BEC-BCS crossover, phase transitions and phase separation in polarized resonantly-paired superfluids

1.1 MB·English

by  Daniel E. Sheehy

BEC-BCS crossover, phase transitions and phase separation in polarized
resonantly-paired superfluids
Daniel E. Sheehy and Leo Radzihovsky
Department of Physics, University of Colorado, Boulder, CO, 80309
(Dated: July 30, 2006)
7
0 We study resonantly-paired s-wave superfluidity in a degenerate gas of two species (hyperfine
0
states labeled by , ) of fermionic atoms when the numbers N↑ and N↓ of the two species are
2 ↑ ↓
unequal,i.e.,thesystemis“polarized”. WefindthatthecontinuouscrossoverfromtheBose-Einstein

n condensate(BEC)limitoftightly-bounddiatomicmoleculestotheBardeen-Cooper-Schrieffer(BCS)
a limitofweaklycorrelatedCooperpairs,studiedextensivelyatequalpopulations,isinterruptedbya
J varietyofdistinctphenomenaunderanimposedpopulationdifference∆N N↑ N↓. Ourfindings
≡ −
3 aresummarizedbya“polarization”(∆N)versusFeshbach-resonancedetuning(δ)zero-temperature
2 phasediagram, which exhibitsregionsof phaseseparation, aperiodicFFLOsuperfluid,apolarized
normal Fermi gas and a polarized molecular superfluid consisting of a molecular condensate and a

] fully polarized Fermi gas. We describe numerous experimental signatures of such phases and the
n
transitions between them, in particular focusing on their spatial structure in the inhomogeneous
o
environmentof an atomic trap.
c
-
r
p I. INTRODUCTION ference δµ = µ µ corresponding to a purely Zeeman
u field h δµ/2.↑− ↓
s ≡
t. A. Background and Motivation This isomorphismmakes a connection to a large body
a of work in a related condensed-matter system, namely
m
BCSsuperconductorsunderanappliedZeemanmagnetic
One of the most exciting recent developments in the
- field, providing further motivation for our study. In con-
d study of degenerate atomic gases has been the observa-
trast to a normal Fermi liquid that exhibits Pauli para-
n tion1–9 of singlet paired superfluidity of fermionic atoms magnetism, a conventional homogeneous BCS state84,85
o interacting via an s-wave Feshbach resonance10–18.
at zero temperature remains unmagnetized until, at a
c
[ A crucial and novel feature of such experiments is critical Zeeman field HZ, it is destroyed in a first-order
the tunability of the position (detuning, δ) of the Fes- transition to the unpairced magnetized normal state. A
2
hbach resonance, set by the energy of the diatomic naturalquestioniswhethertherecanbea“compromise”
v
3 molecular (“closed-channel”) bound state relative to state that exhibits both pairing and nonzero magnetiza-
0 the open-channel atomic continuum, which allows a tion. One proposal for such a state dates back to work
8 degree of control over the fermion interactions that of Fulde and Ferrell(FF)86 and Larkinand Ovchinnikov
7 is unprecedented in other (e.g., solid-state) contexts. (LO)87, and has been the subject of strong interest for
0 As a function of the magnetic-field controlled detun- many years88–94, finding putative realizations in a vari-
6
ing, δ, fermionic pairing is observed to undergo the ety ofsystems rangingfromheavy-fermionsuperconduc-
0
Bose-Einstein condensate to Bardeen-Cooper-Schrieffer tors95 todensequarkmatter92,94. Theseso-calledFFLO
/
t (BEC-BCS) crossover19–29 between the Fermi-surface states, theoretically predicted to be the ground state for
a
m momentum-pairing BCS regime of strongly overlap- a narrow range of applied Zeeman field near the above-
ping Cooper pairs (for large positive detuning) to the mentioned transition, are quite unusual in that, while
-
d coordinate-space pairing BEC regime of dilute Bose- exhibiting off-diagonal long-range order (i.e., superflu-
n condensed diatomic molecules (for negative detuning). idity), they spontaneously break rotational and transla-
o Except for recent experiments30–35 that followed our tional symmetry, forming a crystal of pairing order (i.e.,
v:c otirvigitiyn,almwoosrtkw36o,rakndhaaswfoacvuesoefdreocnentthtehecoarseetiocfala37n–8e3quaca-l awhseurpeekrsFoσliids96t–h9e9)spwinit-hσ FlaetrtmiciewvaevcetvoercstoQr.≈ kF↑ − kF↓,
i
X mixture of two hyperfine states (forming a pseudo-spin The observation of such magnetized superfluidity in
1/2 system), observed to exhibit pseudo-spin singlet su- condensed-matter systems has been elusive for a vari-
r
a perfluidity near an s-wave Feshbach resonance. Here we ety of reasons, primarily because an applied physical
presentanextensivestudyofsuchsystemsforanunequal magnetic field H couples not only to spin polarization
numberofatomsinthetwopairinghyperfinestates,con- (i.e., the Zeeman effect) but also to the orbital motion
siderably extending results and calculational details be- of charged electrons. The latter coupling leads to the
yond those reported in our recent Letter36. Associating Meissner effect, in which a charged superconductor ex-
thetwopairinghyperfinestateswithup( )anddown( ) pels an externally applied magnetic field. For type I
↑ ↓
projectionsofthepseudo-spin1/2,the densitydifference superconductors, for sufficiently large H > H , the en-
c
δn=n n between the two states is isomorphic to an ergy cost of expelling the field exceeds the condensation
impose↑d−“ma↓gnetization” m δn (an easily controllable energy of the superconducting state and the system is
≡
experimental “knob”), with the chemical potential dif- driven normal via a first-order transition. Because this
2
thermodynamic critical field H =∆ 4πN(ǫ ) is much as a function of the magnetic field B near the resonance
c F
smaller than100 the critical (Clogston limit84) Zeeman position B . Equivalently, γ is the ratio of the inter-
0
p
field HcZ = ∆ N(ǫF)/χP (where χP is the Pauli mag- atomicspacingn−1/3(wherenistheatomdensity)tothe
netic susceptipbility), i.e., Hc/HcZ = √4πχP ≪ 1, in effective range r0 characterizing the energy dependence
condensed-mattersystemstheeffectsofapurelyZeeman of two-body scattering amplitude in the open channel.
fieldareexpectedto be obscuredbythe orbitaleffects of As was first emphasized by Andreev, Gurarie and Radz-
the physicalappliedmagnetic field101. The effects ofim- ihovsky26,29 in the context of pairing in an unpolarized
purities88,91,thatarealwayspresentincondensed-matter (symmetric) Fermi gas,in the limit of a vanishingly nar-
systems, can further complicate the realization of FFLO row resonance, γ 0, the two-channel model is exactly
→
andothermagnetizedsuperfluidstates(see,however,re- solvable by a mean-field solution. Consequently, for a fi-
cent work95). nitebutnarrowresonanceγ 1,thesystemadmitsade-
≪
In contrast, trapped atomic systems are natural set- tailedquantitative analyticaldescriptionforanarbitrary
tings where paired superfluidity at a finite imposed value of detuning (throughout the BEC-BCS crossover),
magnetic moment, ∆N = N N , can be experi- withitsaccuracycontrolledbyasystematicperturbative
mentally studied by preparing↑m−ixtu↓res with different expansion in γ.
numbers N ,N of two hyperfine-state species, with This important observation also holds for asymmet-
the (Legend↑re-c↓onjugate) chemical potential difference ric mixtures with an arbitrary magnetization. Hence
δµ = µ µ a realization of the corresponding effec- in the narrow Feshbach resonance limit we can accu-
tive (pu↑re−ly) Z↓eeman field. rately study this system by a perturbative expansion in
Beyond the aforementioned isomorphism in the γ about a variational mean-field solution. To implement
weakly-paired BCS regime, atomic gases interacting via this we compute the energy in a generalized BCS-like
a Feshbachresonanceallow studies of magnetizedpaired variationalstate, and minimize the energy over the vari-
superfluidity as a function of Feshbach resonance de- ational parameters. Explicitly, the variational state is
tuning across the resonance and deep into the strongly- parametrizedbythebosoniccondensateorderparameter
pairedmolecularBECregime,inaccessibleincondensed- b , where Q is a center of mass wavevector that allows
Q
matter systems. for a periodically modulated condensate. Our choice of
the variational ground state is sufficiently rich as to in-
clude the normal state, the gapped singlet BCS state as
B. Theoretical framework and its validity well as its molecular BEC cousin, a magnetized paired
superfluidstatewithgaplessatomicexcitations,aFFLO
state102,103, and inhomogeneous ground states that are
The goal of our present work is to extend the study
a phase-separated coexistence of any two of the above
of s-wave paired resonant superfluidity to the case of
pure states. Our variationalansatz is not, however,gen-
an unequal number of the two hyperfine-state species,
eralenoughtoallowfortheveryinterestingpossibilityof
∆N/N = (N N )/N = 0, namely, to calculate the
phase diagram↑ −as a↓funct6ion of detuning, δ, and polar- auniformbutanisotropicpairedsuperfluidgroundstate,
e.g.,anematicsuperfluid45,49,norofmoreexoticmulti-Q
ization, ∆N/N (or, equivalently, the chemical potential
FFLO superfluids102,103.
difference δµ).
AnappropriatemicroscopicmodelofsuchaFeshbach- Most of our work focuses on the narrow resonance
resonantly interacting fermion system is the so-called (γ 1) limit, studied within the two-channel model.
≪
two-channel model10,24–26,29 that captures the dynam- Althoughexperimentsdonotlieinthisregime,theanal-
ics of atoms in the open channel, diatomic molecules ysisprovidesvaluableandquantitatively trustworthypre-
in the closed channel and the coupling between them, dictions for the behavior of the system (throughout the
which corresponds to the decay of closed-channel di- phasediagram),atleastinthisonenontriviallimit. How-
atomic molecules into two atoms in the open channel. ever, typical present-day experiments1–9 fall in a broad
As we will show, the model is characterized by a key Feshbach resonance, γ 101 104 1, regime104 (see
≃ − ≫
dimensionless parameter Appendix A). To make contact with these experiments
we also extend our results to the broad resonance limit,
√8 Γ complementingourtwo-channelanalysiswithaneffective
0
γ = , (1.1) single-channel model. For a broad resonance, we can ig-
π ǫ
r F norethedispersionoftheclosed-channelmolecularmode
determinedbytheratiooftheFeshbachresonancewidth and integrate it out, thereby reducing the two channel
Γ 4mµ2a2 B2/¯h2 (where µ is the Bohr magneton), model to an effective single (open) channel model with
to0t≈he FermBibegnewrgyǫ ,anddesBcribesthe strengthofthe a tunable four-Fermi coupling related to the atomic s-
F
atom-molecule coupling that can be extracted from the wavescatteringlength. Althoughinthisbroadresonance
two-body scattering length observed13,17 to behave as regime our mean-field variational theory is not guaran-
teed to be quantitatively accurate, we expect that it re-
B mains qualitatively valid. Indeed, we find a reassuring
w
a =a 1 , (1.2)
s bg − B B qualitative consistency between these two approaches.
0
−
(cid:0) (cid:1)
3
Furthermore, to make detailed predictions for cold- ganized into a detuning-polarization phase diagram (see
atom experiments we extend our bulk analysis to in- Figs. 3 and 4) and our presentation logically splits into
clude a trap, V (r). We do this within the local density predictions for a bulk system and for a trapped system.
T
approximation (LDA). Much like the WKB approxima-
tion, this corresponds to using expressions for the bulk
system, but with an effective local chemical potential A. Bulk system
µ(r) = µ V (r) in place of µ. The validity of the
T
−
LDA approximationrelies onthe smoothness ofthe trap In this somewhattheoretically-mindedpresentationof
potential, namely that VT(r) varies slowly on the scale ourresults,wefocusonthetwo-channeldescription,out-
of the longest physical length λ (the Fermi wavelength, lining the system’s phenomenology as a function of de-
scattering length, effective range, etc.) in the problem, tuning, δ 2µ (B B ) (with B the magnetic field at
B 0 0
i.e.,(λ/VT(r))dVT(r)/dr 1. Itsaccuracycanbeequiv- which the≃Feshbach−resonance is tuned to zero energy),
≪
alently controlled by a small parameter that is the ratio andchemicalpotentialdifferenceh,orimposedpolariza-
of the single particle trap level spacing δE to the small- tion ∆N/N.
est characteristic energy E of the studied phenomenon
c
(e.g, the chemical potential, condensation energy, etc.),
by requiring δE/Ec 1. 1. BCS regime: δ 2ǫF
≪ ≫
For large positive detuning, the closed-channel
C. Outline molecules are energetically costly, and the atom density
is dominated by open-channel atoms, exhibiting a weak
Therestofthepaperisorganizedasfollows. InSec.II attractionmediated by virtual closed-channelmolecules.
we summarize our main results. In Sec. III, we re- Consequently, for a weak chemical potential difference
view the standard two-channel and one-channel models (or Zeeman field) h we find a standard s-wave singlet
offermionsinteractingviaans-waveFeshbachresonance. (non-polarized) BCS superfluid ground state, that, as a
Focusing on a narrow resonance described by the two- function of detuning, undergoes a BCS-BEC crossover
channel model, we compute the system’s ground state that is identical to the well-studied crossover at van-
energy in Sec. IV. By minimizing it we map out the ishing h. Deep in the BCS regime, for fixed chemical
detuning-polarization phase diagram. Focusing first on potential µ, this singlet paired state becomes unstable
uniformly-paired states (i.e., ignoring the FFLO state), when the Zeeman field h overwhelms the superconduct-
wedothisforthecaseofpositivedetuning(theBCSand ing gap ∆ (µ). For simplicity, for now ignoring the
BCS
crossover regimes) in Sec. V, and for negative detuning FFLO state, the BCS superfluid then undergoes a first-
(the BEC regime) in Sec. VI, finding a variety of phases order transition to the partially-polarized normal Fermi
andthetransitionsbetweenthem. InSec.VII,werevisit gas state at
the positive-detuning regime to study the periodically-
∆ (µ)
paired FFLO state. In Sec. VIII, we complement our BCS
h (µ,δ) = , (2.1a)
c
analysis of the two-channel model with a study of the √2
one-channelmodel,findinganexpectedqualitativeagree- δ 2µ
ment. Tomakepredictionsthatarerelevanttocold-atom = 4√2e−2µe−γ√−ǫFµ, (2.1b)
experiments, in Sec. IX we extend our bulk (uniform-
consistent with established results first found by
system)analysistothatofatrap. WeconcludeinSec.X
Clogston84 and Sarma85. As is generically the case for
with a discussion of our work in the context of recent
first-ordertransitions,thermodynamicquantities exhibit
and future experiments and relate it to other theoretical
jump discontinuities as h crosses h . In particular, we
c
studiesthathaverecentlyappearedintheliterature. We
find the discontinuities in the atomic, molecular and to-
relegate many technical details to a number of Appen-
tal densities are, respectively,
dices.
c∆2 1 ∆
∆n na na BCS 1 ln BCS >0,(2.2a)
a ≡ SF − N ≃ √µ − 2 8e 2µ
−
II. SUMMARY OF PRINCIPAL RESULTS
∆2 (cid:0) (cid:1)
∆n nm nm BCS >0, (2.2b)
m ≡ SF − N ≃ g2
Havingmotivatedourstudyanddiscusseditstheoreti-
calframeworkandvalidity, belowwe presentasummary ∆n nSF nN,
≡ −
of the main predictions of our work, thereby allowing a 2∆2 c∆2 1 ∆
BCS + BCS 1 ln BCS >0, (2.2c)
readernotinterestedinderivationseasyaccesstoourre- ≃ g2 √µ − 2 8e 2µ
−
sults. Asdiscussedabove,thequantitativevalidityofour (cid:0) (cid:1)
calculationsisguaranteedinthenarrowresonanceγ 1 where na and na are the free atomic densities in the
limit, with all expressions given to leading order in≪γ. superfluiSdFand norNmal states, respectively, nm and nm
SF N
Our results (summarized by Figs. 1-6) are naturally or- are their molecular analogs and n is the total (whether
4
free or paired) atom density. Here, c m3/2/√2π2¯h3 is is an inhomogeneous mixture of phase-separated super-
≡
definedbythethree-dimensionaldensityofstatesN(ǫ)= fluidandnormalstates,withrespectivevolumefractions
c√ǫ. The species imbalance (magnetization) m=(N x(h,δ) and 1 x(h,δ) varying according to
N )/V also exhibits a jump discontinuity ↑− −
↓ n n [h,µ (h)]
N c
x(h,δ)= − , (2.4)
∆m≡mSF −mN ≃−2chc√µ<0, (2.3) nSF[δ,µc(h)]−nN[h,µc(h)]
across h . to satisfy the imposed total atom number constraint
c
h/ǫF n=N/V =xnSF+(1 x)nN. Here,µc(h)isthecritical
−
88 chemicalpotential as a function of h, implicitly givenby
(a)
Eq. (2.1b), specifying the SF-N first-order phase bound-
77
Spin-Polarized Normal (N) ary. The coexistence region is bounded by hc2(δ) from
66 above, corresponding to a Zeeman field below which a
SF
M pure partially-polarized normal Fermi gas phase cannot
55
h satisfy the imposed number constraint while remaining
c2
44 the ground state.
hm hc1 A more careful analysis, that includes a periodically-
33
pairedFFLO state in the variationalansatz,shows that,
PS 22 in fact, deep in the BCS regime, for detuning δ > δ
2ǫ ,thefirst-ordertransitionisactually(fromthesing∗le≈t
11 FFLO F
BEC-BCS Superfluid(SF) BCSstate)intoamagnetizedsuperfluidFFLOstate. As
? δ
illustratedinFig.1bandconsistentwithmanydeep-BCS
δˆc--1100 --88 --66 --44 --22 δˆM δˆ∗ 44ǫF studies86,87,89, we find that this fragile state only sur-
h/ǫF vives over a narrow sliver of Zeeman fields (or imposed
00..1122 polarization),103 undergoing a continuous105–108 transi-
(b) tion at h (δ) to a partially-polarized normal Fermi
FFLO
00..11 N gas ground state. Our work is an extension of these ear-
lier BCS studies to a varying detuning (controlling the
00..0088
PS strength of the attractive interactions), with our main
(cid:8)(cid:25)(cid:8)hFFLO
result in this regime the location of the critical detuning
00..0066
(cid:8)(cid:8)* γ
00..0044 hc1 hc2 (cid:8)(cid:25)(cid:8)FFLO δ ǫF 2 ln(0.159γ) , (2.5)
∗ ≈ − 2
00..0022 (cid:2) (cid:3)
accurately predicted for a narrow resonance (γ 1),
SF ≪
δ belowwhichtheFFLOstateisunstableforanyhor∆N.
δˆ∗ 22..33 22..44ǫF As seen in Fig. 1b, above δ , deep in the BCS regime,
the boundaries h , h , and∗h display exponential
FIG.1: (Coloronline)(a)Globalphasediagramasafunction c1 c2 FFLO
behavior [defining ∆ ∆ (ǫ )]:
of Feshbach resonance detuning at fixed total atom number F BCS F
≡
N andchemicalpotentialdifferencehwithinthetwo-channel
1 δ2∆2
model,showinghomogeneoussuperfluid(SF),magnetizedsu- h (δ) ∆ exp F , δ 2ǫ , (2.6a)
perfluid(SFM),FFLOandspin-polarized normal(N)phases c1 ≈ √2 F − 8γ2ǫ4F ≫ F
as well as a regime of phase separation (PS, shaded). (b) 1 (cid:2) δ∆2 (cid:3)
Zoom-in of the positive-detuning BCS regime, showing the h (δ) ∆ exp F , δ 2ǫ , (2.6b)
regime of phase separation (gray) and FFLO state (red on- c2 ≈ √2 F − 16γǫ3F ≫ F
line). SeeFig.41forasimilar phasediagram withintheone- (cid:2) η2δ∆2(cid:3)
channel model. δˆ∗ δ∗/ǫF, δˆM δM/ǫF and the tricritical hFFLO(δ)≈η∆F exp − 8γǫ3F , δ ≫2ǫF, (2.6c)
point δˆ δ /ǫ ind≡icatelimiting≡detuningvaluesfor FFLO, F
c c F (cid:2) (cid:3)
≡
SFMand phase separated states, respectively. never crossing with increasing detuning, the latter two
boundaries asymptoting to a ratio h /h = η√2
FFLO c2
≈
In the more experimentally-relevant (to cold atoms) 1.066 previously found by Fulde and Ferrell86.
fixed total atom number N ensemble, the density dif-
ference between the normal and superfluid states makes
it impossible for the normal state to satisfy the im- 2. Crossover regime: 0<δ<2ǫF
posed number constraintimmediately aboveh (N,δ)
c1
≡
h (µ (N),δ). [Here,µ arethe superfluid(SF) and Asthedetuningisloweredbelowapproximately2ǫ ,it
c SF SF,N F
normal (N) state chemical potentials corresponding to becomes favorable (for low chemical potential difference
the imposed N.] Consequently, we find that in the fixed h) to convert a finite fraction of the Fermi sea (between
number ensemble the transition at h opens up into a µ δ/2andǫ )intoBose-condensedmolecules. Sincein
c F
≈
coexistence region h (δ) < h < h (δ) in which the gas this crossover regime the pairing strength (proportional
c1 c2
5
to the growing molecular density) is no longer driven by a (δ,h)/a (δ,0)
m m
Fermisurfacepairingandisthereforenolongerexponen- 1
tiallyweak,theresponsetothechemicalpotentialdiffer-
ence,h,(ortheimposedpolarization,∆N)changesqual- 0.8
itatively from the weakly-paired BCS regime discussed
above. 0.6
Consequently, the FFLO state, driven by mismatched
Fermi-surfacepairing,isnolongerstableforδ <δ [given
0.4
byEq.(2.5)],leadingtoadirectfirst-ordertransit∗ionbe-
tween a paired superfluid and a partially polarized nor-
0.2
mal Fermi gas (see Fig. 1).
As above,forimposedatomnumber the phasebound- 2h
ary splits into a superfluid-normal coexistence region 1.05 1.1 1.15 1.2 1.25 1.3 δ
bounded by hc1(δ) and hc2(δ). The lower boundary, FIG. 2: (Color online) Plot of molecular scattering length
h (δ), is still determined by the value of the gap, ∆. a (δ,h) [Eq. (2.14)], normalized to its value at h = 0, as a
c1 m
However, because the gap is in turn set by the molecu- function of 2h/δ.
| |
lar condensate ∆ = g n (δ), that in this regime is no
m
longerexponentiallysmallbutgrowsasapower-lawwith
p molecules,with the remaining unpairedatoms forming a
detuning reduced below 2ǫ , h (δ) is also a power-law
F c1
fully-polarized Fermi sea that carries the imposed mag-
in 2ǫ δ. We note, however, that while h (δ) is sig-
F c1
− netization.
nificantlylargerthanits exponentiallysmallvalue inthe
Asillustratedinthe phasediagram,Fig.1,forδ <δ
BCS regime, it nevertheless remains small compared to M
a spin-singlet (unpolarized) molecular (BEC) superfluid
µ for a narrowresonance,justifying a linear response (in
undergoes a second-order quantum phase transition at
h) approximation for the normal-state energy near h .
c1
The upper boundary, h (δ), (no longer in the lin-
c2 h (δ) δ /2, δ <δ , (2.11)
earPauli-paramagneticregime,i.e., hc2 µ) is increased m ≈| | M
≪6
even more dramatically beyond that of Eq. (2.6b), due to the magnetized SF superfluid. The transition turns
M
to the superfluid-normal density difference enhanced by into a (sharp at low T) crossover at any finite tempera-
alargemoleculardensitynm thathereisafinitefraction ture,sincethenthe magnetizationis finiteforarbitrarily
of the total atom density. This considerably spreads the small h, even below h . In the narrow-resonance limit,
m
coexistence region for δ below 2ǫF, with hc1(δ) given by thephaseboundaryhm(δ)isdeterminedbythevanishing
ofthemajority-specieschemicalpotentialµ =h δ /2.
γ δ 3/2 Physically,thiscorrespondstothecondition↑thatt−he|g|ain
h (δ) ǫ 1 , δ <δ <2ǫ , (2.7)
c1 F M F
≃ 3 − 2ǫ intheZeemanenergy h∆N compensatesthelossofthe
r r F −
(cid:0) (cid:1) molecular binding energy δ /2 per atom.
and the critical detuning δ given by | |
M Forintermediatenegativedetuningδ <δ <δ ,upon
c M
increasingthe Zeemanfield h the magnetizedsuperfluid,
δ 1.24ǫ √γ. (2.8)
M ≈− F SFM, undergoes a first-order transition at hc(δ) [given
by the implicit Eq. (6.43b) in the main text]. In the
Here, h (δ) is implicitly given by
c2
more experimentally-relevant fixed atom number N en-
ǫ3/2 1 δ +h (δ) 3/2 (2.9) semble this transition at hc(δ) opens up (as is standard
F ≃ 2 2 c2 fora first-ordertransition)into a phase-separatedregion
h(cid:0) δ (cid:1) 3/2 δ ofcoexistencebetweentheSFMandafully-polarizednor-
hc2(δ) Θ hc2(δ) , δ <2ǫF, malatomicgas. Thisphase-separatedregimeisbounded
− 2 − 2 −
(cid:0) (cid:1) (cid:0) (cid:1)i by
which is well-approximatedby
h (δ) 0.65δ , δ <δ <0, (2.12)
c1 c
δ ≈ | |
hc2(δ)≃22/3ǫF− 2, δ <22/3ǫF, (2.10) hc2(δ) 22/3ǫF+ |δ|, δ <0, (2.13)
≈ 2
over most of the range of interest.
that are continuations of the boundaries [Eqs. (2.6a),
(2.6b), (2.7) and (2.10)] found in the BCS and crossover
regimes above. We find that the SF -phase separation
M
3. BEC regime: δ<0
instability, initiated at h , is signaled by an enhanced
c1
compressibility and a corresponding suppression (with
In the BEC regime, a new uniform magnetized super- increasing h) of the molecular scattering length
fluid(SF )phaseappears,fordetuningsbelowδ ,when
M M
the population imbalance ∆N = N N is imposed. π2ǫFγ2 2h
The SFMground state consists of clo↑se−d-ch↓annel singlet am ≃ 16√mδ 3/2F4 δ , (2.14)
| | (cid:16)| |(cid:17)
6
plotted in Fig. 2. The function F (x) is defined in tively given by
4
Eq. (6.17) of the main text. In the narrow resonance
limit, γ ≪ 1, indeed hc1 is determined by the vanishing ∆Nc2 3∆F exp δ∆2F , (2.15)
of am, as given in Eq. (2.12) above. N ≈ 2√2ǫF − 16γǫ3F
As illustrated in Fig. 1a, for sufficiently large nega- ∆N 3η∆ (cid:2) η2δ∆2(cid:3)
tive detuning, δ < δ , the h (δ) and h (δ) bound- FFLO F exp F . (2.16)
c c1 c2 N ≈ 2ǫ − 8γǫ3
aries cross and the first-order SF -N transition and the F F
M (cid:2) (cid:3)
corresponding phase-coexistence region are eliminated.
As seen from the phase diagram these two boundaries
The SF then undergoes a direct continuous transition
M crossatδ ,therebyeliminatingtheFFLOstateforlower
at hc2(δ) into a fully-polarized normal state. detuning.∗For a narrow Feshbach resonance (γ 1) at
≪
even lower detuning (see Fig 3(a))
∆N
11N δ 22/3ǫ , (2.17)
p F
≃
(a)
N the normalized critical polarization ∆N /N reaches
00..88 c2
unity according to
PS
00..66 ∆Nc2 ∆NNc2 ≈1−ǫF−23(δ−δp)3/2Θ(δ−δp). (2.18)
00..44
SFM ∆Nc1 Thisimpliesthatfortheselowerdetunings,δ <δp,onlya
fullypolarizednormalFermigascanappear. Incontrast,
00..22 FFLO
as illustrated in Fig. 4, for a broad resonance (large γ)
? ∆N /N < 1, consistent with experiments30 that find
δ c2
δˆc--1100 --88 --66 --44 --22 δˆM δˆ∗ 44ǫF ∆Nc2/N ≈0.7 near unitarity.
Forsufficientlynegativedetuning,δ <δ ,theuniform
∆N M
N magnetic superfluid state, SF , appears at low popula-
M
(b) tion imbalance. For δc <δ <δM it becomes unstable to
N
phaseseparationandcoexistencewiththefully-polarized
00..1155 atomic Fermi gas for ∆N >∆N , with
c1
∆N δ 3/2
c1
00..11 ∆NFFLO 0.029 . (2.19)
N ≈ ǫ
F
(cid:8)FFLO (cid:16) (cid:17)
00..0055 PS ∆Nc2 (cid:8)(cid:25) As this population imbalance is approached, the molec-
ular scattering length a (δ,∆N) drops precipitously
m
and the correspondingmolecular condensate Bogoliubov
δ sound velocity
δˆ∗ 22..33 22..44 ǫF
FIG.3: (ColorOnline)Globalphasediagram(a)asafunction u(δ,∆N) u (δ) 1 ∆N F 1+ 25/3 ∆N 2/3 ,
ofFeshbachresonancedetuningatfixedpopulationdifference ≃ 0 − N 4 δˆ N
a∆NnNdt=heNNF↑↑F−+LNNO↓↓.reTghimeere(gtiomoethoifnpthoasseeesoenpathraistisocnal(eP)Sin)teisrvgernaeys, (where δˆ δ/ǫF) vaqnishes accrordin(cid:16)g to | | (cid:0) (cid:1) (2(cid:17).20)
≡
along a thin line at positive detuning, indicated with a thick
dark (red online) line. (b) Zoom in of the BCS regime at ∆Nc1 1/2 ∆N 1/2
u 0.38u 1 1 , ∆N ∆N .
positivedetuningshowingtheregimeofphaseseparationand ∼ 0 − N − ∆N → c1
c1
theFFLO regime. (cid:0) (cid:1) (cid:0) (cid:1) (2.21)
Finally,forδ <δ ,theSF stateisstableforanypopula-
c M
Itisnowstraightforwardtoconvertthephasediagram tionimbalanceuptothefullypolarizedlimitof∆N =N,
and our other predictions to the more experimentally- whereitisidenticaltothefullypolarized(singlespecies)
relevant ensemble of fixed total atom number N and Fermi gas.
imposed atomic species difference (polarization) ∆N =
N N . As can be seen in Fig. 3, in this ensemble
fo↑r −posit↓ive detuning (BCS and crossover regimes), the B. Trapped system
singlet BCS superfluid is confined to the detuning axis,
andisunstable tophaseseparationandcoexistencewith To make a more direct connection with atomic-gas
a normal atomic gas for any imposed population imbal- experiments we summarize the extension of the above
ance. The upper boundaryof the coexistenceregionand bulk results to the case of a trapped gas. This is par-
the phase boundary of the FFLO phase are then respec- ticularly simple to do in the case of a trap potential
7
(a) m(r)/m0
∆N
11N 0.2
00..88 Normal 0.15
Phase Separation
00..66
0.1
00..44 (cid:8)(cid:25)(cid:8)FFLO
0.05
00..22 SFM r
0.2 0.4 0.6 0.8 1 R0
1
δ˜c --22 δ˜M 00 δ˜∗ 11 22 −kFas (b)
N
FIG. 4: Polarization ∆N/N vs. detuning 1 phase
∝ −kFas
diagram oftheone-channelmodel(appropriateforresonance
width γ ) within mean-field theory showing regimes of
FFLO, s→upe∞rfluid (SF), magnetized superfluid (SF ), and SF rc R0
M
phaseseparation(PS).Notethathere(incontrasttoanarrow
Feshbachresonanceγ 1,Fig.3)atunitarity,(k a )−1 =0,
F s
≪
theboundarybetweenN and PSis at ∆N/N <1, consistent
with experiments30.
FIG. 5: a) Local magnetization m(r) radial profile confined
that varies smoothly on the scale of the Fermi wave-
to the normal outer shell of the cloud, r > r , for coupling
length, i.e., when the Fermi energy is much larger than (k a )−1 = 1.5 and ∆N = 0.15 (dashed) ancd ∆N = 0.20.
the trap level spacing, the regime of most atomic gas F| s| N N
(b)Sequenceof shells, with increasing radius, implied bythe
experiments. In this regime, the trap potential can be magnetization profiles in (a).
easily taken into account via the local density approxi-
mation (LDA). Namely, we treat the trapped system as
a locally uniform one (thereby taking advantage of our (ignoring for simplicity the narrow sliver of the FFLO
bulk results), but with a local chemical potential given state), the normal and (singlet BCS-) SF phases trans-
by µ(r) = µ VT(r). The true chemical potential µ late into two shells of a trapped cloud. Because the bulk
−
and chemical potential difference h still appear and are h (µ) [see Eq. (2.1b)] is an increasing function of µ and
c
determined by constraints of the total atom number N µ(r) is a decreasing function of r, the SF phase (that
and species imbalance ∆N. For simplicity we consider in the bulk appears at high chemical potential and low
an isotropic trap with VT(r)= 12mΩ2Tr2, but our results h) forms the cloud’s inner core of radius rc(δ,h,N) set
canbeeasilygeneralizedtoanarbitraryanisotropictrap. by h (µ(r )) = h. The normal phase forms the outer
c c
The phasetransitionsandcoexistencediscussedabove shell with inner radius r (δ,h,N) and outer shell radius
c
are strikingly accentuated by the trap. To see this, R (δ,h,N) determined by the vanishing of the normal
0
notethat(withintheLDAapproximation)thelocalphe- phase chemical potential, µ (R ,δ,h,N) = 0. The re-
N 0
nomenologyof atrappedcloudis that ofthe bulk one at sulting magnetization density profile m(r), confined to
aneffectivechemicalpotentialµ(r). Hence,aradialslice (and thereby vividly imaging) the outer normal shell, is
throughatrappedcloudisaneffectivechemicalpotential illustratedinFig.5a. Followingouroriginalprediction36,
“scan” through the bulk-system phase diagram at fixed this shell structure has recently been experimentally ob-
µ and h (the latter displayed in the main text, Fig. 32). served in Refs. 30,31, and subsequently calculated theo-
Consequently, as we first predicted in our earlier pub- retically by a number of groups.53,55,58,59
lication36, a trapped cloud consists of a combination of On the BEC side of the resonance, the appearance of
shells of the various superfluid and normal phases, with an additional phase, the SF , allows for the possibility
M
the exact structure determined by the value of detuning of a triple-shell cloud structure SF SF N with in-
M
→ →
and population imbalance (or, equivalently, the Zeeman creasing radius at low population imbalance (small h).
fieldh). Acriticalvalueofthechemicalpotentialµc sep- The inner (Rf1) and outer (RTF) boundaries of the
arating two phases in the bulk system translates, within SF shelloccurwherethepopulationimbalancem(r)be-
M
LDA,into a criticalradiusrc,givenbyµ(rc)=µc (more comes nonzero at the SF-SFMboundary and where the
generally a hypersurface), that is the boundary between molecular density n (r) vanishes at the SF -N bound-
m M
two corresponding phase shells in a trapped gas. ary, respectively. The radii of the shells are given by the
More concretely, on the BCS side of the resonance bulkcriticalfieldsh (µ(R ))=handh (µ(R ))=h.
m f1 c TF
8
(a) aharmonic(moregenerally,power-law,ratherthanbox-
like) confining potential, that leads to a large density
n¯ (r) variation in the gas. Consequently, following the above
m
0.1 arguments, within LDA in a realistic trap the gas can
only exhibit a thin shell of FFLO phase, whose width
δr (among other parameters, N, ∆N, and δ) is set by
the inverse trap curvature and proportional to the dif-
0.05 ference in critical chemical potentials for the transition
into the FFLO and normal phases. Clearly, to resolve
the abovementioned Bragg peaks, the Bragg peak width
m¯(r) ¯h/δr must be much smaller than Q, or, equivalently, the
r FFLO shell width δr must be larger than the FFLO pe-
0.5 1 1.5 RTF0 riod 2π/Q. Since FFLO state is confined to a narrow
sliver of the phase diagram and to be meaningful (i.e.,
(b) LDA reliable) the shell width must be much larger than
N
the period 2π/Q, its direct observation in density pro-
files may be difficult for present-day experiments. Simi-
SFM
lar conclusions aboutthe effect of the trapon the FFLO
phase have been found in recent work by Kinnunen et
SF
Rf1RTF Rf2 al.53 that does not rely on the LDA approximation.
On the other hand, the identification of the FFLO
state through less direct probes maybe possible. For ex-
ample, its spontaneousbreaking oforientationalsymme-
FIG. 6: a) Normalized molecular density n¯ (r) and normal- try for an isotropic trap should be detectable in noise
m
ized magnetization m¯(r) of a trapped fermion gas as a func- experiments110–112 sensitivetoangle-dependenceofpair-
tion of radius r (normalized to the radius of the unpolarized ing correlations across a Fermi surface.
cloud, with parameters given in Sec. IXB). b) Schematic of
the sequence of shells, with increasing radius, implied by the
curvesin (a). Furthermore, gapless atomic excitations available in
the FFLO and SF states should be observable through
M
Bragg spectroscopy and reflected in thermodynamics
Thecloud’sradialprofilefornm(r)andm(r)forthiscase such as power-law (rather than nearly activated paired-
is illustrated in Fig. 6. For larger population imbalance superfluid) heat capacity. The latter should also ex-
sRtfru1citsudrrei.veAntteovzeenroh,igrehseurlt∆inNg,inRaTdFovuabnleis-hsheesllleSaFdMin→g tNo ShFibitFaFlLatOe,ntFFheLaOt peNa1k05a,caronsdsStFhe fiNrspt-hoarsdeertrSanFsMit→ionNs,,
a fully polarized, normal cloud of radius Rf2. and→anearlymean-→fieldsingularity→acrossthecontinuous
Another interesting prediction of our work36 is the SF SF transition.
M
possibilitytorealizeinresonantly-interactingdegenerate →
Fermi gases the enigmatic FFLO state86,87. In an ideal-
ized box-like trap the molecular occupation distribution Finally,wenotethatouraboveresults(thatarequan-
n = b b of this inhomogeneous superfluid exhibits titativelyaccurateatleastinthenarrowresonancelimit)
q h †q qi
finitemomentumpairingpeakssetbyFermisurfacemis- show no evidence of a homogeneous but magnetized su-
match Q(δ) k k perfluid near the unitary limit a , nor in the
∼ F↑− F↓ BCSandcrossoverregimesdefinedsby→pos∞itive detuning.
√2λ∆ (ǫ ) We find unambiguously that this magnetized superfluid
BCS F
Q(δ) , (2.22)
phase only appears on the BEC side for sufficiently neg-
≈ ¯hv
F
ative detuning, δ < δ , embodied in the SF ground
M M
(here λ 1.200) corresponding to reciprocal lattice state. We do, however, find that the corresponding in-
vectors of≃this supersolid102. This will translate into versescatteringlength(k a ) 1 = 2δˆ /πγ shifts to-
F sM − M
−
spontaneous Bragg peaks (observable by projecting onto ward the unitary limit with increased resonance width
a molecular condensate1) appearing at h¯Qt/m in the (saturating at (k a ) 1 1 within mean-field theory)
F sM −
≃
cloud’s shadow images after expansion time t, akin to and upon increasing temperature. Whether this is suffi-
that exhibitedby a superfluid releasedfromalattice po- cient or not to explain the putative existence of such a
tential109, where, in contrast to the FFLO state, Bragg phase, as claimed in the Rice experiments31 (that were
peaks areexplicitly inducedby the periodicity ofthe op- based on an apparent observation, in the unitary limit,
tical potential. of a critical population imbalance for the transition to
However, realistic magnetic and optical traps produce phase separation), remains an open question.
9
III. TWO-CHANNEL MODEL OF S-WAVE (a)
FESHBACH RESONANCE
The two-channel model of fermions interacting via an
s-wave Feshbach resonance, briefly discussed in Sec. IB,
describesopen-channelfermions(cˆ )andclosed-channel
kσ
molecular bosons (ˆb ) coupled by molecule-atom inter-
q
conversion10,24–26,113. Itischaracterizedbythefollowing
(b)
model Hamiltonian:
ǫ
H= ǫkcˆ†kσcˆkσ+ 2q +δ0 ˆb†qˆbq = +
k,σ q
X X(cid:0) (cid:1)
g
+√V ˆb†qcˆk+q2↓cˆ−k+q2↑+cˆ†−k+q2↑cˆ†k+q2↓ˆbq ,(3.1)
Xk,q(cid:16) (cid:17)
where V is the system volume (that we shall generally FIG.7: (a)Feynmandiagramcorrespondingtotheatomscat-
set to unity) ǫ k2/2m, m is the atom mass and the tering amplitude, with solid lines indicating atoms and the
k
≡ single (double) dashed lines indicating the bare (full) molec-
“bare”detuning δ is relatedto the position ofthe Fesh-
0
ularpropagator. (b)Aself-consistentequationforthemolec-
bachresonanceδ inawaythatwedeterminebelow. The
ular propagator.
molecule-atominterconversiontermischaracterizedbya
coupling g that measures the amplitude for the decay of
a closed-channel s-wave singlet diatomic molecule into a
respectively given by
pair of open-channel fermions.
Although within some approximation the model , d3p m
H δ = δ g2 , (3.3)
Eq. (3.1), can be derived from a more microscopic start- 0− (2π¯h)3p2
ing point ofpurely fermionic atomsinteracting via a van Z
g4m3
der Waals potential and including exchange and hyper- Γ . (3.4)
fine interactions (see e.g., Ref. 23), the model’s ultimate 0 ≡ 16π2¯h6
justification is that, as we willshow below, it reproduces
TheintegralinEq.(3.3)iscutoffbytheultravioletscale
the Feshbach resonance phenomenology. Namely, for
Λ 2π/d, set by the inverse size d of the closed-channel
positive detuning δ > 0 the model exhibits a resonance ≃
(molecular) bound state, so that
at rest-energy δ, whose width is controlled by g, and ex-
hibits a true molecularbound state atnegativedetuning g2mΛ
δ =δ , (3.5)
δ < 0. Associated with the resonance the open-channel 0− 2π2¯h2
s-wave scattering length a (δ) diverges and changes sign
s
as 1/δ, when the resonance at δ is tuned through zero giving the relationbetween the bare andphysicaldetun-
−
energy. ing.
The s-wave scattering length114 a = f (0) is then
s 0
−
given by
A. Scattering theory in the vacuum
Γ 1
0
a (δ)= ¯h . (3.6)
s
− m δ
The above properties follow directly from the scatter- r
ingamplitudef (E),thatcancanbeeasilycomputedex- Thus, to reproduce the experimentally observed depen-
0
actlyfortwoatomscatteringinthevacuum26,29. Thelat- dence of the scattering length on the magnetic field B,
ter follows from the T-matrix f (E) = (m/2π¯h2)T(E) Eq. (1.2), we take10 δ 2µ (B B ), representing the
0 B 0
− ≈ −
that, in a many-body language, is given by the diagram Zeeman energy splitting (at large two-atom separation)
in Fig. 7a, with the molecular propagator exactly given betweentheopenandclosedchannels. Therelevantmag-
by the RPA-like geometric series of fermion polarization netic moment is naturally dominated by the two-atom
bubblesillustratedinFig.7b. Withthedetailsofthecal- electronic spin state (hence the proportionality constant
culationsrelegatedto Appendix A (see alsoRefs.26,29), of the Bohr magneton µ above), that is approximately
B
we find at low E asingletforthe closedchannelandatripletforthe open
channel. Detailed multi-channel calculations (see, e.g.,
¯h √Γ0 Ref.18)allowa moreaccuratedeterminationofparame-
f (E)= , (3.2)
0 −√mE δ+i√Γ0√E terswhennecessary. Equatingas inEq.(3.6)withaesxp.,
−
Eq.(1.2),allowsusto determinethe Feshbachresonance
where δ is the renormalized (physical) detuning and Γ energy width Γ (or equivalently the parameter g) aris-
0 0
isaparametercharacterizingthewidthoftheresonance, inginthemodelintermsoftheexperimentallymeasured
10
2
“width” B defined by the dependence of the scattering mr E
w 0
length on B in Eq. (1.2): 2
4mµ2a2 B2 1.5
B bg w
Γ . (3.7)
0 ≈ ¯h2 1
The bound states of the model are determined by the 0.5
real negative-energy poles of f0(E), that can be shown |r0|
to appear only at negative detuning115, δ <0. For small -2 -1 1 2 − as
negative detuning, δ Γ0, the pole is at low energy, -0.5
| | ≪
E Γ , so that E in the denominator of f (E) can be
0 0
neg≪lected relative to √Γ E. At more negative detuning, -1
0
δ Γ , one can instead ignore the √Γ E and the pole
0 0 -1.5
| |≫
is simply given by the detuning δ. Together these limits
FIG. 8: Plot of the negative-energy bound-state [E < 0,
give: Eq.(3.9b)]andpositive-energyresonance[E >0,Eq.(3.12a)]
positions as a function of r /a .
δ2 = h¯2 , for δ Γ , −| 0| s
Ep(δ) −Γ0 −2ma2s | |≪ 0 (3.8)
≈ (δ, for δ Γ0,
| |≫ by
a standard result114 consistent with the observed phe-
E (δ)=E (δ) iΓ(δ), (3.11)
nomenology of Feshbach resonances13,17. The complete p r −
E (δ) interpolatingbetweenthese limiting expressionsis
p with
given by
1 2h¯2 r
0
E (δ)=δ Γ = | | 1 , (3.12a)
Γ 2δ 4δ r − 2 0 mr2 a −
E (δ) = 0 1+ | | 1+ | | , (3.9a) 0 | s|
p − 2 Γ0 −s Γ0 Γ0 4δ 2h(cid:0)¯2 (cid:1)
h i Γ(δ)= 1= 2r /a 1, (3.12b)
2h¯2 r 2r 2 rΓ0 − mr02 | 0| | s|−
= 1+ | 0| 1+ | 0| , (3.9b) p
−mr02 as −s as where the imaginary part Γ measures the resonance
h i
width (i.e., molecular decay rate is Γ/¯h).
obtainedbysolvingthequadraticequation1/f (E )=0.
0 p
In Eq. (3.9b) we expressed E in terms of the scatter-
p
ing length as and the effective range parameter r0 = B. Scattering at finite density: small parameter
2h¯/√mΓ . We note in passing that, unlike the case of
0
−
a non-resonant short-range potential where r > 0 and
0 AsalreadydiscussedintheIntroduction,atfiniteden-
measures the range of the potential (hence the name), sity n=k3/3π2 the two-channelmodel admits a dimen-
F
here, for a resonant interaction, r < 0 is negative, with
0 sionless parameter γ 1/(k r ) that is the ratio of the
F 0
its magnitude characterizing the closed-channel molecu- average atom spacing∝n 1/3 t|o|the effective range. The
−
lar lifetime τ ¯h/√Γ δ at positive detuning26,29. In
0 parameterγ controlsaperturbativeexpansion(aboutan
∼
terms of a and r the scattering amplitude takes the
s 0 exactlysolvablenon-interactingg =0limit)ofanyphys-
standard form114
ical quantity and is given by
1
f0(k)= a−s1+r0k2/2 ik, (3.10) γ g2N(ǫF) = √8 Γ0 = g2c = 8 1 , (3.13)
− − ≡ ǫF π rǫF √ǫF πkF|r0|
where k = 2mrE/¯h2 and mr = m/2 is the two-atom relatedtotheratiooftheFeshbachresonancewidth(con-
reduced maqss. trolledbytheatom-moleculecouplingg)totheFermien-
As can be seen by analyzing Eq. (3.2), the physical ergy. Thekeyobservationisthatγ isindependentofthe
bound state disappears115 for positive δ. However, as scatteringlengtha anddetuningδ,andassuch,onceset
s
illustrated in Fig. 8, a positive energy resonance does small, remains small throughout the crossover, even for
not appear until δ reaches the threshold value of δ = a Feshbach resonance tuned through zero. Hence, two-
Γ /2, corresponding to a (δ ) = r . This absenc∗e of channel model predictions for a narrow Feshbach reso-
0 s 0
a resonance for a range|of po∗si|tive|de|tuning 0 < δ < δ nance, (defined by γ 1, i.e., width of the resonance
is a property unique (due to the absence of a centrifuga∗l much smaller than the≪Fermi energy) are quantitatively
barrier) to an s-wave resonance, contrasting with finite accurate throughout the BEC-BCS crossover.
angular momentum resonances29,116. This contrasts qualitatively with a one-channel model
For larger positive detuning, δ > Γ /2 (a < r ), a characterizedby adimensionless gasparameterna3 that
0 | s| | 0| s
finite-width resonance appears at a complex E (δ) given divergesforaFeshbachresonancetunedtozero(i.e.,“on
p