Destroying superfluidity by rotating a Fermi gas at unitarity

0.21 MB·English

by  I. Bausmerth

Destroying superfluidity by rotating a Fermi gas at Unitarity
I. Bausmerth,1 A. Recati,1,∗ and S. Stringari1
1Dipartimento di Fisica, Universit`a di Trento and CNR-INFM BEC Center, I-38050 Povo, Trento, Italy
WestudytheeffectoftherotationonaharmonicallytrappedFermigasatzerotemperatureunder
the assumption that vortices are not formed. We show that at unitarity the rotation produces a
phase separation between a non rotating superfluid (S) core and a rigidly rotating normal (N) gas.
The interface between the two phases is characterized by a density discontinuity nN/nS = 0.85,
independent of the angular velocity. The depletion of the superfluid and the angular momentum
of therotating configuration are calculated as a function of the angular velocity. The conditions of
8 stabilityarealsodiscussedandthecriticalangularvelocityfortheonsetofaspontaneousquadrupole
0 deformation of theinterface is evaluated.
0
PACSnumbers:
2

n
a The effect of the rotation on the behavior of a super- the Feynman critical angular velocity for the formation
J fluid is a longstanding subject of investigation in con- ofquantizedvorticeswhosenucleationisinhibitedbythe

3 densed matter as well as in nuclear systems [1, 2]. Due presenceofa barrier. Underthese conditionsnew phys-
2
totheirrotationalityconstraintimposedbytheexistence

of the order parameter a superfluid cannot rotate like a
]
t normal fluid. This implies the quenching of the moment
f
o ofinertiaatsmallangularvelocitiesΩandtheoccurrence
s ofquantizedvorticesathigherΩ. Thesepeculiarfeatures
.
t have been experimentally studied in a systematic way in
a
superfluid helium and, more recently, in Bose-Einstein
m
condensates (see e.g. [3]).
-
d The recent realization of ultracold Fermi gases with
n tunable interaction close to a Feshbach resonance has
o openednewopportunitiestoinvestigatetheeffectsofsu-
c
perfluidity along the so called BCS-BEC crossover (for
[
recent reviews see, for example, [4, 5]). While on the

3 BEC side of the resonance (small and positive values of
v the scattering length) the physics of a Fermi superfluid
3
approaches the behavior of a Bose-Einstein condensed
5
gasofdimers,ontheoppositeBCSside(smallandnega-
6
tivevaluesofthescatteringlength)superfluidityismuch
0
. more fragile because of the weakness of Cooper pairs. A
1
further challenging regime is given by the unitary gas
1
7 of infinite scattering length where the system exhibits a FIG. 1: Upper panel: without rotation the system is fully
0 universal behavior and, in many respects, behaves like a superfluid. Lower panel: the rotation of the trap favours the
: strongly interacting fluid. Quantized vortices have been formation of a phase separated state, with a superfluid core
v
recently observed on both sides of the resonance [6], in- surroundedbyanormalpartexhibitingthebulgeeffectofthe
i
X cluding unitarity, providing a unifying picture of the be- rotation (Ω=0.45ω⊥).
r havior of the system along the crossover.
a
PrevioustheoreticalworkonsuperfluidrotatingFermi ical phenomena occur like, for example, the spontaneous
gaseshas mainly focused onthe dynamics[7] andonthe breakingof rotationalsymmetry causedby the energetic
instability [8, 9] of configurations with high vortex den- instability of the surface modes [10] and, at even higher
sity. The purpose of this Letter is to investigate the be- angular velocities, the occurrence of dynamic instabili-
haviorofthegaswhen quantized vortices are not formed. ties [11]. These phenomena have been observed experi-
Experimentally this scenario is achievable through an mentally in BEC’s [12], confirming in a qualitative and
adiabatic ramping of the the angular velocity of the ro- quantitative way the correctness of the irrotational hy-
tating trap starting from a configuration at rest. In the drodynamic picture of rotating superfluids.
case of trapped BEC’s this procedure has permitted to The key point of the present work is that trapped su-
reachvalues of angular velocity significantly higher than perfluid Fermi gases can behave quite differently from
BEC’sbecausepairsareeasilybrokenbytherotation. In
the following we will restrict our discussion to the most
relevant unitary gas. While at small angular velocities
∗Electronicaddress: [email protected] the superfluid is unaffected by the rotation of the trap
2
[13],wepredictthatathigherangularvelocitiestherota- the superfluid and normal part, as well as with respect
tionresultsinaphaseseparationbetweenanonrotating tothepositionofthebordersurface. Noticethatthispic-
superfluid core and a rigidly rotating normal component ture ignores surface tension effects, a plausible assump-
(see Fig.(1)). The mechanism of pair breaking is very tioninthelimitoflargesamples. Thesuperfluidvelocity
intuitive. In fact near the borderof the cloud, where the obeystheirrotationalityconstraintandcanbewrittenas
densitynissmall,theenergycostfordestroyingsuperflu- v = Φ. Varying the energy with respect to the veloc-
S
∇
idity is also small, being proportionalto n2/3. Viceversa ity potential Φ yields the continuity equation
thecentrifugalenergygainedbythenormalphasecanbe
(( Φ Ω r)n )=0, (4)
large, being proportional to Ω2R2 where R is the radius ∇· ∇ − × S
of the cloud. Notice that this mechanism cannot occur while variation with respect to the superfluid density n
S
in the BEC regime due to the high energy cost needed leads to
to break the dimers. Furthermore, differently from the ~2
nucleation of vortices, the formation of the normal part µ=ξ (6π2n )2/3+VS(r), (5)
S S
2m
isnotinhibitedbythepresenceofabarrierasalltherel-
evantenergyscalesarevanishinglysmallneartheborder whereVS(r)=V(r)+1mv2 mΩ(r v ) istheeffective
2 S− × S Z
and the energy gain is ensured as soon as one starts ro- harmonic potential felt by the superfluid.
tating the trap. The appearanceof a normalpartdue to Usingthe sameprocedureforthe normalpartwithout
rotationis notpeculiar of ultra-coldatomic Fermi gases. the irrotationality constraint we have v = Ω r, i.e.
N
×
For example, superfluidity is known to be unstable in thenormalcomponentrotatesrigidly. Thevariationwith
nuclei rotating with high angular velocity [2]. Such an respect to the normal density yields an equation similar
effectisalsorelatedtotheso-calledintermediatestatein to (5)
type-I superconductors, where the role of the rotation is
~2
played by the external magnetic field (see, e.g., [14]). µ=ξ (6π2n )2/3+VN(r), (6)
N N
We are interested in a Fermi gas confined by a har- 2m
monic potential V(r) = m(ω2x2 +ω2y2 +ω2z2)/2, ro- where the effective harmonic potential VN(r) is now
x y z
tating with angular velocity Ω along z. We study the quenched by the rigid rotation according to (ωN)2 =
x
problem in the frame rotating with the trap where the ω2 Ω2, (ωN)2 =ω2 Ω2.
x− y y−
potential is static and the Hamiltonian is H −ΩLZ. In By varying the energy (1) with respect to RS(θ,φ) we
the local density approximation the energy of the rotat- eventually find the equilibrium condition for the coexis-
ing configuration at zero temperature can be written as tence of the two phases in the trap. The resulting equa-
tion implies that the pressure of the two phases be the
E = dr ǫ(n)+V(r)+ 1mv2 mΩ(r v)Z µ n , same: n2S(∂ǫS/∂nS) = n2N(∂ǫN/∂nN). Using Eqs.(2) and
Z (cid:20) 2 − × − (cid:21) (3), one then predicts a density discontinuity at the in-
(1) terface given by
where ǫ(n) is the energy density per particle, v the ve-
locity field, µ the chemical potential and n the density. nN ξS 3/5
=γ with γ = =0.85, (7)
BytermingR (θ,φ)theinterfaceseparatingthesuper- n (cid:18)ξ (cid:19)
S S N
fluid from the normal component, the integral in Eq.(1)
independent of the angular velocity. The equal pressure
splits in two parts. The internal superfluid core occu-
condition (7) combined with Eqs.(5) and (6), results in
pies the region r < R (θ,φ) and the surrounding nor-
S the useful relationship
mal phase is confined to R (θ,φ) <r <R (θ,φ), where
S N
R (θ,φ)istheThomas-Fermiradiusofthegaswherethe (µ VS(r))=γ(µ VN(r)), (8)
N
− −
density vanishes. The energy densities in the two phases
whichdetermines the surface R (θ,φ) separatingthe su-
are given, respectively, by S
perfluid and the normal part.
3 ~2 In the following we assume ωx = ωy ω⊥ and we
ǫS =ξS52m(6π2nS)2/3, (2) consider the solution vS = 0, correspond≡ing to a non
rotating axi-symmetric superfluid and hence to VS =V.
In this case we find
3 ~2 −1
ǫN =ξN52m(6π2nN)2/3, (3) RS2(θ)= 2mµ(cid:18)ωz2cos2θ+ω⊥2 sin2θ+ 1 γ γΩ2sin2θ(cid:19) ,
−
(9)
where n (n ) is the superfluid (normal) density and
S N
whereθisthepolarangle. OntheothersidetheThomas-
the dimensionless parameters ξ = 0.42 and ξ = 0.56
S N
Fermi radius R of the normalgas is fixed by the condi-
account for the role of interactions in the two phases. N
tion µ=VN yielding
Their value has been calculated in [15, 16] employing
Quantum Monte Carlo simulations. R2(θ)= 2µ ω2cos2θ+ω2 sin2θ Ω2sin2θ −1 R2(θ).
The equilibrium is found by minimizing the energy N m z ⊥ − ≥ S
with respect to the densities and the velocity fields of (cid:0) (cid:1) (10)
3
The value of µ is fixed by the normalization condition andthetotalnumberN asfunctionoftheangularveloc-
ity. The higher the angular velocity, the more particles
n dr+ n dr=N , (11) prefertostayinthenormalphaseandthusthesuperfluid
S N
Z Z is depleted. At small angular velocities the depletion of
r<RS(θ) RS(θ)<r<RN(θ)
the superfluid follows the law N /N =1 ( γ )5/2 Ω5.
S − 1−γ
whereN isthetotalnumberofparticles. WhileforΩ=0
the system is completely superfluid, for Ω > 0 it phase
N (cid:144)N
S
separatesintoasuperfluidandanormalcomponentchar-
1
acterized by the density jump (7) at the interface. This
behavior shares interesting analogies with the the phase
0.8
separationbetweenasuperfluidandanormalcomponent
exhibited by polarized Fermi gases [17, 18, 19] where a 0.6
jump in the density at the interface is also predicted to
occur[20]. Bytomographictechniques[19]itisnowadays 0.4
possible to measure the density ”in situ”, thus the pre-
dicteddiscontinuity,andhencethevalueofξ /ξ ,should 0.2
S N
be observable experimentally.
The radii R and R coincide at θ = 0 which means W(cid:144)Ω¦
S N 0.2 0.4 0.6 0.8 1
the absence of the normal part along the z-axis. In the
plane of rotation (θ = π/2) the difference between the
two radii is instead maximum and becomes larger and
larger as Ω increases. In particular the radius of the FIG. 3: Depletion of the superfluid as a function of the trap
superfluid is always smaller than the radius of the cloud angular velocity for an axi-symmetric configuration (vS.
intheabsenceofrotation,whiletheradiusofthenormal
gas is always larger due to the bulge effect produced by Another important observable is the angular momen-
the rotation. tum LZ. For an axi-symmetric configuration the su-
perfluid does not carry angular momentum which is
1 1 then provided only by the normal component: LZ =
0.75 Ω dr (x2+y2)nN. The total angular momentum then
L0.8 (cid:144)RRS00.5 ivnacRlureeaasetsΩw=ithωΩ (asnede Fevigen.(t4u)a).llyTrheeacahnegsultahremriogmidenbtoudmy
0 0.25 ⊥
H of a rotating configuration has been measured in BEC’s
0nS0.6 0 0.25 0.5 0.75 1 by studying the precession phenomena exhibited by the
(cid:144) W(cid:144)Ω¦
L surface excitations [21].
r0.4
H
n
0.2 LZ(cid:144)LZ,RIG
1
0
0.2 0.4 0.6 0.8 1
r(cid:144)R 0.8
0
0.6
FIG. 2: Density profile for θ = π/2 of the rotating Fermi
0.4
gas at Ω = 0.45ω⊥ (full line). The profile in the absence of
rotation is also shown (dashed line). Inset: Superfluidradius
0.2
versusΩ/ω⊥.
W(cid:144)Ω¦
0.2 0.4 0.6 0.8 1
InFig.(2)weplotthedensitiesn andn asafunction
S N
of the radial coordinate at θ = π/2 in a spherical trap
for Ω = 0.45ω . The densities and the radial coordi-
⊥
nate have been renormalized with respect to the central FIG. 4: Angular momentum in units of the rigid value as
density n0 and the Thomas-Fermi radius R of the su- a function of the trap angular velocity for an axi-symmetric
S 0
perfluidatrest. TheinsetshowsthesuperfluidradiusR configuration.
S
renormalizedby R as a function of the angular velocity
0
Ω/ω⊥. While for small values of the angular velocity (Ω <
From the knowledge of the density profiles and from 0.2ω )thesuperfluidisrobust,itisremarkablethateven
⊥
theradiiEqs.(9)and(10)wecancalculatethenumberof atangularvelocitiesfarfromthecentrifugallimitthede-
particles in each phase. In Fig.(3) we show the ratio be- pletion of the superfluid and hence the angular momen-
tweenthe numberofparticlesN inthesuperfluidphase tum of the system are sizable.
S
4
A major issue concerns the conditions of stability of gence of dynamic instabilities. In the case of a rotating
the rotating configurationdiscussed in this work. Let us BEC a dynamic instability takes place at values of Ω
firstconsiderthequestionofenergeticstability. We have slightly larger than ω /√2 and corresponds to the ap-
⊥
shownthat,intheframerotatingwiththetrap,thephase pearance of imaginary components in the frequency of
separatedconfigurationisenergeticallyfavouredwithre- somehydrodynamicmodes[11]. Inthe caseofthe rotat-
spect to the configuration where the whole gas is at rest ingFermigasdiscussedinthisworkadynamicinstability
and superfluid. This is true for any value of the angu- might be associated with the Kelvin-Helmholtz instabil-
lar velocity. When Ω exceeds a critical value of order ity of the interface between two fluids in relative motion
(~/mR2)ln(R/d), where d is the healing length fixed, at (see.e.g [24]). However, if the densities of the two flu-
unitaritybytheinterparticledistance,quantizedvortices idsaredifferent,anexternalforcestabilizesthetwo-fluid
become an even more favourable configuration. This en- systemagainstthe appearance ofcomplex frequencies in
ergetic instability is not howeverexpected to be a severe the low energy excitations of the interface [24]. This is
difficulty if one increases the angular velocity in an adi- actually our case where the density of the two phases
abatic way because the presence of a barrier inhibits the exhibits the gap (7) and the system feels the external
accesstothe vorticalconfigurationasprovenexperimen- force produced by the harmonic confinement. We conse-
tally in the case of BEC’s [22]. At higher angular veloc- quently expect that the system be dynamically stable at
ities the axi-symmetric superfluid configuration is even- least for moderately small values of the angular velocity.
tually expected to exhibit a surface energetic instability,
In conclusion we have shown that an ultracold Fermi
undergoing a continuous shape deformation, similarly to
gasatunitaritycanseparateintoasuperfluidandanor-
what happens in Bose-Einstein condensates [10, 12, 23].
mal component as a consequence of the adiabatic ramp-
This effect is accounted for by the solution v = 0 of
S 6 ing of the trap rotation. The formation of the rotating
Eq.(4), corresponding to a spontaneous breaking of ro-
normalcomponentrequiresthatthe traptransfersangu-
tational symmetry in the superfluid component. In the
lar momentum to the gas within experimentally accessi-
case of Bose-Einsteincondensates a quadrupole instabil-
bletimes. Itsrealizationwouldopentheuniquepossibil-
itytakesplaceatΩ =ω /√2andforlargervaluesofΩ
cr ⊥ ity of exploring the Fermi liquid behaviour of a strongly
the solutionv =0correspondstothe socalledovercrit-
S interacting gas at zero temperature. Important effects
ical branch [10]. A different value of Ω is predicted for
cr to investigate are, for example, the zero sound nature of
the rotating Fermi gas, due to the new boundary condi-
the collective oscillations and the behaviour of viscosity.
tion imposed by the presence of the normal component.
The detailedstudy ofthe energeticinstability associated
This condition requires that the velocity of the super-
withthespontaneousbreakingofrotationalsymmetryas
fluid be tangential to the interface. In the simplest case
well as the effects of the rotation on the polarized phase
ofa 2Dconfigurationwe find the value Ω =0.45ω for
cr ⊥ (N =N ) will be the object of a future work.
theemergenceofaspontaneousquadrupoledeformation. ↑ 6 ↓
The experimental measurement of Ω would provide a We acknowledge stimulating discussions with Frederic
cr
crucial test of the consequences of the phase separation Chevy, Stefano Giorgini and Lev Pitaevskii. We also
caused by the rotation of the unitary Fermi gas. acknowledge support by the Ministero dell’Istruzione,
Anevenmorechallengingquestionconcernsthe emer- dell’Universit`a e della Ricerca (MIUR).
[1] R.J. Donnelly, Quantized Vortices in Helium II (Cam- [10] A.Recati,F.Zambelli,andS.Stringari,Phys.Rev.Lett.
bridge UniversityPress, 1991). 86, 377 (2001).
[2] D.M. Brink and R.A. Broglia, Nuclear Superfluidity: [11] S. Sinha and Y. Castin, Phys. Rev. Lett. 87, 190402
Pairing in Finite Systems (Cambridge University Press, (2001). A similar scenario is also obtained for a rotating
2005). Fermigasundertheassumptionthatthesystemremains
[3] I.Coddington,P.C.Haljan,P.Engels,V.Schweikhard,S. fully superfluid at high angular velocities (G.Tonini et
Tung,andE.A.Cornell,Phys.Rev.A70,063607(2004). al., Eur. Phys. J. D 39, 283 (2006))
[4] W. Ketterle, M. Inguscio and C. Salomon (editors) Ul- [12] K.W. Madison, F. Chevy, V. Bretin, and J. Dalibard,
tracoldFermiGases(ProceedingsoftheVarenna’Enrico Phys. Rev.Lett. 86, 4443 (2001).
Fermi’ SummerSchool, 2007). [13] M.CozziniandS.Stringari,Phys.Rev.Lett.91,070401
[5] S. Giorgini, L.P. Pitaevskii, and S. Stringari, cond- (2003).
mat/0706.3360. [14] P.G. deGennes, Superconductivityof metals and alloys,
[6] M.W. Zwierlein, J.R. Abo-Shaeer, A. Schirotzek, C.H. (Westview Press, 1999).
Schunck,and W. Ketterle, Nature435, 1047 (2005). [15] J. Carlson, S.Y. Chang, V.R. Pandharipande, and K.E.
[7] M. Antezza, M. Cozzini, S. Stringari, Phys. Rev. A 75, Schmidt, Phys.Rev.Lett. 91, 050401 (2003).
053609 (2007). [16] G.E. Astrakharchik, J. Boronat, J. Casulleras, and S.
[8] H. Zhai and T.-L. Ho, Phys. Rev. Lett. 97, 108414 Giorgini, Phys. Rev.Lett. 93, 200404 (2004).
(2006). [17] M.W. Zwierlein, C.H. Schunck, A. Schirotzek, W. Ket-
[9] G. Moller and N.R. Cooper, cond-mat/0704.3859. terle, Nature 442, 54 (2006); Y. Shin, M.W. Zwierlein,
5
C.H. Schunck, A. Schirotzek, W. Ketterle, Phys. Rev. Lett. 85, 2223 (2000).
Lett.97, 030401 (2006). [22] P. Rosenbusch, D.S. Petrov, S. Sinha, F. Chevy, V.
[18] G.B.Partridge,W.Li,R.I.Kamar,Y.Liao,R.G.Hulet, Bretin,Y.Castin,G.Shlyapnikov,andJ.Dalibard,Phys.
Science 311, 503 (2006). Rev. Lett.88, 250403 (2002).
[19] Yong-il Shin, Christian H. Schunck, Andre Schirotzek, [23] F.Dalfovo andS.Stringari, Phys.Rev.A63,011601(R)
Wolfgang Ketterle, cond-mat/0709.3027. (2001).
[20] C. Lobo, A.Recati, S.Giorgini, S. Stringari, Phys. Rev. [24] L.D. Landau and E.M. Lifshitz Fluid Mechanics (2nd
Lett97,200403(2006);A.BulgacandM.McNeilForbes, edn.) (Butterworth-Heinemann,2000).
Phys.Rev.A 75, 031605(R) (2007).
[21] F. Chevy, K.W. Madison, and J. Dalibard, Phys. Rev.