V ol . 1 0 1 ( 2002)
A CT A P HY SIC A P O LO N IC A A
No . 3
P ro ceed in gs of t h e I n t er n at iona l Co n f er ence \ Qu an tu m O p t i cs V " , Ko ƒci el isko, Po la nd , 200 1
R eson an ce Su p er Ûu id i t y in a Qu antu m
De gen erate Ferm i G as
S. K ok k el man s , M. H olla n d, R . W als er
Ê
JI L A , U ni v er sit y of Col or ado and Na ti onal Insti t ute of Standa rds and T echno lo gy
Bo ul der, Col ora do 80 309-0440, U. S.A.
and M. Chi of al o
Scuol a No r mal e Superi ore, Pi azza dei Caval i eri 7, 56126 Pi sa, Ita l y
W e consi der t he sup erÛuid ph ase transition that arises when a F eshbach
resonance pairin g occurs in a dilute F ermi gas. T his is related to the phenomenon of sup ercondu cti v it y describ ed by the seminal Bardeen { Coop er{
Schrie ˜er theory . In supercondu ctivi ty , the phase transition is caused by a
couplin g betw een pairs of electrons w ithin the medium. T his coupli ng is perturbative and leads to a critical temp erature T c w hich is small compared to
the Fermi temp erature T F . Even high- T c sup erconductors typically ha ve a
critical temp erature which is two orders of magnitude below T F . H ere w e describ e a resonance pairing mechanism in a quantum degenerate gas of potassium ( 4 0 K ) atoms w hich leads to sup erÛui di ty in a no vel regime | a regime
that promises the unique opp ortunity to exp erimentally study the cross-over
from the Bardeen À C oop er{Schrie˜er phase of w eakly- coupled fermions to
the Bose{Einstei n condensate of strongly- boun d comp osite bosons. W e Ùnd
that the transitio n to a superÛuid phase is possible at the high critical temp erature of about 0 :5 T F . I t should b e straightf orw ard to verif y this prediction, since these temp eratures can already be achieved exp erimentally .
PACS numb ers: 03.75.Fi, 67. 60. {g, 74.20. { z
1. I n t r o d u ct io n
The pheno m enon of superÛuidi ty i s closely rel ated to Bo se{ Ei nstei n condensati on (BEC), as was shown i n the founda ti on of the mi croscopi c theo ry of
superÛui d 4 He i n the 1960' s. In b osoni c Ûuids the pha se tra nsiti on i s m ar ked by
the app earance of a m acroscopi c numb er of bosons in the l owest qua ntum sta te.
In ferm i oni c system s the occurrence of superconducti vi ty and superÛuidi ty i n suÊ
corr esp on din g au t h o r; e-m ail : servaa s. kok kel ma ns@ji la. color a do .edu
(3 87)
388
S. K okkelman s, M. H ol l and, R. Wal ser , M. C hi ofal o
p erconducto rs and l i qui d 3 He, i s due to the ri se of a pai ri ng Ùeld and thereby , i n
a general i zed sense, to a condensati on of Co oper pa i rs.
The study of superÛui d pha se tra nsiti ons i n ferm i on and b oson system s ha s
pl ayed an i m p orta n t ro le in the developm ent of m any areas of qua ntum physi cs.
Thei r chara cteri sti cs determ i ne the observed pro perti es of som e of the most di sti nct
system s im agina bl e, i ncl udi ng the cosm ol ogy of neutro n sta rs, the no n- vi scous Ûow
of superÛuid l iqui d heli um , the non- resisti ve currents in superconducto rs, and the
structure and dyna mi cs of m i croscopi c elementa l nucl ei. Recentl y, physi ci sts ha ve
succeeded i n dem onstra ti ng the creati on of weakl y i ntera cti ng qua ntum Ûuids by
cool i ng di l ute gases to tem p eratures i n the na nokel vi n scal e. For these near i deal
gases, reachi ng such i ncredi bl y lo w tem pera tures is requi red i n order to cross the
thresho l d for superÛuid pro p erti es to emerge. These system s o˜er great o pp ortuni ti es for study since they can b e created i n ta bl e-to p exp eri ments, mani pul ated by
l aser and m agneti c Ùelds whi ch can be contro l l ed wi th hi gh preci sion, and di rectl y
observed usi ng conventio na l opti cs. Furtherm ore thei r m icro scopi c b eha vi or can
b e understo od theo reti cal l y fro m Ùrst pri nci pl es. Observ ati ons of Bo se{Ei nstei n
condensati on [1], and demonstra ti ons of the near i deal degenerate Ferm i gas [2],
are b ecomi ng fai rl y routi ne i n ato m ic physi cs | som ethi ng whi ch woul d ha ve b een
ha rd to foresee even ten years ago.
The pheno m enol ogy of superÛuid di lute gases can b e qui te di sti nct fro m tha t
of condensed matter systems. In thi s l etter, we present a stri ki ng i l l ustra ti on of thi s
Fig. 1.
A log{lo g plot show ing six distinct
temp erature
normalized
Tc
regimes for quantum Ûuids. T he transitio n
is show n as a function of the relev ant gap energy
2Â
. Both quantities are
by an e˜ective F ermi temp erature
TF
. F or the BC S systems in region ( a ),
and the systems in the cross- over region ( b ),
2Â
is the energy needed to break up a
fermion pair, and
Ê
TF
Ê
is the Fermi energy . F or the systems in region ( c ), which are all
strongly bound comp osite bosons and exhibi t BEC phenomenol ogy,
2Â
is the smallest
energy needed to break the comp osite boson up into two fermions, i. e. ionizati on to a
charged atomic core and an electron, and
Ê
TF
is the ionic F ermi temp erature.
Resonance Super Ûuidi t y i n a Quant um Degenerate Fer mi Gas
389
p oi nt by predi cti ng the existence of a Feshbach resona nce superÛuidi ty i n a gas of
ferm i oni c pota ssium ato ms. Thi s system has an ul tra hi gh cri ti cal pha se tra nsi ti on
tem p erature i n close pro xi m it y to the Fermi tem p erature. Thi s is a no vel regi me for
qua ntum Ûuids, as i l l ustra ted i n Fi g. 1, where our system a nd others whi ch exhi bi t
superÛui di ty or BEC are com pared. Sim pl y by m odi fying a contro l pa ram eter, i n
thi s case the streng th of m agneti c Ùeld, the system we consider can p otenti al l y
expl ore the cross-o ver regi m e b etween the Ba rdeen{ Co op erÀ Schri e˜er (BCS) [3]
tra nsiti on of weakl y-coupl ed ferm i on pa i rs and the Bo se{Ei nstei n condensati on of
stro ngl y- b ound com p osite pa rti cl es [4]. Thi s i s an i ntri gui ng regi m e for qua ntum
Ûuids as i t bri dges the physi cs of superconduc to rs and sup erÛuid 3 He, a nd the
physi cs of superÛui d 4 He and b osoni c al kal i gases. Non- resona nt pai ri ng appl i ed
to a di l ute ga s yi elds a T c tha t depends exp onenti al ly on the i nv erse scatteri ng
l ength [5], as wi l l b e poi nted out i n the fol l owing section. Typi cal l y thi s results i n
a cri ti cal tem p erature of order T c ¤ 1 0 4 T F or T c ¤ 10
K, whi ch i s way out
of reach i n current exp eri ments. Ma ny qua li ta ti ve features of the na ture of the
superÛui d pha se tra nsiti on are m odi Ùed i n the presence of a resonance coupl i ng,
i ncl udi ng the parti cipa ti on of al l ferm i ons i n the pa iri ng Ùeld and the form ati on of
a Bo se{Ei nstei n condensate of mol ecules at the cri ti cal p oi nt. Thi s i s i l l ustra ted
À
390
S. Kokkel mans, M. H ol l and, R. Wal ser, M. C hi ofal o
i n Fi g. 2, where the na ture of a resona nce pai ri ng m echani sm i s com pa red to the
case of non- resona nce pai ri ng.
The system we study consi sts of an ensembl e of ferm ioni c 4 0 K ato ms equal l y
di stri buted between the two hyp erÙne sta tes whi ch ha ve the l owest i nterna l energy
i n the presence of a m agneti c Ùeld. W e cal cul ate the ful l therm odyna mi cs for
thi s resona nt superÛuid system vi a a reno rm al ized l ow energy Ùeld theory [6].
W e trea t expl i citl y a short ra nge quasib ound resona nt sta te by extendi ng the
theo ry gi ven i n Refs. [7]. The pa ra meters of the theo ry are uni quel y speciÙed
by the kno wn dependence o f the scatteri ng pro p erti es on magneti c Ùeld. In a
di l ute gas, al l i ntera cti ons are assumed to o ccur thro ugh bi nary col l i sions descri bed
by a two -b o dy i ntera cti on. The pro perti es o f the scatteri ng are determ i ned by
the p ositi ons of the b ound sta tes i n the i ntera cti on potenti al s. In the l ow energy
reg im e, onl y the hi ghest b ound states pl ay an i m p orta n t rol e, and the scatteri ng
i s com pl etely descri bed by the s -wa ve pha se shi ft chara cteri zed by the scatteri ng
l ength a . In a two -b ody potenti al , a b ound sta te may l i e near thresho l d and give
ri se to a very l arg e val ue of the scatteri ng l ength. Thi s occurs, fo r exam pl e, i n
the tri pl et p otenti al of 6 Li whi ch yi elds a scatteri ng l ength o f ab out À 2 0 0 0 a 0 [8].
Acco rdi ng to the conv enti onal BCS theo ry , thi s woul d im pl y a m uch l arg er val ue
for the cri ti cal tem p erature [9] tha n the typi cal va lue for no nresona nt scatteri ng
m enti oned previ ously. Mo reo ver, i n a m ul ti -channel system, a b ound sta te may
cro ss the thresho ld energy as a functi on of m agneti c Ùeld and enter the conti nuum ,
resul ti ng i n a Ùeld- dependent Feshbach scatteri ng resonance [10]. As thi s occurs,
a dra m ati c mo di Ùcati on of the scatteri ng l ength i s observed (see Fi g. 3).
Fig. 3.
Scattering
length as a function
of magnetic Ùeld, for a collisi on betw een
atoms prepared in the two low est hyp erÙne states. T he resonance Ùeld value is
1 96
gauss and the w idth is equal to 7.7 gauss [16]. T he asymptotic
by a Feshbac h resonance.
40
B
K
=
b ehavior is caused
Resonance Super Ûuidi t y i n a Quant um Degenerate Fer mi Gas
391
2 . P r o b l em s wi t h B C S t h eo r y cl o se t o r eso n an ce
The BCS theo ry of superconduc ti vi ty appl i ed to a di l ute gas considers bi na ry
i ntera cti ons b etween parti cles in two di sti ngui shabl e qua ntum sta tes, say j " i and
j # i . For a uni form system , the ferm i oni c Ùeld o perato rs m ay b e Fouri er- expanded
i n a b ox wi th p eri odi c b ounda ry condi ti ons gi vi ng wavev ector- k dependent creati on and a nni hi l ati o n opera to rs a k ¥ and a kk ¥ for states j ¥ i . At l ow energy, the bi na ry scatteri ng pro cessesare assumed to b e compl etely chara cteri zed by the - wave
scatteri ng l ength
i n term s of a conta ct qua sip otenti al
, where
i s the numb er density . The Ham i lto nia n descri bi ng such a system i s gi ven by
y
(1 )
where
i s the ki neti c energy,
i s the m ass, and the constra int
g ives mom entum conserv ati on.
For a negati ve scatteri ng length, the therm odyna m ic pro p erti es of the gas
show a sup erÛuid pha se tra nsi ti on at a cri ti cal tem p erature
whi ch ari ses due to
an i nsta bi l i ty to wards the form atio n of Coo per- pai rs. W hen the gas i s di l ute, as
chara cteri zed by the i nequa l i ty
(o r equi v alently
, where
is
the Ferm i wavenum ber), the appl i cati on of mean- Ùeld theo ry gi ves a wel l -known
sol uti on for the ra ti o of
to the Ferm i tem p erature
[5]
exp
(2 )
The exact pref acto r to the exponenti al depends on the preci se fo rm of the ana l yti c
i nteg ra l appro xi mati ons made i n the deri vati on. Several pa pers ha ve p oi nted out
tha t the presence of a scatteri ng resonance i n di lute al kal i gases can b e used to
obta i n a very l arg e negati ve val ue for the scatteri ng l ength [9]. Thi s pro m i ses the
opp ortuni ty for the system to enter the hi gh- superÛuidi ty reg im e as the ra ti o i n
Eq. (2 ) appro aches uni ty . Howev er, di rect appl i cati on of the BCS theo ry close to
resona nce then b ecom es speculati ve due to the p otenti al brea kdo wn of a num b er
of underl yi ng assumpti ons:
1. Exa ctl y on resonance the theo ry fai l s as the scatteri ng l ength pa ssesthro ugh
and the Ham i lto ni an i n Eq. (1 ) canno t b e deÙned.
2. For the m ean Ùeld appro ach to be accura te i t i s requi red tha t there are many
pa rti cl es i nside a vol ume associ ated wi th the spati al scal e of a Coop er- pai r.
Thi s condi ti on b egi ns to break do wn a s
appro aches
.
3. The theo ry of the di l ute gas i s form ula ted on a p erturba ti on appro ach ba sed
on an expansion in the sm al l para m eter
. W hen thi s para m eter appro aches uni ty the p erturba ti on theory fai l s to converg e.
These p oints show tha t care shoul d b e ta ken i n appl yi ng Eq. (2 ) near the
p oi nt of resona nce where the ba sis for the conven ti onal m ean- Ùeld theo ry is no t
wel l f ounded.
392
S. Kokkel mans, M. H ol l and, R. Wal ser, M. C hi ofal o
D espite these li m i tatio ns, on general gro unds, one woul d expect to be abl e to
deri ve a reno rm a li zabl e l ow-energ y e˜ecti ve Ùeld theory even i n clo se pro xi m i ty to
a resona nce. Thi s statem ent i s ba sed on the i denti Ùcati on tha t at rel evant densiti es
the ra ng e of the i nterpa rti cle p otenti al i s al ways orders of ma gnitude smal ler tha n
the i nterpa rti cle spaci ng. Here we present a theo ry of superÛuidi ty i n a gas of di l ute
ferm i oni c ato m s whi ch ha ndl es correctl y the scatteri ng resona nce and pl aces the
tra nsiti on tem p erature to the superÛui d sta te i n the experi m enta l ly accessibl e
ra ng e.
W hi l e the scatteri ng l ength a usual l y chara cteri zes the ra ng e of the i nterato m i c p otenti al fo r a col li sion, thi s i s a p oor appro xi m ati on i n the vi cini ty of
a scatteri ng resonance. The scatteri ng pro p erti es are com pl etel y determ i ned by
the p ositi ons of the b ound sta tes i n the i ntera cti on potenti al s. In a mul ticha nnel
system , a b ound sta te may cross the thresho ld as a functi on of magneti c Ùeld
and enter the conti nuum , resul ti ng i n a Ùeld-dep endent Feshbach scatteri ng resona nce [10]. As thi s occurs, the scatteri ng l ength becom es stro ngl y dep endent on
the Ùeld, and exactl y at thresho l d it chang es sign by passing thro ugh Ï 1 .
3 . R eso n an ce p ai r i n g t h eo r y
W hen such resona nce pro cesseso ccur, i t i s necessary to form ul ate the Ha mi l to ni an by separa ti ng out the resona nce sta te and trea ti ng i t expl i ci tl y. Thi s i s m oti vated by the m i croscopi c i denti Ùcati on of tw o typ es of scatteri ng contri buti ons:
one fro m the scatteri ng resonance, and one fro m the ba ckg ro und no n- resonant
pro cesses tha t incl udes the contri buti ons fro m al l the other b ound sta tes. The
no n- resonant contri buti ons gi ve ri se to a ba ckgro und scatteri ng l ength a b g whi ch
i s a go od chara cteri zati on of the p otenti al ra nge. The corresp ondi ng qua sip otenti al in tha t case i s given by U bg = 4 ¤ ñh 2 a bg
. The Feshbach resona nce occurs
due to a coupl ing wi th a mol ecular sta te, tha t i s l ong -l i ved i n com pari son wi th
chara cteri sti c no n- resona nt col l i sion ti mescal es. Thi s sta te i s a com p osite b oson
whi ch i s descri b ed by bosoni c anni hi l ati on op erato rs
. It i s para m eteri zed by a
detuni ng energy fro m thresho l d, deno ted by
, tha t i s dep endent o n the value of
the m agneti c Ùeld. The coupl i ng streng th of
to the two -pa rti cle conti nuum i s
wel l chara cteri zed by a sing l e coupl i ng consta nt , i ndep endent of . These considera ti ons im ply tha t the Ha mi lto ni an gi ven i n Eq. (1 ) is no t su£ cient to account for
the i m porta nt resona nce pro cessesand must b e extended to i ncorp ora te expl i citl y
the coupl i ng b etween the ato m ic and m ol ecular gases
...
+
+
(3 )
Evo luti on generated by thi s Ham i lto ni an conserves the pa rti cle num ber
. Note tha t the Ham i lto ni an does not conta i n
Resonance Super Ûuidi t y i n a Quant um Degenerate Fer mi Gas
393
expl i citl y, and tha t the Ùeld dep endence of the scatteri ng i s com pl etel y characteri zed by the pa ra meters: g ; ¡ , and U bg . The m agni tude of g i s deri ved i n
the fol l owi ng way. W e deÙne ç as the pro duct of the m agneti c Ùeld wi dth of
the resonance and the magneti c m oment di ˜erence of the Feshbach sta te and the
conti nuum sta te. For l arg e val ues of ¡ , the b oson Ùeld bkk can b e adi aba ti cal l y eli m p
i na ted fro m the theo ry, and then g =
çU bg i s requi red i n order for the scatteri ng
pro perti es to ha ve the correct dependence on m agneti c Ùeld .
The essential p oi nt i s tha t thi s Ha mi l to nia n, founded on the m icro scopi c basis
of resonance scatteri ng , i s wel l- b ehaved at al l detuni ngs ; even for the pa tho l ogi cal
case of exact resonance. The di l uteness cri teri on is now gi ven by constra i nts whi ch
requi re both the p otenti al ra nge and the spati a l extent of the Feshbach resonance
sta te, to be m uch smal l er tha n the i nterpa rti cle spacing (e.g.
1).
W e appl y thi s Ham i l to nia n to deri ve the self- consistent mean-Ùelds for gi ven
therm o dyna m i c constra i nts by form ula ti ng a Hartree{ Fo ck{ Bo gol i ub o v theory .
The m ean- Ùelds present incl ude the ferm i on numb er
, the m ol ecule Ùeld
ta ken to b e a classical Ùeld, and the pai ri ng- Ùeld
. It i s wel l known tha t such a theo ry m ust b e renorm al i zed
i n order to rem o ve the ul tra vi ol et di vergence whi ch ari ses fro m the i ncorp ora ti on
of second- order vacuum contri buti ons. Thi s i mpl ies repl acing the physi cal pa ra meters i n the Ham i lto ni an,
, and , by reno rm al i zed val ues so tha t observ abl esare
i ndep endent of a hi gh m omentum cut- o˜ used i n the f orm ul ati on o f the e˜ecti ve
Ùeld- theo ry [13]. In order to di agona l i zethe Ham i l tonia n, we construct Bo gol i ub o v
qua siparti cles accordi ng to the general canoni cal tra nsform ati on [14]
a
cos
e sin
e sin
cos
(4 )
G i ven sing l e pa rti cl e energies,
, where
is the chem i cal potenti al , and the gap pa ra meter i n the qua sipa rti cl e spectrum
,
the two tra nsf orm ati on a ngl es are speciÙed as ta n
and
exp i . The corresp ondi ng qua siparti cle spectrum i s
. D ro ppi ng term s of hi gher order tha n qua dra ti c i n the ferm i on op erato rs, gi ves the resul ti ng m any- b ody Ham i lto ni an
(5 )
whi ch i s no w i n di agona l f orm .
394
S. Kokkel mans, M. H ol l and, R. Wal ser, M. C hi ofal o
4 . Th er m o d yn am i c so l u t io n s
The next ta sk i s to cal cul ate the therm odyna mic soluti ons. Equi l i bri um p opul ati o ns for the qua sipa rti cles are gi ven by the Ferm i { Di ra c di stri buti on. The
ferm i on num b er and pa i ri ng Ùeld are not onl y i nputs to the Ha mi l to ni an, but al so
determ i ne the qua sipa rti cl e spectrum . Theref o re, they m ust b e self- consistent wi th
the values deri ved by sum mi ng the rel evant equi l i bri um density m atri x el ements
o ver al l wave num b ers. In pra cti ce, at a gi ven tem p erature, chem i cal p otenti al , and
m ol ecule num b er ¢ m , thi s requi res an i tera ti ve m etho d to l ocate self- consistent val ues for f and p . The va lue of ¢ m i s cal cul ated by m inim izi ng the gra nd p otenti al
` G = À k b T ln
at Ùxed tem pera ture and chem ical p otenti al , wi th
denoti ng Bo l tzm ann' s consta nt. The parti ti on functi on
T r exp
i s found fro m Eq. (5 ). Thi s pro cedure i s m athem ati cal ly equi va lent to mi nim izi ng the Hel mho l tz free-energy at Ùxed tem p erature and density and corresponds
uni quel y to the m axi mum entro py sol uti on. Thi s sol uti on ha s an associated pa rti cl e num b er,
, ta ken at consta nt tem p erature and vol um e, whi ch
m ust m atch the actua l pa rti cle density of the gas, so tha t the Ùnal step i s to adj ust the chemi cal potenti al unti l thi s condi ti on i s sati sÙed. The who l e pro cedure i s
rep eated o ver a rang e of tem p eratures to determ ine the l o cus of therm odyna m ic
equi l i bri um p oi nts. For l arg e p ositi ve detuni ngs, where the m ol ecule Ùeld coul d b e
el im i nated fro m the theo ry enti rel y, regul ar BCS theo ry emerges. For thi s case,
when the scatteri ng l ength is negati ve the b ehavi our of the cri ti cal tem p erature
on
i s gi ven by the usual exponenti al l aw [5].
In thi s pa p er, we use ferm i oni c K ato ms as an exam pl e of the a ppl i cati on of thi s theo ry . The va lues of our i ntera cti on pa ra meters
and
657 K are obta i ned fro m [15]. W e Ùx the to ta l density to b e
cm , a typi cal experi m enta l val ue expected for thi s quantum degenera te gas i n an opti cal tra p. W e set the detuni ng to b e
, so tha t the
qua si-b ound state i s detuned sli ghtl y abo ve the ato m i c resonance. For a tem p erature a bo ve , the grand potenti al surf ace i s shaped l i ke a b owl , and the value of
whi ch m i ni m izes the grand p otenti al i s
, associated wi th the self- consistent
sol uti on
. For
, the gra nd po tenti al surf ace i s shaped l i ke a Mexi can
ha t, and i ts m i ni mum i s gi ven by
wi th a no n- zero ampl itude and an undeterm i ned pha se. The superÛuid pha se tra nsiti on theref ore l eads to a sponta neousl y
bro ken sym metry . The val ue of
can b e cl earl y found fro m Fi gs. 4 and 5, where
we show the chem ical p otenti al , the m ol ecular density , a nd the gap as a functi on
of tem p erature. W e Ùnd fo r our para m eter set f or K and al most zero detuni ng a
rem ark abl y hi gh val ue fo r the cri ti cal tem pera ture
, i .e.
0 6 K.
Furtherm ore, we Ùnd a weak dependence of
on the density , so tha t the
val ue o f
has m ore-or- l ess the sam e density b ehavi or as
. W hen we i ncrease
the detuni ng to
(thi s corresponds to a magneti c Ùeld detuni ng of
0.5 G s away fro m the Feshbach resonance), the val ue o f
dro ps to appro xi matel y
.
Resonance Super Ûuidi t y i n a Quant um Degenerate Fer mi Gas
Fig. 4.
C hemical p otential
as a function
395
of temp erature for the system of resonance
pairing (solid line). T he second order phase transition
occurs at
T c
clear cusp is visibl e. T he dashed line show s the chemical p otential
¤
0: 5T F
, w here a
of a non- interacting
F ermi gas.
Fig. 5.
T he temp erature at the phase transition
the molecular Ùeld. T his amplitude
in the region
T <
Tc.
F or
T =
0,
is also visibl e from the amplitud e of
is non- zero only w hen the broken symmetry
of the total gas sample. T he inset show s the b ehavi our of the gap Â
critical
temp erature
Tc
exists
the molecules form a Bose condensed fraction of 1.5%
can b e related to the value of the gap at
T =
= U p
0.
À
g¢m
. T he
For comparison,
in sup erconducto rs the analogous gap is simply the bindin g energy of a fermion pair.
The system of 4 0 K ato m s, equal l y di stri buted am ong the two l owest hyp erÙne sta tes, i s a go od candi date for demonstra ti ng the superÛuid pha se tra nsiti on.
It not onl y exhi bi ts a Feshbach resona nce, but al so, the i nel asti c bi na ry col l i sio n
events are energeti cal l y forbi dden. Three- b ody i ntera cti ons are hi ghl y suppressed,
since the asym pto ti c three- b ody wa ve functi on shoul d consist of a pro duct of
three s -wa ve two -b ody scatteri ng wa ve functi ons. In a three- b ody i ntera cti on,
two - particl es are al ways i n the same i ni ti al hyp erÙne sta te, and theref ore the correspondi ng s -wa ve sta te i s forbi dden. The onl y three- bo dy rel axa ti on coul d come
396
S. Kokkel mans, M. H ol l and, R. Wal ser, M. C hi ofal o
fro m asym pto ti c p - wa ves, but these ha ve very l i ttl e contri buti on at the l ow tem p eratures considered. Al tho ug h the deta i l ed three- b ody col li sion pro bl em i s an
i ntri cate one, thi s asym pto ti c sta ti sti cal e˜ect shoul d l ead to a l arg e suppression
of the vi bra ti ona l rel axati on of quasi- bound m ol ecules.
Current exp eri m ental techni ques for ul tra cold gases do not pro duce sampl es
whi ch are spati al l y uni form . An opti cal di p ol e tra p m ay b e needed to conÙne the
hi gh Ùeld seeking ato ms, and the condi ti ons for the sup erÛuid pha se tra nsi ti on
wo ul d be sati sÙed Ùrst i n the tra p center where the densi ty i s hi ghest. The presence of the quasi-b ound m ol ecules m ay b e a very useful aspect all owing di rect
observ ati on of the pha se tra nsiti on thro ug h i magi ng the mol ecular Ùeld.
In concl usion, we ha ve shown tha t resona nce pa i ri ng i n an al kal i gas yi elds
a qua ntum Ûuid tha t can underg o a superÛuid pha se tra nsiti on at a tem p erature
compa ra bl e to the Ferm i tem perature. Thi s extra ordi na ry pro p erty pl aces thi s
system i n a reg im e whi ch li es i n b etween BCS- l ike sup erconducto rs, a nd b osoni c
system s whi ch may underg o BEC. Since the tra nsiti on tem p erature i s larg er tha n
the lo west tem p eratures alrea dy achi eved i n a degenerate Ferm i gas, i t shoul d b e
p ossibl e to study thi s new typ e of qua ntum m atter and to qua nti tati vely com pa re
wi th our predi cti ons.
Ac kn owl ed gm en t s
W e tha nk J. Co oper, E. Cornel l , D . Ji n, C. W i eman, and B. D eMa rco for
very sti m ula ti ng di scussions. Supp ort i s ackno wledg ed for M. H. fro m the Na ti onal
Science Founda ti o n, for S.K. fro m the U. S. Departm ent of Energ y, O£ ce of Ba sic
Energ y Sciences vi a the Chem i cal Sciences, G eosciences and Bi osciences D i vi sion,
for M. C. fro m the INFM and SNS, Pi sa (Ita l y), and for R .W. fro m the Austri an
Aca dem y of Sciences.
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