEC) [2]. The region ofinfinite scattering (or a1s  0) is particularly interesting,where the Fermi gas exhibits universal behavior, in thesense that the interaction energy scale at T  0 is inde-pendent of atomic details and is given by Fermi energy [3].The physics of these Fermi gases is also related to those ofstrongly interacting nuclei which also have large scatteringlengths. Universality implies that the superfluid transitiontemperature Tc is comparable to Fermi temperature TF. Infact, Tc=F  0:2 [4] at resonance, the highest of all knownfermion superfluids.Although all vortices of s-wave BCS superfluids havetopologically invariant properties like quantized circula-tion of h=2M, other features like vortex core size, circulat-ing current, and the bound state spectrum depend ondynamical details. It is natural to ask how the propertiesof a vortex change as one goes across the Feshbach reso-nance from the BCS to the BEC side, and how unitaritymanifests itself in these properties. The goal of this Letteris to answer these questions using the Bogoliubov–de Gennes (BdG) approach [5] at T  0 in a three dimen-sional system for a wide resonance [6]. Because of theconnection between strongly interacting atomic Fermigases with other strongly interacting Fermi systems, ourresults will have implications beyond cold atom physics.We find that(A) On the BCS side, the order parameter  for asingle vortex exhibits two length scales: an initial rise onthe scale of k1F [7] for which we give an elementaryanalytical argument, and an eventual approach to its bulk06=96(9)=090403(4)$23.00 09040value 0 on the coherence length scale   @vF=0 wherevF is the Fermi velocity. At unitarity where 0  F @2k2F=2M,  reduces to k1F , and the two length scalescoincide. On the BEC side the  reaches its bulk valueover the coherence length  1= nasp , where n is thedensity. We also find that the density n is depleted inthe vortex core, and its value n0 at the center is dramati-cally reduced as one goes from the BCS to the BEC limit.(B) The circulating current j around the vortex corehas a peak value jmax which is nonmonotonic across theresonance and reaches a maximum precisely at unitarity.Its scale is set by the critical velocity vc which is deter-mined by pair breaking on the BCS side, a single-particleeffect, but by collective excitations on the BEC side. jmax isone of the very few properties of atomic Fermi gases thatvaries nonmonotonically across resonance, in contrast tomost thermodynamic properties.(C) We find that unitarity represents the most robustsuperfluid state in the entire BCS-BEC crossover. Notonly does one obtain the highest Tc at unitarity (which is,however, not too different from the Tc value [10] for all1=kFas > 0), but one also obtains the highest critical ve-locity vc  vF.(D) We find that as we go from the BCS to the BEClimit, the number of fermionic bound states in the vortexcore decreases, with a corresponding increase in both theenergy of the lowest bound state and their level spacing.Remarkably, we find that a bound state is observed waypast unitarity, deep into the bosonic regime, which isunique to the molecular BEC. We also find that motionalong the vortex core broadens a bound state of angularmomentum ‘ into a band with kz dispersion.The BdG approach was first used to study vortices inBCS superconductors in the classic work of Caroli,de Gennes, and Matricon [5]; see also Ref. [9]. In thesuperfluid atomic Fermi gases, the BdG approach hasbeen used in Ref. [11] in the BCS limit and in Ref. [12].The latter work is in 2D and is implicitly restricted to‘‘narrow resonances,’’ which have very different physicsfrom the wide resonances relevant to all current experi-3-1 © 2006 The American Physical Society 0 0.4 0.8 1.2 0  5  10  15∆(ρ)/∆0kF ρ-1.01.00.0kF-1ξFIG. 1. The order parameter normalized by its value far fromthe vortex for three different couplings 1=kFas  1; 0; 1. Notethe two length scales in the BCS limit 1=kFas  1.PRL 96, 090403 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending10 MARCH 2006ments [6,13]. The vortex problem has also been studiedusing hydrodynamic [14] and microscopic density func-tional [15] approaches. However the results (B), (C), and(D) mentioned above have not been reported before.Bogoluibov–de Gennes approach. For a wide reso-nance, it is sufficient to use the single channel model [6].The BdG equations in this case areT^ rr T^ !unrvnr  En unrvnr (1)where T^  @2r2=2M,  is the chemical potential,En the eigenvalues, and the eigenfunctions satisfyZd3rumrunr  vmrvnr	  mn: (2)The order parameter r and the chemical potential  aredetermined by the self-consistency equation r gPnunrvnr and the average density n 2PnRd3rjvnrj2. ThePn is restricted to 0  En  Ecwhere Ec is a high energy cutoff (discussed below). Fromnow on we measure all energies in units of F and lengthsin units of k1F , so that the bare attraction g has units ofF=k3F. The cutoff Ec and the corresponding gEc shouldbe chosen such that [2,10]1kFas  8Fgk3F 2EcFs(3)so that the low energy effective interaction is described bythe scattering length as.We work in cylindrical coordinates r  ; ; z in agauge in which r  ei. Working in a cylindricalbox of radius R and height L, our normalized wave func-tions are of the form unr  unei‘eikzz=2Lpandvnr  vnei‘1eikzz=2Lpso that (1) decouplesinto different l and kz sectors [9]. We further expand theradial wave functions un Pjcnjj‘ and vn Pjdnjj‘1 in the orthonormal basis set j‘ 2pJ‘	j‘=R=RJ‘1	j‘	 where 	j‘ is the jth zero ofJ‘x. The BdG eigenvalue equation now reduces to amatrix diagonalization problemT‘ ‘‘1 T‘1 !jj0cnj0dnj0  En cnjdnj (4)where Tjj0‘  	2j‘=R2  k2z jj0 and jj0‘ Rdj‘j0‘1. While the different ‘ and kzsectors are completely decoupled in (4), they are coupledthrough the self-consistency equations [16]. Since the BdGequations are invariant under En ! En, unr ! vnr,vnr ! unr, one can simply get the positive energyeigenfunctions for negative ‘ by looking at the negativeenergy eigenfunctions for positive ‘.Ideally one would like to take Ec ! 1 and obtainsolutions independent of this cutoff. In practice, the sizeof Hilbert space grows like RLEcpand to make the09040calculation manageable we choose Ec  9F, R 25k1F , and L  10k1F . We have checked that for thischoice of cutoff, (larger than that in Ref. [12]), our resultsin the uniform case are no more than 5% different from theinfinite cutoff answers at unitarity.Vortex core structure. The order parameter  isplotted in Fig. 1 for various values of 1=kFas. The weakoscillations in , most prominent in the BCS regime,are likely finite size effects with no physical significance[17]. Another very interesting aspect of the BCS regime,clearly visible in Fig. 1, is the presence of two length scalesk1F and  in the T  0 result for . This is in markedcontrast to the Ginzburg-Landau result where GL tanh=. While this effect was, in fact, recognized in theearly superconductivity literature [8] in an Eilenbergercalculation, we provide an elementary analytical derivationhere. Close to the origin, one may ignore  relative to thekinetic energy terms in the BdG equations and find u‘ A1‘ J‘kF, v‘  A1‘ J‘1kF where A‘ are constants.For small  we thus find   gu0v0  A20 kF.We next determine A0 as follows. For large ,   0and u0  1=B0p  coskF exp= and v0 1= B0p  sinkF exp= (up to irrelevant phaseshifts) where B0 is another constant. Matching the largeand small  solutions at   , we get A0  B0k1F 1=2[18]. Finally, using the normalization condition (2) we canfix the constant A0  =kF1=2 which leads to the result  0kF for small . Thus in the BCS limit, theinitial slope of  is set by k1F although the eventualapproach to its uniform value 0 is on a second scale of  @vF=0  k1F .We see that outside of the BCS regime there appears tobe a single length scale in  as seen in Fig. 1. As thecoupling increases toward unitarity the order parametervalue at large , 0 increases toward F and the scale shrinks to k1F . The overall behavior of  across theresonance is shown in Fig. 2(a). At the same time thedensity profile n around the vortex evolves as shown3-2PRL 96, 090403 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending10 MARCH 2006in Fig. 2(b). Near the center of the vortex n ’ n0 a2. The density at the center n0 is a strongly decreasingfunction of 1=kFas as shown in the inset of Fig. 2(b)dropping from approximately 0:8n at 1=kFas  1, cor-responding to a nearly ‘‘full’’ core, to 0:1n at 1=kFas 1, which is a nearly ‘‘empty’’ core.Circulating current. The current circulating aroundthe vortex core is j  svs, where s is the local superfluiddensity and vs  @=2M^. Far away from the vortexcore, j 1= since s is constant. On the other hand,near the vortex core, s 2, so that j . With decreas-ing , the current initially increases until it reaches amaximum, jmax, at . For  <  the kinetic energycost of current flow is much larger than the condensationenergy and the superfluid order parameter is suppressed atthe center of the vortex. In this sense, the maximum currentdensity at  gives us an estimate of the critical currentbeyond which superfluid order is destroyed.Formally the current density is given byj    2@2LmXnlkzl 1Xjdnjjl12^: (5)The profiles of j for various 1=kFas are shown inFig. 3(a). They all show a peak, as explained above. Wefind that [19] the location  of the maximum currentshows a weak nonmonotonic behavior in the vicinity ofunitarity, which is roughly consistent with the nonmono-tonic behavior of the coherence length  as a function of1=kFas predicted earlier; see Fig. 3 of Ref. [20].The peak current jmax, however, shows a nonmonotonicbehavior as a function of 1=kFas, peaking at unitarity asshown in Fig. 3(b). We may understand this nonmonotonicbehavior as follows. On the BCS side, the critical velocity,determined by depairing, is 0=kF and thus jmax nvF0=F. The critical current thus increases as onemoves toward unitarity from the BCS side as 0 increases.At unitarity, universality dictates that the critical velocity 0 0.5 1 1.5 2 0  4  8∆( ρ)/ εFkF ρ1/(kF as) = 1.00.50.0- 0.5- 1.0(a) 0 0.25 0.5 0.75 1 1.25 0  4  8n(ρ)/nkF ρ(b)1.00.50.0- 0.5- 1.01.0-1.00.5n(0)/n(kF as)-1FIG. 2. (a) The order parameter profile and (b) the densityprofile for different values of 1=kFas. Inset: the density at thecenter of vortex as function of 1=kFas.09040must be of order vF so that jmax  nvF. As one goes towardthe BEC side the critical current is now determined not bypair breaking but rather by the collective excitations. TheLandau criterion suggests vc  vFkFaspwhich leads to ajmax which decreases with increasing 1=kFas. We thus findthe interesting result that the mechanism for destruction ofsuperfluidity, as reflected in the maximum current jmax,changes as one goes across the resonance from pair break-ing on the BCS side to collective on the BEC side.Spectrum. We now turn to the fermionic bound statesin the vortex core and their evolution through the BCS-BEC crossover. The BCS-limit results are very well-known[5]. Caroli, de Gennes, and Matricon [5] showed that foreach ‘ and kz there is a bound state, with energy less than0, in the core of the vortex. The lowest energy fermionicexcitation (with ‘  0 and kz  0) has a ‘‘minigap’’20=2F  0. As we increase 1=kFas the spectrumchanges as shown in Figs. 4(a) and 4(b). For clarity weshow the spectrum only as a function of angular momen-tum ‘ at fixed kz  0 (kz dispersion is discussed later).As long as the chemical potential > 0 (which includesunitarity), the bound (continuum) states are those withenergies smaller (larger) than 0. The bound state spec-trum at unitarity [Fig. 4(a)] is not qualitatively differentfrom the BCS limit, except that both the minigap and thelevel spacing are larger, and therefore one has fewer boundstates. Remarkably, the minigap continues to follow20=2F even through unitarity as shown in Fig. 4(c).Once the chemical potential < 0, as in the BECregime shown in Fig. 4(b), the continuum of fermionicexcitations exists for E  jj2  201=2 [2]. We can stilldefine a fermionic bound state by demanding that thecorresponding wave function decays exponentially tozero away from the vortex core. If such states exist, thenit can be easily shown that their energy must lie in theinterval jj  E< jj2  201=2. Remarkably, we findsuch a bound state well into the BEC regime as evidentfrom Fig. 4(b). It is interesting that the off-diagonal poten-tial  produces an Andreev bound state even at an 0 0.1 0.2 0  4  8j(ρ)/nv FkF ρ1/(kF as) = 0.0 1.0- 1.0(a) 0.1 0.2-1  0  1j max/(nvF)(kF as)-1(b)FIG. 3. (a) The current distribution for three different values of1=kFas (b) The peak current jmax vs 1=kFas.3-3 0 0.4 0.8 1.2-15 -5  5  15El1/(kF as) = 0.0∆kz = 0(a) 0 1 2 3-15 -5  5  15El1/(kF as) = 1.0 (µ2+∆2)1/2 |µ|kz = 0(b) 1 2-0.5  0  0.5E min(kF as)-1∆2/2εF(c) 0 0.4-0.5  0  0.5E minkzE(0)/(1-kz2)1/2(d)FIG. 4. The BdG spectrum in the kz  0 sector as a function of‘ for (a) 1=kFas  0:0, i.e., unitarity and (b) 1=kFas  1:0,i.e., on BEC side. (The smaller dots correspond to bound statesliving on the boundary) (c) The minigap (‘  0, kz  0) as afunction of 1=kFas. (d) The kz dependence of the bound stateenergy for ‘  0 at unitarity.PRL 96, 090403 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending10 MARCH 2006energy larger than the maximum value of the potential 0,but we must emphasize that this bound state is below thecontinuum which starts at jj2  201=2. The fermionicbound states are a unique consequence of composite natureof the bosons and are absent in atomic BEC.As a result of motion along the vortex axis, each state forfixed angular momentum ‘ shown in Figs. 4(a) and 4(b)actually broadens into an energy band with kz dispersion.Our calculated kz dependence of the bound state energy forthe l  0 bound state at unitarity is shown in Fig. 4(d). Theenergies (which are discrete due to the finite L along z)continue to follow the BCS-limit prediction [5] E0=1k2z1=2 even at unitarity.Finally, we note that there is a branch of bound states[shown with smaller dots in Figs. 4(a) and 4(b)] which liesbelow the fermionic energy gap in the bulk but differs fromthe vortex core states (which exist only for ‘  0) in that itexists both for positive and negative ‘. From their wavefunctions [19] we deduce that these states are not related tothe vortex core, but are, in fact, trapped near   R due tothe suppression of the order parameter by the hard wallboundary condition [21].Conclusions. The physics of a vortices in the BEC toBCS crossover has led to interesting results (A) through(D) summarized in the introduction. The study of the orderparameter profile in the vortex core and the circulatingcurrent may require new experimental methods whichwill allow more detailed diagnostics. The existence offermionic bound states should be detectable by spectro-09040scopic means, or through the damping of sound, or througheffects of dissipation in vortex dynamics. We hope that theunique properties of vortices that we point out here willstimulate further experimental and theoretical studies onnew methods to probe strongly interacting Fermi gasesacross Feshbach resonance.This work is supported by NASA Grant No. NAG8-1765and NSF Grant No. DMR-0426149.3-4[1] M. W. Zwierlein et al., Nature (London) 435, 1047 (2005).[2] A. J. Leggett in Modern Trends in the Theory of Con-densed Matter, edited by A. Pekalski and R. Przystawa(Springer-Verlag, Berlin, 1980); M. Randeria in BoseEinstein Condensation, edited by A. Griffin, D. Snoke,and S. Stringari (Cambridge, Cambridge, England, 1995).[3] See K. O’Hara et al., Science 298, 2179 (2002); T. L. Ho,Phys. Rev. Lett. 92, 090402 (2004), and references therein.[4] C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett. 92,040403 (2004); M. W. Zwierlein et al., Phys. Rev. Lett. 92,120403 (2004).[5] C. Caroli, P. de Gennes, and J. Matricon, Phys. Lett. 9, 307(1964).[6] That current experiments are for wide resonances is em-phasized in R. Diener and T. L. Ho, cond-mat/0405174;See also D. S. Petrov et al., cond-mat/0502010.[7] While it was recognized in earlier work [8,9], this factnevertheless does not seem to be widely appreciated.[8] L. Kramer and W. Pesch, Z. Phys. 269, 59 (1974).[9] F. Gygi and M. Schlu¨ter, Phys. Rev. B 43, 7609 (1991);J. D. Shore et al., Phys. Rev. Lett. 62, 3089 (1989).[10] C. A. R. Sa´ de Melo, M. Randeria, and J. R. Engelbrecht,Phys. Rev. Lett. 71, 3202 (1993).[11] N. Nygaard et al., Phys. Rev. Lett. 90, 210402 (2003).[12] M. Machida and T. Koyama, Phys. Rev. Lett. 94, 140401(2005).[13] A. Bulgac, cond-mat/0505524.[14] G. Tonini and Y. Castin, cond-mat/0504612.[15] A. Bulgac and Y. Yu, Phys. Rev. Lett. 91, 190404 (2003).[16]   g=2LPnjj0lkzcnjdnj0jlj0l1 and n 2=R2LPnjlkzd2nj are the self-consistency equations.[17] We believe that the smooth  profile seen in Ref. [9],which used a BCS interaction and a very large system size,shows that the oscillations seen in the BCS limit of ourdilute gas model are a finite size artifact, and not Friedel-like oscillations as suggested in Refs. [11,12].[18] Equating A10 J0kF  B01=2 coskFe1, and us-ing J0x  cosx=xpfor x  1, we have A0  B0k1F 1=2.[19] R. Sensarma, M. Randeria, and T. L. Ho (unpublished).[20] J. R. Engelbrecht, M. Randeria, and C. A. R. Sa de Melo,Phys. Rev. B 55, 15 153 (1997).[21] Y. Ohashi and A. Griffin, Phys. Rev. A 72, 013601(2005).

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Vortices in Superfluid Fermi Gases through the BEC to BCS Crossover

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Vortices in Superfluid Fermi Gases through the BEC to BCS Crossover

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