Universal Spin Transport in a Strongly Interacting Fermi Gas

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by  Ariel Sommer

Universal Spin Transport in a Strongly Interacting Fermi Gas
Ariel Sommer,1 Mark Ku,1 Giacomo Roati,2 and Martin W. Zwierlein1
1Department of Physics, MIT-Harvard Center for Ultracold Atoms, and
Research Laboratory of Electronics, MIT, Cambridge, Massachusetts 02139, USA
2LENS and Dipartimento di Fisica, Universita` di Firenze, and INO-CNR, 50019 Sesto Fiorentino, Italy
Transport of fermions is central in many fields of physics. Electron transport runs modern tech-
nology, defining states of matter such as superconductors and insulators, and electron spin, rather
than charge, is being explored as a new carrier of information [1]. Neutrino transport energizes
supernova explosions following the collapse of a dying star [2], and hydrodynamic transport of the
quark-gluon plasma governed the expansion of the early Universe [3]. However, our understanding
1
ofnon-equilibriumdynamicsinsuchstronglyinteractingfermionicmatterisstilllimited. Ultracold
1
gases of fermionic atoms realize a pristine model for such systems and can be studied in real time
0
with the precision of atomic physics [4, 5]. It has been established that even above the superfluid
2
transitionsuchgasesflowasanalmostperfectfluidwithverylowviscosity[3,6]wheninteractions

n are tuned to a scattering resonance. However, here we show that spin currents, as opposed to
a mass currents, are maximally damped, and that interactions can be strong enough to reverse spin
J currents,withoppositespincomponentsreflectingoffeachother. Wedeterminethespindragcoeffi-

4 cient,thespindiffusivity,andthespinsusceptibility,asafunctionoftemperatureonresonanceand
showthattheyobeyuniversallawsathightemperatures. Atlowtemperatures,thespindiffusivity

] approaches a minimum value set by (cid:126)/m, the quantum limit of diffusion, where (cid:126) is the reduced
s
Planck’s constant and m the atomic mass. For repulsive interactions, our measurements appear to
a
g exclude a metastable ferromagnetic state [7–9].
-
t
n
a Understanding the transport of spin, as opposed to cientsofaFermigascanbeestimatedongeneralgrounds.
u the transport of charge, is of high interest for the novel At the Feshbach resonance, the scattering cross-section
q field of spintronics [1]. While charge currents are unaf- σ between atoms of opposite spin is given by the square
.
t fected by electron-electron scattering due to momentum of the matter wavelength, in the degenerate regime σ ∼
a
conservation,spincurrentswillintrinsicallydampdueto 1/k2,wherek =(6π2n)1/3 istheFermiwavevectorand
m F F
collisions between opposite spin electrons, as their rela- n is the density of atoms in each spin state. The mean
-
d tive momentum is not conserved. This phenomenon is free path between collisions is thus l = 1/nσ ∼ 1/kF
n known as spin drag [10, 11]. It is expected to contribute or about one interparticle spacing, the smallest possible
o significantly to the damping of spin currents in doped meanfreepathinagas. Theaveragespeedv ofatomsis
c semiconductors [12]. The random collision events also on the order of the Fermi velocity (cid:126)k /m. In estimating
[ F
lead to spin diffusion, the tendency for spin currents to thespindiffusivityD ≈vlthedensity-dependentfactors
s
1 flowsuchastoevenoutspatialgradientsinthespinden- cancel, giving D ≈ (cid:126)/m. This value for D represents
s s
v
sity, which has been studied in high-temperature super- a universal quantum limit to spin diffusivity in Fermi
0
conductors [13] and in liquid 3He-4He solutions [14, 15]. gases. Away from resonance, the scattering cross sec-
8
7 Creatingspincurrentsposesamajorchallengeinelec- tion decreases, increasing Ds. For temperatures T much
0 tronic systems where mobile spins are scattered by their greater than the Fermi temperature TF = (cid:126)2kF2/2mkB,
1. environment and by each other. However, in ultracold the scattering cross section will be given by the square
0 atoms we have the freedom to first prepare an essen- of the thermal de Broglie wavelength and thus decr√ease
1 tially non-interacting spin mixture, separate atoms spa- as σ ∝ 1/T, while the velocity will increase as v ∝ T,
:1 tially via magnetic field gradients, and only then induce causingDs toincreaseasDs ∝T3/2. Finally,inadegen-
v stronginteractions. Pastobservationsofspincurrentsin erate Fermi gas the average velocity will remain on the
Xi ultracold Fermi gases [16, 17] were made in the weakly- order of the Fermi velocity, but the effective scattering
interacting regime. Here we access the regime near a cross section will scale as σ ∝ T2 due to Pauli blocking,
r
a Feshbach resonance [5], where interactions are as strong causingDstoincreaselikeT−2asthetemperatureislow-
as allowed by quantum mechanics (the unitarity limit). ered. ForaFermigas,wethusexpecttheminimumDsto
Wemeasurespintransportproperties,thespindragcoef- occurattemperaturesneartheFermitemperature. Cor-
ficientΓsd andthespindiffusivityDs,ofastronglyinter- respondingly, the coefficient Γsd characterizing spin drag
acting Fermi gas composed of an equal number of atoms is expected to reach a universal maximum value on reso-
in two different spin states. In the strongly-interacting nanceandfortemperaturesneartheFermitemperature,
regime, spin drag is expected to reach a universal max- given by the Fermi rate EF/(cid:126), where EF = (cid:126)2kF2/2m is
imum value, and spin diffusion is expected to reach a the Fermi energy.
universal minimum. In the experiment, we prepare an equal mixture of
The universal behaviour of the spin transport coeffi- the two lowest hyperfine states (“spin up” and “spin
2
establishing a spin current. Measurements are made by
a
selectively imaging the two spin components.
Figure 1 shows the collision between the two spin do-
mains on resonance. The clouds bounce off each other
m
m and essentially completely repel. Due to the axial trap-
1.3 ping potential, the clouds return after the collision, and
we observe several oscillations in the displacement d =
(cid:104)z (cid:105) − (cid:104)z (cid:105), where (cid:104)z (cid:105) is the center of mass of the
↑ ↓ ↑(↓)
spin up (down) cloud. After the oscillations have de-
cayed, the displacement decreases to zero monotonically,
on a timescale on the order of one second, an extremely
b
longtimecomparedtothetrappingperiod(44ms). The
underlying explanation for spin current reversal and the
slowrelaxationcanbefoundintheextremelyshortmean
m
m free path and the high collision rate between opposite-
1.3 spinatomsatunitarity. Accordingtotheaboveestimate,
the spin diffusivity is approximately (cid:126)/m, which for 6Li
is (100 µm)2/s. The atom clouds in the experiment have
a length on the order of 100 µm, and it takes them on
the order of a second to diffuse through each other. So
Time (1 ms per frame) we are indeed observing quantum-limited spin diffusion.
The initial bounces will occur when the mean free path
c 0.3 d of a spin up atom in the spin down cloud is smaller than
m) 0.2 m)0.2 thespindowncloudsize,i.e. whenthemixtureishydro-
m 0.1 m
Position (--000...021 Position (-00..02 drsecyganitoatnemr,eicdit.bIinsacstktheiaenndtoomftohqreueiscplkiiklnyeluydpifftrhueasgtiinotgnhiewnthsoperitneheiutspcpaianntopdmroowpins-
-0.3 agate ballistically. Indeed, we see bounces occur already
0 40 80 120 160 400 800 0.0 0.4 for interaction strengths away from the Feshbach reso-
Time (ms) V/kB (µK) nance where the mean free path is on the order of the
axial cloud size (see Supplementary Information).
FIG.1. Observationspincurrentreversalinaresonant The relaxation dynamics close to equilibrium give di-
collisionbetweentwooppositelyspin-polarizedclouds rect access to the spin transport coefficients. The spin
of fermions. (a)showsthetotalcolumndensityand(b)the
drag coefficient Γ is defined as the rate of momentum
differenceincolumndensitiesofthetwoclouds(red: spinup, sd
transfer between opposite spin atoms [10], and is there-
blue: spin down), in 1 ms intervals during the first 20 ms
fore related to the collision rate. The relaxation of the
after the magnetic field is set to the Feshbach resonance at
834 G. The collision leads to the formation of a high-density displacement d near equilibrium then follows the differ-
interface between the two spin states. (c) The separation ential equation [18]
between the centers of mass of the two spin states initially
oscillatesatafrequencyof1.63(2)ωz,whereωz =2π×22.8Hz Γsdd˙+ωz2d=0.
isthetrapfrequencyintheaxialdirection(seeSup. Info. for
Fittinganexponentialwithdecaytimeτ tothedisplace-
discussion). Evenafterhalfasecond,thereisstillsubstantial
spin separation. The diffusion time indicates a diffusivity on ment gives the spin drag coefficient as Γsd =ωz2τ.
theorderof(cid:126)/m. (d)Showstheharmonictrappingpotential The spin drag coefficient is found to be greatest on
along the axis of symmetry. resonance, and thus spin conduction is slowest (see Sup-
plementary Information). On dimensional grounds, Γ
sd
must be given by a function of the reduced tempera-
ture T/T times the Fermi rate E /(cid:126). At high tem-
F F
down”)offermionic6Liinacylindricallysymmetricatom peratures, we expect the spin drag coefficient to obey
trap [4, 5]. The confinement along the axis of symme- a universal scaling Γ ∼ nσv ∼ T−1/2. In fig. 2 we
sd
try is harmonic, with frequency ω . We separate the show the spin drag coefficient as a function of T/T .
z F
two spin components along the axis of symmetry of the We observe T−1/2 scaling for T/T > 2, finding Γ =
F sd
trap (see Methods Summary) and turn on strong inter- 0.16(1)E(cid:126)F(T/TF)−1/2. At lower temperatures, we ob-
actions between unequal spins by quickly ramping the serve a crossover from classical to non-classical behav-
magneticfieldtoaFeshbachresonancelocatedat834G. ior: the spin drag coefficient reaches a maximum of ap-
The confining potential of the trap forces the two clouds proximately 0.1E /(cid:126) near the Fermi temperature. We
F
of opposite-spin atoms to propagate towards each other, interpret this saturation of the spin drag coefficient as
3
a consequence of Fermi statistics, as σ and v approach where n (↓) is the density of spin up(down) atoms.
↑
constantvalues[16]determinedbytheFermiwavevector We calculate the trap-averaged spin current as
k . The maximum spin drag coefficient corresponds to J = 1(n +n )d˙, where the densities are evaluated at
F s 2 ↑ ↓
a minimum spin conductivity σ = n on the order the trap center.
of k /(cid:126). This is the slowest spinscondmuΓcstdion possible in Wefindthatspindiffusivityisminimumwheninterac-
F
three dimensions in the absence of localization. tionsareresonant(seeSupplementaryInformation). The
increase in spin diffusivity for positive scattering length
a, as well as the decrease in spin drag, argues against
0.12
the existence of a ferromagnetic state in repulsive Fermi
gases, for which diffusion should stop entirely [7, 9, 22].
0.10
Figure 3 reports the measured spin diffusivity as a func-
F
/Ed tionoftemperatureatunitarity. Inthehigh-temperature
Γs0.ℏ08 limit on resonance, one expects Ds ∼ v/nσ ∝ T3/2. At
high temperatures, we indeed find this temperature de-
0.06 pendence, with a fit giving D = 5.8(2)(cid:126)(T/T )3/2 for
s m F
T/T > 2. Down to our lowest temperatures, the spin
F
diffusivityisseentoattainalimitingvalueof6.3(3)(cid:126)/m.
2 3 4 5 6 789 2 3 4 5 6 789
0.1 1 10 Fermi liquid theory predicts a rapid increase at low tem-
T/T
F peraturesasD ∝(T /T)2. Frommeasurementsonspin
s F
quadrupolar oscillations in highly spin-imbalanced mix-
FIG. 2. Spin drag coefficient of a trapped Fermi gas.
tures it is known [23] that the resulting crossover from
Spindragcoefficientforresonantinteractions,asafunctionof
diffusive to ballistic motion occurs below T/T ≈ 0.1.
the dimensionless temperature. The temperature is normal- F
However,belowT/T ≈0.17abalancedgasissuperfluid
izedbythemaximumFermitemperature,definedintermsof F
themaximumdensityofasinglespinstateinthesystem. We andspintransportpossiblyinhibitedbythesuperfluidto
findagreementbetweenmeasurementstakenatthreedifferent normal interface [21].
valuesoftheaxialtrappingfrequencyω =2π×22.8Hz(red
z
circles),ω =2π×37.5Hz(bluetriangles),ω =2π×11.2Hz
z z
(black squares). The data for T/TF > 2 fits to a T−1/2 law 2
(solid line). Dashed line: a power law fit with an uncon-
strained exponent for T/T < 0.5 to show the trend. The 100
F 8
spin drag coefficient is obtained from the decay rate of the
6
spinseparationandnormalizedbythesquareofthetrapfre-
4
quency and by the maximum Fermi energy. The error bars ℏ
are statistical. mD/s 2
10
8
At low temperatures, the spin drag coefficient de- 6
creaseswithdecreasingtemperature. Notheoreticalpre- 4
diction for the spin drag coefficient exists in this regime.
2
For highly-imbalanced spin populations, a decrease in
2 3 4 5 6 789 2 3 4 5 6 789
the spin drag coefficient is expected due to Pauli block- 1 10
T/T
ing [19], which reduces the collision rate in the gas by a F
factorof (T/T )2 atsufficientlylow temperatures. How-
F FIG.3. SpindiffusivityofatrappedFermigas. Spindif-
ever, for systems with equal spin populations, it was
fusivityonresonance(solidcircles)asafunctionofthedimen-
shownthatpairingcorrelationsenhancetheeffectivecol- sionless temperature T/T . At high temperatures, the spin
F
lisionrateforcollectiveexcitations, leadingtoacollision diffusivityobeystheuniversalT3/2 behaviour(solidline). At
rate that increases dramatically as the temperature is low temperatures the diffusivity approaches a constant value
lowered [20]. The observed reduction of Γsd at low tem- of 6.3(3)(cid:126)/m for temperatures below about 0.5TF, establish-
peratures contrasts with this prediction, and is qualita- ingthequantumlimitofspindiffusionforstronglyinteracting
Fermi gases.
tively consistent with the onset of Pauli blocking.
Comparing the relaxation rate to the gradient in spin
density allows us to also measure the spin diffusivity D .
s When comparing these results to theoretical calcula-
At the center of the trap, where the trapping forces van-
tions, it is important to account for the inhomogeneous
ish, the spin current density J is given by the spin dif-
s densitydistributionsandvelocityprofilesthatresultfrom
fusion equation [21]
the trapping potential. For a homogeneous system on
∂(n −n ) resonance, and at high temperatures compared to the
Js =−Ds ↑∂z ↓ , Fermi temperature, we predict Ds = 1.11m(cid:126)(T/TF)3/2
4
and Γsd =0.90E(cid:126)F(T/TF)−1/2 (see Supplementary Infor- a
mation). Themeasuredspindragcoefficientissmallerby 3.0
afactorof0.90/0.16(1)=5.6(4)whilethespindiffusivity E)F 2.5
2
is larger by about the same factor, 5.8(2)/1.11=5.3(2). n/
3 2.0
This factor reflects the inhomogeneity of the system and of
agrees with an estimate from the Boltzmann transport nits 1.5
u
equation (see Supplementary Information). (κ 1.0
Finally,themeasuredtransportcoefficientsgiveforthe , χs 0.5
first time access to the temperature-dependence of the
spin susceptibility χs(T) in strongly interacting Fermi 2 3 4 5 6 7891 2 3 4 5 6 78910
gases. Defined as χ = ∂(n↑−n↓), the spin susceptibility T/T
describes the spin rsespo∂n(sµe↑−toµ↓a)n infinitesimal effective b F
1.4
magnetic field or chemical potential difference µ −µ
↑ ↓
applied to the gas, and is a crucial quantity that can 1.2
discriminate between different states of matter [8]. In 1.0
κ
a magnetic field gradient, particles with opposite spin /χs 0.8
are forced apart at a rate determined by the spin con- 0.6
ductivity σ , while diffusion acts to recombine them. 0.4
s
The balance between the processes of diffusion and con- 0.2
ductionthereforedeterminestheresultingmagnetization
2 3 4 5 6 789 2 3 4 5 6 789
gradient, a connection expressed in the Einstein relation 1 10
T/T
F
χ =σ /D [9]. In calculating this ratio from observable
s s s
quantities, the relaxation time τ cancels, as both σ and
s FIG. 4. Spin susceptibility on resonance determined
D are proportional to 1, yielding
s τ fromtheEinsteinrelation. (a)Compressibility(solidblue
circles) and spin susceptibility (open red circles) normalized
1 ∂(n −n )
χ = ↑ ↓ , bythecompressibility 3n ofanidealFermigasatzerotem-
s mdω2 ∂z 2EF
z perature. FortemperaturesbelowtheFermitemperature,the
susceptibility becomes suppressed relative to the compress-
where ∂(n↑−n↓) is evaluated near the trap center. The
∂z ibility, due to interactions between opposite-spin atoms. The
inhomogeneous trapping potential has practically no ef-
spin susceptibility coincidentally matches the compressibility
fect on the measurement of χs, as all quantities involved of a non-interacting Fermi gas (dashed line) in the range of
refer to the vicinity of the center of mass. temperaturesthatwecouldaccess. (b)Redcircles: spinsus-
Figure 4 reports our findings for the spin susceptibil- ceptibility divided by the compressibility obtained from the
ity at unitarity, as a function of the dimensionless tem- same clouds. At temperatures above the Fermi temperature,
the ratio of spin susceptibility to compressibility approaches
perature T/T . At high temperatures, we observe the
F unity (dashed line).
Curie law χ = n/T. In this classical regime of uncor-
s
related spins, the susceptibility equals the (normalized)
compressibility of the gas κ = ∂n/∂µ that we also di-
rectly obtain from our profiles.
and therefore our spin susceptibility data agree down to
At degenerate temperatures, the measured suscepti-
this point with the expected behavior for a Fermi liquid.
bility becomes smaller than the compressibility. This is
In conclusion, we have studied spin transport in
expected for a Fermi liquid, where χ = 3n 1 and
s 2EF 1+F0a strongly interacting Fermi gases. The spin diffusivity
κ = 23EnF 1+1F0s [8] with Landau parameters F0s and F0a was found to attain a limiting value of about 6.3(cid:126)/m,
describing the density (s) and spin (a) response. The establishing the quantum limit of diffusion for strongly
spinsusceptibilityisexpectedtostronglydecreaseatsuf- interacting Fermi gases. Away from resonance the
ficiently low temperatures in the superfluid regime, as diffusivity increases. This casts doubt on the possibility
pairs will form that will not break in the presence of an of stabilizing a ferromagnetic gas on the repulsive side
infinitesimal magnetic field. For the temperature range of the Feshbach resonance [7], which would require a
of our experiment this behaviour is not yet observable. vanishing diffusivity [9]. The observed slow relaxation of
It is currently under debate whether the strongly inter- spin excitations is a likely explanation for the surprising
acting Fermi gas above the superfluid transition temper- – possibly non-equilibrium [21] – profiles in imbalanced
ature is a Fermi liquid [24] or a state with an excitation Fermi gases observed at Rice [27], which did not agree
gap (pseudo-gap) [25, 26]. The opening of a gap in the with equilibrium measurements at MIT [28, 29] and at
excitation spectrum would be revealed as a downturn of the ENS [23]. Our measurements of the temperature
thespinsusceptibilitybelowacertaintemperature. Such dependence of the spin diffusivity at low temperatures
a downturn is not observed in χ down to T/T ≈ 0.2, disagree with the expected behavior of a Fermi liquid,
s F
5
while the spin susceptibility that we measure is consis-
tent with a Fermi liquid picture. An interesting subject
of further study is whether spins are still able to diffuse
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7
Universal Spin Transport in a Strongly Interacting Fermi Gas
Supplementary Information
DEPENDENCE ON INTERACTION STRENGTH
We study the dependence of the spin transport properties of the system on interaction strength by ramping to a
variable final field in the vicinity of the Feshbach resonance and measuring the subsequent evolution of the system.
Figure 5 shows the results of colliding two clouds at different fields, revealing the transition from transmission of the
cloudsthrougheachothertoreflectionofthecloudsasthemeanfreepathbecomessmallerthanthecloudsize. When
the scattering length is set to zero (Fig. 5.a), the center of mass separation oscillates at the trap frequency ω . On
z
resonance(Fig. 5.g),theobservedoscillationfrequencyof1.63(2)ω isintermediatebetweenthefrequency1.55ω of
z z
the axial breathing mode of a unitary Fermi gas in the hydrodynamic limit [5] and the non-interacting value of 2 ω ,
z
asthesystemcontainsahydrodynamicregionatthecenter,andisnon-interactinginthespin-polarizedwings. Figure
6 shows the spin transport coefficients versus interaction strength. The spin drag coefficient exhibits a maximum on
resonance (6.a), while the spin diffusion coefficient is minimum on resonance (Fig. 6.b).
a e
b m) f m)
µ µ
n ( n (
o o
ati ati
c ar g ar
p p
e e
S S
d
h
FIG. 5. Collision between spin up and spin down clouds with varying interaction strength. After separating the
spincomponents,themagneticfieldwasrampedtoavariablevalueneartheFeshbachresonancetoreachdifferentinteraction
strengths. Theinteractionparameterk a,withk =(6π2n)1/3 andnthecentraldensityperspincomponent,wasdetermined
F F
byaveragingthevaluesofk obtainedfromimagestakenafter200msofevolutiontime(notshown). Thevaluesofk awere
F F
(a) 0, (b) 0.08, (c) 0.13, (d) 0.19, (e) 0.26, (f) 1.2, (g) ∞, and (h) -1.5
a 0.08 b 3
0.07
0.06 2
EF0.05 ℏ
/Γsd0.04 mD/s
ℏ 10
0.03 9
8
0.02 7
6
0.01
5
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8
(kFa)-1 (kFa)-1
FIG.6. Spindragcoefficient(a)andspindiffusivity(b)acrosstheFeshbachresonance. Thespin-separatedcloudswerecooled
at 834G to T/T ≈0.16 (blue triangles) and to T/T ≈0.40 (red cirlces) before ramping to the final field. The largest spin
F F
drag and smallest spin diffusivity occur at the Feshbach resonance, where 1/k a=0. The solid line in (a) is a Lorenztian fit.
F
To reduce clutter, the 0.4 T data are not shown in (a).
F
8
IMAGE ANALYSIS
The experiment takes place in a trapping potential of the form
1
V(r,z)= mω2z2+V (r),
2 z r
(cid:112)
where r = x2+y2. Here z is the symmetry axis of the trap, and we image along the y axis.
From each run of the experiment we obtain two-dimensional column densities n2d(x,z) of both spin states σ =↑,↓.
σ
Fittingatwo-dimensionalGaussiantoeachcolumndensityprovidesameasurementofthecenterofmassofeachspin
state. We subtract the z components of the centers of mass of the two spin states to obtain the separation parameter
d=(cid:104)z (cid:105)−(cid:104)z (cid:105). Thethree-dimensionaldensitiesn (r,z)areobtainedusinganumericalimplementationoftheinverse
↑ ↓ σ
Abel transformation (similar to [31]):
n (r ,z)=j(cid:88)max n2σd(xj+1,z)−n2σd(xj,z) ln(cid:34)j+1+(cid:112)(j+1)2−i2(cid:35),
σ i xj+1−xj j+(cid:112)j2−i2
j=i
For each line of constant z, the values of n (r ,z) with small i are sensitive to noise in n2d(x ,z) for small j. To
σ i σ j
reduce noise, we fit a one-dimensional Gaussian to n2d(x ,z), for each value of z and for j <4 (corresponding to the
σ j
innermost 5 µm of the clouds or less than a tenth of the Fermi-radius), and use this fit (with sub-resolution sampling
to reduce discretization error) in the above formula.
The temperature is determined by analyzing the densities n (r,z) for |z| < 60 µm (as the 1/e radii of the clouds
σ
range from 100 to 400 µm, this restriction excludes the more polarized, large z, regions of the clouds). We obtain the
temperature of each cloud by fitting the density versus potential energy V to
n=λ−3f(βµ−βV),
(cid:113)
whereβ = 1 , λ= 2π(cid:126)2 , T isthetemperature, µisthechemicalpotential,andf isauniversalfunctiondefining
kBT mkBT
the equation of state of the unitary Fermi gas [30], with µ and T as the fit parameters.
MEASUREMENT OF TRANSPORT COEFFICIENTS
Near equilibrium, d decays exponentially, and we fit the measured values to
d(t)=d e−t/τ.
0
The spin drag coefficient is then
Γ =ω2τ.
sd z
To non-dimensionalize this quantity we multiply by (cid:126) and divide by the Fermi energy E = (cid:126)2 (3π2n((cid:126)0))2/3, where
F 2m
n((cid:126)0)=n ((cid:126)0)+n ((cid:126)0) is the total three-dimensional density at (r,z)=(0,0).
↑ ↓
After fitting to d(t) to obtain τ, we obtain the spin diffusivity from each spin up-spin down image pair as
n((cid:126)0)d
D = ,
s 2gτ
where g = ∂(n↑−n↓)| is the spin density gradient. The spin density gradient is obtained from the slope b of a
∂z z=0,r=0
linear fit to the polarization p(z)= n↑−n↓ as a function of z at r =0 using the formula
n↑+n↓
g =b·(n ((cid:126)0)+n ((cid:126)0)).
↑ ↓
The Einstein relation provides the spin susceptibility as
n((cid:126)0)
χ =σ /D =
s s s 2mD Γ
s sd
g
= .
mdω2
z
Although derived using transport coefficients, the final expression for χ does not actually depend on the relaxation
s
time τ, so it gives an independent value of χ from each run of the experiment.
s
9
THEORETICAL CALCULATION OF TRANSPORT COEFFICIENTS
ThetransportcoefficientsintheclassicallimitT (cid:29)T maybecalculatedusingtheBoltzmanntransportequation.
F
The Boltzmann transport equation describes the evolution of the semi-classical distribution functions f ((cid:126)r,(cid:126)v,t), with
σ
σ =↑,↓:
∂f F(cid:126)
σ +(cid:126)v·∇ f + ·∇ f =I [f , ]
∂t r σ m v σ coll ↑ ↓
Define the joint distribution function f((cid:126)r ,(cid:126)r ,(cid:126)v ,(cid:126)v )=f ((cid:126)r ,(cid:126)v )f ((cid:126)r ,(cid:126)v ). Any quantity of the form χ((cid:126)r ,(cid:126)r ,(cid:126)v ,(cid:126)v )
↑ ↓ ↑ ↓ ↑ ↑ ↑ ↓ ↓ ↓ ↑ ↓ ↑ ↓
can be averaged over f [32]:
1 (cid:90)
(cid:104)χ(cid:105)= d3r d3r d3v d3v (χf),
N N ↑ ↓ ↑ ↓
↑ ↓
where N is the number of atoms with spin σ.
σ
We generalize to arbitrary scattering length a, and non-uniform drift velocity, the calculation by Vichi and
Stringari [18] of the equation of motion for the relative coordinates of the two spin components in a harmonic trap of
angular frequency ω along the z axis. From the Boltzmann equation,
z
∂ (cid:104)z −z (cid:105)=(cid:104)v −v (cid:105)
t ↑ ↓ z↑ z↓
and
∂ (cid:104)v −v (cid:105)=−(cid:104)v −v (cid:105)Γ −ω2(cid:104)z −z (cid:105),
t z↑ z↓ z↑ z↓ sd z ↑ ↓
where the spin drag coefficient is
(cid:104)(v −v )I [f](cid:105)
Γ = z↑ z↓ coll .
sd (cid:104)v −v (cid:105)
z↑ z↓
The full collision integral reads
1N +N (cid:90) (cid:90) d3p (cid:90) d3p (cid:90) dσ
(cid:104)(v −v )I [f](cid:105)=− ↑ ↓ d3r ↑ ↓ dΩ |(cid:126)v −(cid:126)v |(v −v )×
z↑ z↓ coll 2 N N (2π(cid:126))3 (2π(cid:126))3 dΩ ↑ ↓ z↑ z↓
↑ ↓
(cid:2)(1−f )(1−f )f(cid:48)f(cid:48) −f f (1−f(cid:48))(1−f(cid:48))(cid:3).
↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓
The integral describes the scattering of particles ↑ and ↓ into new states ↑(cid:48) and ↓(cid:48). We assume that the distribution
functionsnearequilibriumaref ((cid:126)r,(cid:126)v)=f0((cid:126)r,(cid:126)v−(cid:126)v ),wheref0 istheequilibriumdistributionofbothspinstates
σ σ,drift
(assumed to be in the classical limit) and (cid:126)v =±v ((cid:126)r)zˆis the drift velocity of spin σ. The collision integral can
σ,drift d
then be calculated (for details, see for example [33]) and yields
16 (cid:18)k T(cid:19)1/2
Γ = σ(T/T ) B n ((cid:126)0)/α,
sd 3 a πm 0
with
(cid:90) 1
σ(T/T )=4πa2 du u5e−u2,
a 1+ T u2
Ta
(cid:82) n v d3r
α=n0((cid:126)0)(cid:82) n02vd d3r,
0 d
n =(cid:82) f0d3v is the equilibrium density of each spin state, and T = (cid:126)2 is the temperature scale associated with
0 a kBma2
the scattering length a. The limiting values of the temperature-dependent average scattering cross section σ(T/T )
a
are
σ(T (cid:28)T )=4πa2
a
10
and
2π(cid:126)2
σ(T (cid:29)T )= =λ2,
a mk T
B
with the thermal de Broglie-wavelength λ. For unitary interactions, T = 0, and the latter limit applies. Define the
a
Fermi energy as E = (cid:126)2 (6π2n ((cid:126)0))2/3. In the unitary limit where σ =λ2 we find
F 2m 0
√
(cid:126)Γ 32 2 (cid:114) E 0.90(cid:114) E
sd = F = F .
E 9π3/2α k T α k T
F B B
For a uniform system α=1, while for a harmonically trapped system with a uniform drift velocity we find α=23/2.
However, the drift velocity profile cannot be uniform: Even if it started out uniform, spin currents would get damped
faster in the center of the overlap region of the two clouds, where the collision rate is high, than in the wings,
where it is low. A non-uniform drift velocity profile will develop. The nature of the linearized Boltzmann equation
allows for a variational principle where trial functions replacing the true distribution function f ((cid:126)r,(cid:126)v) yield upper
σ
bounds on the actual spin drag coefficient [33]. Minimizing Γ for a trial class of non-uniform drift velocities
sd
v (r)=v (1−n (r)γ/n (0)γ) yields γ →0. In this case α=25/2 ≈5.7.
d d0 0 0
The spin diffusivity measured in our experiment for T (cid:29)T is then
F
1 n ((cid:126)0)
D = 0
s χ mΓ
s sd
9π3/2α (cid:126) (cid:18) T (cid:19)3/2
= √
32 2 m TF
(cid:126) (cid:18) T (cid:19)3/2
=1.1α .
m T
F
The bulk value for α = 1 represents a lower bound on the diffusivity. It is known from similar calculations that
in the high-temperature limit, this lower bound should be within a few percent of the actual bulk value [33]. The
effect of inhomogeneities in a harmonic, cylindrically symmetric trap increases the diffusivity by α ≈ 5.7. The
high-temperature result for the diffusivity has been obtained independently by Georg Bruun [34].