Oberving Condenation in Atomic Fermi Gae (Term Eay for 498ESM, Spring 2004) Ruqing Xu Department of Phyic, UIUC (May 6, 2004) Abtract Oberving condenation in a ga of fermion ha been another intereting topic after the realization of the Boe-Eintein Condenation in atomic gae. The condenation of fermion may happen either in the BEC or the BCS manner, depending on how the fermion are paired to each other. Thi paper give a brief review on the recent breakthrough in thi area.
1 Introduction For boon particle, the well-predicted phenomenon of Boe-Eintein Condenation (BEC) ha been firt realized experimentally in the ytem of trapped alkali atom in 1995. After that, people have been naturally thinking about it equivalent phenomenon in fermion. The Pauli principle wouldn t allow the fermion to condenate into a ingle quantum tate like boon do, therefore the only way for fermion to form a condenate i to pair-up. One imple way for them to pair-up i jut forming diatomic molecule, which are alway boonic and can certainly make the BEC if the temperature i low enough. Another way, which i more complicated, i that the fermion may form weakly-bounded pair like Cooper-pair of electron in a Type I uperconductor, which ha been well-explained by the BCS theory; in thi way the ytem will alo condenate in low temperature, and people ha already ued thi to explain the uperfluidity of 3 He. Whether the fermion will pair-up in the firt way or the other, depend on whether the effective interaction between the particle are repulive or attractive[1]. A repulive interaction correpond to the molecular BEC and an attractive interaction would produce Cooper pair. In comparion, a boon ga can only condenate when their interaction are repulive, otherwie the whole ytem would collape and end with an exploion. Thu it i the Pauli principle that give the fermion the unique way of forming Cooper-pair. Realizing uch kind of condenate in atomic Fermi gae i of ignificant importance for phyic theorit. Ultracold dilute atomic gae are ideal ytem to probe and control quantum emergent phenomena. They offer a unique way to tudy the univeral behavior of trongly-interacting many-body ytem, uch a high-t c uperconductor. An ultimate goal for experimentalit ha been the realization of uperfluidity in atomic Fermi gae. However, realizing and probing uch a condenation i much more challenging than the imple BEC for boon. The required temperature for a BCS-typed condenation i only a mall fraction of the Fermi temperature T F (about 2 10 TF in a high-t c uperconductor), till lower than the currently available lowet temperature in cooled atomic gae. Alo, in ultra-low temperature identical fermion tend to avoid head-on colliion which i eential for the evaporate-cooling proce and therefore the ytem i hard to be cooled down further. Another eriou problem i how to probe the Cooper pair when they have been formed in an atomic ga.
Fortunately, people have olved the firt two problem. In 1999, DeMarco and Jin olved the cooling problem[2]. In 2001 theorit pointed out that a uperfluid phae tranition may occur at a high critical temperature ( ~ 0.5T F for 40 K atomic ga) when a Fehbach reonance paring occur in a dilute Fermi ga[3]. After thee breakthrough people have made ignificant progree toward the fermionic condenation and, in lat year (2003), three experimental group ha reported their realization of the molecular BEC in atomic Fermi gae[4-6]. After realizing the molecular BEC, people have in principle been able to create the BCS-typed condenation ince they can already tune the effective interaction between the trapped atom from repulive to attractive by taking advantage of the Fehbach reonance. But the problem i how to prove the exitence of the condenate on the BCS ide? A paper in Jan. 2004[7] reported a recent experiment which might be the firt time for people to produce and prove the exitence of uch a degenerate quantum ytem. The ret part of thi report will mainly focu on thi recent experiment. We will firt give a brief decription of the experiment and it reult; then ome baic concept, like the -wave cattering and the Fehbach reonance, will be reviewed in detail to have a better undertanding of the phyical procee involved in thi experiment. After that we will dicu the remaining intereting problem aroe from the experiment. 2 The Experiment 2.1 Decription of the Experiment In Jan. 2004 the team of JILA, NIST and Univerity of Colorado at Boulder declared that they oberved condenation of fermionic atom pair in the BCS-BEC croover regime[7]. In thi experiment, a dilute ga of fermionic 40 K atom were trapped and cooled in a magnetic trap and then loaded in to a far-off reonance optic dipole trap. The 40 K atom ha a total atomic pin f = 9/2 in it ground tate. In the initial tage, an incoherent mixture of the 9 /2, 7/2 and 9 /2, 9/2 wa prepared in order to realize -wave colliion in the ultracold Fermi ga (we will explain thi in more detail later); in thi way the ga wa evaporated and cooled to temperature far below the Fermi temperature( T F 0.6µ K ). Then a magnetic field Bhold wa applied to the ytem with it magnitude near to the Fehbach reonance ( B 0 = 202.10 ± 0.7G ). Note that the interaction between the atom are effectively repulive when B < B, which correpond to the Boe-Eintein Condenation of diatomic hold 0
molecule; and effectively attractive when B hold > B 0, which correpond to the BCS-type condenation of Cooper pair. In order to probe the condenation after the magnetic field i applied, the JILA team ued an idea of projection : they weep the magnetic field rapidly down by 10G, which put the ga far on the BEC ide of the reonance, where it i weakly interacting. The invere peed of thi weep i 50 µ / G, which, they believed, wa fat enough to prevent forming of a molecular condenate but alo ufficiently low to convert the original Cooper pair into bounded molecule. Along with thi weep, the ga wa imultaneouly releaed from the trap and allowed for free expanion; then after 17m of expanion the molecule are electively detected uing radio-frequency photodiociation immediately followed by pin-elective aborption imaging; -- thee technique were baically employed to determine the fraction of molecule having zero-momentum, which hould correpond to the fraction of initial condenation. Figure 1 and 2 how the main reult of the JILA team experiment. Figure 1 preent the meaured condenate fraction N / 0 N a a function of the magnetic-field detuning from the reonance, B = Bhold B0. The data were taken with two different hold time t hold, which mean the time in which the ytem tayed with the magnetic field B hold before the rapid weep. Figure 2 how the time-of flight image for the fermionic condenate with three different B hold value. Thee reult clearly, and for the firt time, demontrated that a condenate did form at the BCS ide nearby the Fehbach reonance, where the BEC of diatomic molecule i impoible. The reearcher called it a fermionic condenate. Figure 1. Meaured condenate fraction a a function of detuning from the Fehbach reonance B = Bhold B0. The region between the dahed line i the region of the Fehbach reonance. Circle correpond to t hold = 2m and triangle correpond to t = 30m hold
Figure 2. Time of flight image taken after the projection of the fermionic ytem onto a molecular ga. B =0.12, 0.25 and 0.55 G(left to right) on the BCS ide of the reonance. Two month after thi report, another team in MIT reported a imilar obervation which wa performed with a different type of atom: 6 Li[8]. In contrat to the previou experiment, where the condenate fraction wa at mot 15%, the MIT team reported a high fraction of condenate, which wa up to 80%. 2.2 Some Baic Concept In The Experiment To undertand thi new experiment in more detail, we need to get into ome more baic idea/concept. 2.2.1 The -wave cattering The -wave cattering length i a baic quantity in decribing the interaction in dilute ultra-cold atomic gae (either boonic or fermionic). Firtly, a baic aumption in thee problem i that, in a dilute atomic ga, particle interact eentially a binary atom ytem. Specifically, the atom-atom interaction can be decribed a in binary colliion, partly becaue the colliion complex i o hort lived that it interaction with other particle may be neglected. The colliion between two atom can be treated a a tandard quantum cattering problem. Here we can make another important implification. For ultra-cold atom, whoe thermal-energy are very mall, the 2-body cattering with angular momentum l 0 can be neglected, ince for l 0 the probability of finding two atom at a ditance r 0 from each other fall off a ( kr )2, where k i the relative wavevector. Thu we may retrict our dicuion to the l = 0 (-wave) cattering. The o-called -wave cattering length, ψ ( r ) a, i defined a follow, ( a ) r in k r, 0 l
where ( r) ψ i the wavefunction of the cattered -wave; and we can ee that the -wave phae hift δ = ka. The value of a i in general a function not only of the chemical and iotopic pecie involved but of the hyperfine indice of the two atom, and even the external magnetic field. A poitive a correpond to a effectively repulive interaction between the two atom; and a negative a mean attractive. The value of a alo reflect the trength of the interaction. Further, it not hard to how that for fermion, when a > 0 they tend to form diatomic molecule and when a < 0 they prefer to pair-up like Cooper-pair in a uperconductor. A brief derivation on thi could be found in Ref.[1] by A. Leggett in 1980. Interetingly, although the -wave cattering alway work in boonic ytem, it not alway the cae for fermion! Actually, the patial wavefunction of two identical fermion with no other degree of freedom will never take the form of an -wave due to the Pauli principle, which mean identical fermion can only interact through a p-wave cattering which i much weaker than the -wave proce; and thi ha already made a big problem for experimentalit to cool down atomic Fermi gae. Becaue the fermionic atom do not make head-on colliion, one cannot perform the evaporate-cooling which ha been eential in cooling down the boonic atom to achieve the BEC. Fortunately, people have found way to overcome thi difficulty. DeMarco and Jin from the JILA team circumvented thi roadblock in 1999[2]. The trick they invented i uing a mixture of two pin tate of the ame atom, between which the -wave colliion are allowed. Thi method ha proved to be quite effective and that why we aw they include both the 9 /2, 7/2 and 9 /2, 9/2 tate in thi latet experiment (ee the previou ection). 2.2.2 The Fehbach Reonance The ucce of the current experiment in thi area are all directly baed on the ability to tune the value of the -wave cattering length, i.e., the effective interaction trength (and alo ign) between the atom. Such ability i achieved by taking advantage of the phyical phenomenon called the Fehbach reonance. A comprehenive dicuion of Fehbach reonance in atomic condenate ytem can be found in Ref.[9], here let jut have a brief review of the baic idea. Conider a ytem with two degree of freedom, one of which i aociated to the fragmentation of the ytem. If at a reonance the ytem would turn into a
bound tate if the coupling between thee two degree of freedom i et to zero, then uch a reonance i a Fehbach reonance. For a imple example, we can conider a rare ga atom with a vibrationally excited diatomic molecule. When the rare ga atom i far from the molecule, it ee a weakly attractive potential. During the colliion it may excite the molecule into an excited vibrational tate, meanwhile it loe ome energy and fall into the well of the attractive potential. It would tay trapped in thi bound tate if the coupling between the movement of the rare ga atom and the vibration of the molecule were zero; while in reality thi non-zero coupling turn thi bound tate into a Fehbach reonance and i reponible for it finite lifetime. Here in our cae of two-body cattering of alkali atom, the two degree of freedom involved in the Fehbach reonance are the ditance between the atom and the total pin of the ytem. More pecifically, in the preence of an external magnetic field, an atom hould tay in an eigentate f, m f, where f = + i i the total atomic pin, with and i referring to the total electronic and nuclear pin repectively. A two-atom cattering proce would be trivial if the total pin of the ytem F= f 1 + f 2 were conerved. However, the hyperfine interaction term in the Hamiltonian ha the form of ( ) H = cont. i + i, hf 1 1 2 2 which doe not commute with the total pin F. Therefore during the cattering, thi term will give ome probability for the ytem to undergo a pin-flip and thu turn into another channel of molecular diociation. The Fehbach reonance occur if the interaction potential of another pin-flipped binary atom channel, acceible from the incident channel by virtue of the hyperfine interaction, upport a bound tate with energy E m near the continuum level of the incident channel. Figure 3 (taken from Ref.[9]) give a chematic repreentation of uch energy level. The dahed curve correpond to the potential curve of the incident channel and the olid curve correpond to another channel with a different total pin. The middle horizontal line give the Fehbach reonant energy, at which a bound tate exit in the upper potential curve. Since the energy difference between thee different pin channel depend on the external magnetic field, people are able to tune the two-atom cattering proce to be in the upper or lower neighborhood of the reonance by tuning the magnetic field. Ref.[9] give the dependence of the effective cattering length on the external magnetic field a:
a eff = a ( B B ) γ R B where a i the cattering length calculated with the incident channel potential only, B R i the magnetic field value exactly at reonance, i the energy gap between the two channel when r (a hown in Fig.3), and γ i a quantity depending on the matrix element of the hyperfine interaction. From thi formula it clear that when B i tuned to approach the value of either poitive or negative ign. B R, a eff goe to infinity and can take Figure 3. Schematic repreentation of the molecular potential of the incident and intermediate tate channel in a Fehbach reonance (from Ref.[9]). Figure 4. Scattering length veru magnetic field near the Fehbach reonance (from Ref.[10])
Such a dependence of a eff on B ha been verified in experiment. Figure 4 how the meaured -wave cattering length between the m f = 9 / 2 and m = 5/ 2 tate of 40 K atom near the Fehbach reonance[10]. The divergence of f a eff at reonance i clearly hown. 3 Dicuion Although the JILA team obervation wa performed uccefully and ha been widely recognized a an important tep forward, their reult ha alo arien a lot of argument. The mot important quetion i that, did they really meaure the Cooper pair on the BCS ide of the condenation? After their rapid weep of the magnetic field, do thoe molecule they oberved directly correpond to the initial weakly-bounded Cooper pair? According to a brief report on thi experiment in Phyic Today by B. Levi[11], theorit Tin-Lun Ho ha made ome doubt on thi iue. Ho argued that the ize of the pair formed on the BCS ide of the reonance hould be much larger than the ize of the molecule formed at the BEC ide, and uch difference would not allow any overlap to form molecule after the weep of the field. Furthermore, he aid the colliion rate in the trongly interacting regime hould allow enough colliion during the magnetic field weep that the momentum ditribution of the molecular condenate can not repreent that of the original BCS ytem. In the later paper by the MIT group[8], who performed a imilar experiment with 6 Li and oberved the imilar phenomena, the author alo explained their reult in a different way from the JILA team argument. They aid that, at the Fehbach reonance, a molecular tate ha a finite lifetime; and while in the preence of a Fermi ea, it lifetime will be increaed due to Pauli blocking from all the other particle. The molecular level will be populated until it energy become larger than twice the Fermi energy correponding to the total number of atom. The BEC-BCS croover hould occur at thi point intead of the location of the two-atom Fehbach reonance. Therefore, intead of declaring having oberved atomic Cooper pair, the MIT cientit tentatively interpreted their reult a a BEC of pair of atom which are molecular in character and tabilized by the exitence of the Fermi ea ; and they remained that the exact nature of thee atom pair were yet to be elucidated. Then, one week after the publication of the MIT group paper, two
theorit from Netherland reported that they believe the data reported by the JILA group in Ref.[7] can be undertood in term of a BEC of molecule[12]. In concluion, depite the remaining myterie, people have for the firt time made a ucceful attempt to oberve an unexplored territory of the BCS-typed condenation of the trongly-interacting fermionic atom. Thi ha been an important tep toward forming an artificial fermionic uperfluid in atomic ga; and alo opened the way toward tudying the ideal ytem of trongly-interacting fermion which might be helpful to the undertanding of the high-t c uperconductor. Thi area i definitely expected to have much more exciting reult coming out in the near future. Reference [1] A. Leggett, in Modern Trend in the Theory of Condened Matter (Springer-Verlag, Berlin, 1980), pp. 13 27 [2] B. DeMarco, D. Jin, Science 285, 5434 (1999) [3] M. Holland, S. Kokkelman, M. Chiofalo, and R. Waler, Phy. Rev. Lett. 87, 120406 (2001) [4] M. Greiner, C. Regal, and D. Jin, Nature 426, 537 (2003) [5] S. Jochim, M. Bartentein, A. Altmeyer et al., Science 302, 2101 (2003) [6] M. Zwierlein, C. Stan, C. Schunck et al, Phy. Rev. Lett. 91, 250401 (2003) [7] C. Regal, M. Greiner, D. Jin, Phy. Rev. Lett. 92, 040403 (2004) [8] M. Zwierlein, C. Stan, C. Schunck et al., Phy. Rev. Lett. 92, 120403 (2004) [9] E. Timmerman, Phyic Report 315, 199 (1999) [10] C. Regal and D. Jin, Phy. Rev. Lett. 90, 230404 (2003) [11] B. Levi, Phyic Today 57, No.3, 21 (2004) [12] G. Falco and H. Stoof, Phy. Rev. Lett. 92, 130401 (2004)