Table Of ContentNonlinear spin diffusion and spin rotation in a trapped Fermi gas
Tilman Enss
Institut fu¨r Theoretische Physik, Universita¨t Heidelberg, 69120 Heidelberg, Germany
(Dated: February 27, 2015)
Transverse spin diffusion in a polarized, interacting Fermi gas leads to the Leggett-Rice effect,
where the spin current precesses around the local magnetization. With a spin-echo sequence both
the transverse diffusivity and the spin-rotation parameter γ are obtained; the sign of γ reveals
5
1 the repulsive or attractive character of the effective interaction. In a trapped Fermi gas the spin
0 diffusion equationsbecomenonlinear, and theirnumericalsolution exhibitsan inhomogeneous spin
2 stateevenatthespinechotime. Whilethemicroscopicdiffusivityandγ increaseatweakcoupling,
their apparent values inferred from the trap-averaged magnetization saturate in agreement with a
b
recent experiment for a diluteultracold Fermi gas.
e
F
PACSnumbers: 67.85.Lm,75.76.+j,51.10.+y
6
2
I. INTRODUCTION fordilute,homogeneousFermigaseshavebeencomputed
] using kinetic theory [9, 10].
s
a Transverse spin diffusion occurs when the magnetiza- In an infinite homogeneoussystem where the only po-
g tionisorientedalongdifferentdirections,forinstance,in sitiondependencearisesfromthemagneticfieldgradient
-
t a spin helix. It has been observed in spin-echo experi- α, the phase angle φ is directly proportional to the mi-
n ments in helium [1], polarized hydrogen, and, recently, croscopicparameterµ. Instead,forafinitehomogeneous
a
ultracold atomic gases [2–4]. The transverse magnetiza- box the phase φ, andhence the apparentvalue of µ, sat-
u
q tion evolves according to a diffusion equation, but there urateswhenthesystemsizeisreached[11]. Experiments
. are additional terms from the precession of the spin cur- with ultracoldatomic gasestypically employ a harmonic
t
a rents around the local magnetization. This Leggett-Rice trapping potential which is both finite and inhomoge-
m effect [5] is related to the identical spin-rotation effect neous: in this case, the diffusivity D0⊥, the Leggett-Rice
- [6] and leads to reactive spin currents and spin waves, parameterµ, andthe magnetizationmz arestrongly po-
d which have been observed in ultracold Fermi gases [7]. sitiondependent,andthediffusionequation(1)becomes
n
Consider a polarized sample with small transverse mag- nonlinear. In previous studies the evolution has been
o
netization (small tipping angle). The transverse magne- linearized in order to determine the collective mode fre-
c
[ tizationisconvenientlycombinedintoacomplexnumber quencies and decay rates in the trapping potential [12].
2 m+ =mx+imy, which evolveswith a complex diffusion Otherstudies considerthe nonlinearevolutionofthe full
coefficient, phase-spacedistributionforanondegenerateorcollision-
v
less trapped gas [13]. Here, I numerically solve the non-
7
7 ∂m+ = D0⊥ 2m iαx m , (1) linearevolutionequationwithpositiondependentkinetic
6 ∂t 1 iµm ∇ +− 3 + coefficients including medium scattering to obtain the
z
2 − transverse magnetization decay and the growth of the
0 phase φfor the specific trapgeometryused inthe exper-
where D is the transverse diffusivity and µm = γ (at
1. full polar0⊥ization)the dimensionless spin-rotatioznparam- iment.
0 eter. Forsmallµ 0thisistheusualdiffusionequation, Kinetic theory [14] is employed to compute the spin
5 whileforlarge µ →thediffusionequationhasanimaginary evolution of the trapped gas. This method is well con-
1 effective diffusi|vi|ty and resembles the Schr¨odinger equa- trolled in the weak-coupling limit, which is also the pa-
:
v tion [8]. In a spin-echo pulse sequence, the second term rameter regime where finite-size corrections due to the
Xi in Eq. (1) expresses a linear gradient α of the external trappingpotentialaremostpronounced[4]. Ontheother
magnetic field along the x direction, which winds the hand, at strong coupling toward unitarity and at low
r 3
a magnetization into a helix. After a time t , a π pulse is temperature near the superfluid phase transition of an
π
applied, which is equivalent to flipping the sign of α. In attractiveFermigas[15],kinetictheoryisexpectedtore-
the ensuing time evolution, the helix unwinds until the ceive quantitative corrections from the effects of pairing
magnetization is realigned at the echo time t =2t . In and shortquasiparticle lifetimes, which are incorporated
e π
the presence of spin rotation µ = 0, realignment at t for instance when computing transport from the Kubo
e
occurs at a phase angle φ µ w6 ith respect to the ini- formula within a Luttinger-Ward approach[16, 17].
∝
tial orientation at t = 0 [5], and the value of µ can be This paper is structured as follows: in Sec. II the spin
inferred. For a strongly interacting Fermi gas the value evolution equations for a trapped Fermi gas are derived
of µ has recently been measured and used to determine from kinetic theory, while Sec. III compares them to the
thespin-antisymmetricFermi-liquidparameterFa,while known homogeneous limit. Section IV presents the re-
1
the signof µ revealsthe attractiveorrepulsivecharacter sults for the magnetization profiles and apparent diffu-
of the effective interaction [4]. Theoretically, D and µ sivities, and Sec. V concludes with a discussion.
0⊥
2
II. KINETIC THEORY for a Fermi gas in a harmonic trapping potential V(x),
whichcanbeanisotropicwithdifferenttrappingfrequen-
TransportinaninteractingFermigasmaybedescribed cies ωj in spatial direction j. The magnetic field
by kinetic theory for quasiparticles, as long as they are
~
sufficiently long-lived. For a multi-component Fermi gas h (x,t)= Ω(x,t), Ω=Ω +Ω , (8)
p 0 mf
−2
with two or more spin species one has to compute the
W
time evolution of the spin distribution np, which is a Ω0(x,t)=α(t)x3zˆ, Ωmf = ~ m(x,t) (9)
matrix with components np = npσσ′(x,t) in spin space.
The evolutionequationderivedby Landauand Silin [18, is written in terms of the Larmor frequency Ω = Ω +
0
19] reads Ω . For a spin-echo protocol it has two contributions:
mf
(i) Ω = γB(x,t) is due to the external magnetic field
0
∂np 1 1 B(x,t) with gyromagnetic ratio γ. A spatially constant
+ [ ε , n ] [ ε , n ]
∂t 2 ∇p p ∇r p +− 2 ∇r p ∇p p + B is compensated by going to the co-rotating frame in
i ∂np Bloch space, but a magnetic field Bz gradient of slope α
+ [ε ,n ] = (2)
~ p p − ∂t along the x3 direction winds up the local magnetization
(cid:18) (cid:19)coll into a spin spiral. (ii) The second contribution to the
where εp = εpσσ′(x,t) is the matrix of single-particle Larmorfrequency,Ωmf,isamean-fieldtermproportional
energies. The left-hand side constitutes the drift term, to the local magnetization m of a polarized Fermi gas.
whiletheright-handsidedescribesthe changeinthe dis- It leads to the precession of the spin current around m.
tribution caused by collisions. Specifically for the spin- Theevolutionofthefulldistributionfunctionsfp(x,t)
1/2 case, n and ε are 2 2 matrices in spin space, for and σp(x,t) is simplified by considering moments with
instance, inpthe ,p basis.× respect to momentum:
↑ ↓
The spin matrices can be decomposed in terms of the
d3p
identity I and the Pauli matrices σ. The occupation n(x,t)= f (x,t) (10)
number matrix is written as (2π~)3 p
Z
d3p ∂ε
1 Jn(x,t)= pf (x,t) (11)
np = 2(fpI+σp·σ) (3) j Z (2π~)3 ∂pj p
d3p
where fp(x,t) is the particle number distribution func- m(x,t)= (2π~)3 σp(x,t) (12)
tion and σ (x,t) the spin vector distribution in Bloch Z
p d3p ∂ε
space. Similarly, the energy matrix J (x,t)= pσ (x,t) (13)
j (2π~)3 ∂p p
Z j
ε =ε I +h σ (4)
p p p· withbarevelocityv =∂ε /∂p =p /m. Anadditional
pj p j j
combines the spin-independent single-particle energies contribution fp∂hp/∂pj to the spin current is absent for
εp(x,t) and a magnetic field hp(x,t). One may then momentum-independenthp. Thelocalpolarizationisde-
rewrite Eq. (2) as fined as M(x,t)=m(x,t)/n(x,t) where M(x,t) 1.
| |≤
ThespincurrentJ isbothavectorinBlochspace(bold
j
symbol) and a vector in position space (j index): it en-
∂f ∂ε ∂f ∂ε ∂f
p p p p p
+ codes how the magnetization changes as one goes along
∂t ∂p ∂x − ∂x ∂p
j (cid:20) j j j j thej direction. Theevolutionequationsforthemoments
X
∂h ∂σ ∂h ∂σ ∂f read,usingthespecificformofthesingle-particleenergies
p p p p p
+ = (5) (7)–(9),
∂p · ∂x − ∂x · ∂p ∂t
j j j j (cid:21) (cid:18) (cid:19)coll
∂ n+ Jn =0 (14)
and t ∇j j
Xj Jn
∂σp + ∂εp∂σp ∂εp ∂σp ∂tJjn+αn∇jn+ωj2xjn=−τnj (15)
∂t ∂p ∂x − ∂x ∂p
Xj (cid:20) j j j j ∂tm+ jJj +Ω0 m=0 (16)
∇ ×
∂h ∂f ∂h ∂f 2 ∂σ
+ p p p p h σ = p . (6) Xj
∂pj ∂xj − ∂xj ∂pj(cid:21)− ~ p× p (cid:18) ∂t (cid:19)coll ∂tJj +αkPk∇jm+α⊥P⊥∇jm+ωj2xjm
W ∂J
The spin-rotation term hp σp is responsible for the + Ω + m J = j . (17)
Leggett-Rice effect. The sing×le-particle energies are 0 ~ × j ∂t
(cid:16) (cid:17) (cid:18) (cid:19)coll
p2 m The projectors P a (a mˆ)mˆ and P 1 P give
εp(x,t)= 2m +V(x), V(x)= 2 ωj2x2j (7) thecomponentofkthe≡magn·etizationgrad⊥ien≡tpa−rallkeland
j perpendicular to the local magnetization, respectively.
X
3
When deriving these equations, the mean field Ωmf has 101 γ
been retained only in the spin-rotation term and not in
0
the Poisson brackets in Eqs. (5) and (6). This can be 10
βmα
justified as the leading order in a controlled large-N ex- 10-1 τF/τD II
pansion [20]. In general, the evolution of the currents
depends on the second moments of fp and σp which 10-2 τ λ-3|W|
F
in turn depend on higher moments. However, near lo- 3
λ n
calequilibriumthelinearizedBoltzmannequationrelates 10-3
the highermoments tothe lowerones viathe coefficients 1/kFa=-2 (BCS)
-4
α , τ , α , τ , and W, and one obtains a closedset of 10
n n , D
evolutionke⊥quations. These coefficients are discussed in 0 0.5 1 1.5
Sec. IIB.
1
10
0 γ
10
A. Initial conditions βmα
II
τ /τ
10-1 F D
3
In the absence of the external magnetic field gradient -3 λ n
-2 τ λ |W|
α, the local equilibrium Fermi distribution is 10 F
1 10-3
n (x)= (18) 1/k a=0 (UFG)
p± exp(β(εp(x) µ ))+1 -4 F
− ± 10
0 0.5 1 1.5
in terms of the chemical potential µ of the majority
(minority) component. The resulting±density profile of x3 / RTF
the fully polarized gas is
FIG. 1. (Color online) Kinetic coefficients and density pro-
n(x)= λ−3Li3/2( z+e−βV(x)) (19) filesinthetrapontheBCSside(upperpanel)andatunitar-
− −
ity (lower panel) at degenerate temperature (T/TF)i = 0.2:
withthermalwavelengthλ=(2π~2β/m)1/2 andpolylog- total phase-space density λ3n(x) (solid black line), diffu-
arithmLi (z). Thefugacityz =exp(βµ )ofthemajor- sion parameter βmα [dotted (brown) line], diffusive scat-
s + + k
itycomponentisdeterminedbythetotalparticlenumber tering rate τF/τD [solid (red) line], interaction parameter
N in the trap of average frequency ω¯ =(ω ω ω )1/3, τFλ−3|W|[dash-dotted(blue)line],andLeggett-Riceparam-
1 2 3 eter γ =−τDWn/~ [dashed (magenta) line].
1
N = Li ( z ). (20)
−(β~ω¯)3 3 − +
distribution remains independent of x and x , but cur-
1 2
At high temperatures in the Boltzmann regime the den- rents are generated which can change the distribution
sity profile is Gaussian, n(x) exp( βV(x)), while at along the x3 direction. Hence, the spin evolution effec-
∝ −
low temperatures it approaches the Thomas-Fermi pro- tively reduces to a one-dimensional problem for m and
file, J3 along the gradient direction.
x2 3/2
n(x)=n 1 j (21)
0 − R2 B. Kinetic coefficients
(cid:16) Xj TFj(cid:17)
wthiethFTerhmoimeanse-rFgeyrmEi ra=diu(s6NR)T1F/3j~=ω¯ o(2fEaFf/umllyωj2p)o1l/a2riaznedd strTenhgethcooefffithceiesnptisnαcku,r⊥r(exnt)gienneErqa.te(d17b)ypaamraamgneetrtiizzeattiohne
F
gradient. Their valuesare determinedby the Boltzmann
gas. The density profile is shownin Fig.1 both for weak
equationlinearizedaroundthe localequilibriumsolution
coupling on the BCS side (upper panel) and for strong
[9, 10]. There are different contributions from the longi-
couplingatunitarity(lowerpanel). Notethatthephase-
tudinalmagnetizationgradientduetothetrappotential,
space density λ3n (solid black line) is unaffected by s-
andthetransversemagnetizationgradientfromthehelix.
wave scattering in the fully polarized gas.
For a fully polarized gas,
In the spin-echo protocol [4], the gas is initially fully
polarized, m =n or M =1,ata tipping angleθ from n(x)
| | | | α (x)=α (x)= (23)
the z axis in Bloch space, k n mχ(x)
m(x,t=0)=(sinθ,0,cosθ)n(x). (22) P(x)
α (x)= (24)
⊥ mn(x)
This distributionis stationaryfor α=0 and the particle
and spin currents Jjn and Jj vanish. When the exter- whereχ(x)=−λ−3βLi1/2 −z+e−βV(x) isthelocalsus-
nal gradient α is switched on along the x direction, the ceptibility,andP(x)= λ 3β 1Li z e βV(x) the
3 − (cid:0)− − 5/2 −(cid:1) + −
(cid:0) (cid:1)
4
local pressure of a free Fermi gas. In Fig. 1, α is plot- the unitary Fermi gas (UFG, 1/k a = 0) to the repul-
F
ted as the dotted line: for a high temperaturke or low sive fermion branch (1/k a&1) above the BEC ground
F
density in the outer regions of the trap it reaches the state[15]. Atstrongcouplingthemany-bodyT matrixis
Boltzmannlimit α =α =1/βm, while for a high den- needed,whichincludesmediumscattering. Inthe ladder
sity atthe trapcenkter it⊥is enhanced andapproachesthe approximation,whichis the leadingorderof the large-N
low-temperature limit α =v2/3 [5]. In general, α and expansion in the number of fermion flavors [20, 22], the
F
α acquireinteractioncokrrections[9],butforlargeiknitial full T matrix reads
po⊥larization and s-wave scattering these are negligible.
theSedceocnady,otfhtehceosepffiinciceunrtrseτnkt(dxu)eatnodscτ⊥at(txer)inpgar[a9m],etrize T−1(p1,p2d)3=p T0−1(p1n,p2)+n
+ (2π~)3ε +εp,+ ε p+εp1+p2,− +i0. (29)
∂J d3p ∂ε ∂σ Z p1 p2 − p− p+p1+p2
j p p
=
∂t (2π~)3 ∂p ∂t The T matrix is computed numerically, and Fig. 1
(cid:18) (cid:19)coll Z j (cid:18) (cid:19)coll shows the resulting spin diffusion rate τ /τ in units
= PkJj P⊥Jj. (25) of the Fermi frequency 1/τF = EF/~ as tFheDsolid (red)
− τ − τ line: it is highest in the trap center and decreases pro-
k ⊥
portional to the density in the outer regions. At unitar-
The component of the current P J parallel to the lo-
j ity, the scattering rate is about 10 times larger than at
cal magnetization m decays withkthe longitudinal diffu-
weak coupling 1/k a = 2. In the Boltzmann regime,
F
s(B1ioo−nthPtdikmi)ffJeejrτfdko,ercwaayhpsiolewlaitrthihzeetdht,reastntrrsoavnnegsrvslyeerdcseeogmdenipffeournaseitonentFtPeirm⊥mJeijgτ⊥a=s. ~w/hτeDre=Ei(4x√)32πinsβλt3h(cid:2)e1e−xpβoεnBe−n−tia(lβiεnBte)g2reaxlpa(nβdεBεB)E=i(−~2β/εmBa)2(cid:3)
[10]. At unitarity the scattering cross section decreases
[9, 10, 21]: all the states between the majority and the
minority Fermi surfaces of a polarized gas are available with temperature, and ~/τD =4√2nλ3/(3πβ) [23].
for transversescattering and τ can be much lowerthan Third, the mean field Ωmf = Wm(x,t)/~ describes
τ . For the experiment at ha⊥nd [4], however, the low- the precession of the spin current around the local mag-
ekst temperatures reached at the trap center are around netization and is given by a momentum average of the
T/TF & 0.3, and previous studies have shown that in real part of the many-body T matrix T(p1,p2) over the
this temperature range τ and τ are nearly equal [10]. momentumstatesbetweenthemajorityandtheminority
It is therefore justified tokwork w⊥ith a single decay time Fermi surfaces, weighted by the velocity squared [10],
τ =τ τ [10],
D
⊥ ≈ k 1 d3p d3p
1 2
W = v (v v )(n n )
~ sinh(βh) 1 C m (2π~)3(2π~)3 1j 1j− 2j 1+− 1−
τ = C (2π~)8 d3p1...d3p4 ⊥| |Z (n n )Re (p ,p ). (30)
⊥ ⊥ Z × 2+− 2− T 1 2
×δ(3)(p1+p2−p3−p4)δ(εp1 +εp2 −εp3 −εp4) At weak coupling W = 0(0,0) = 4π~2a/m agrees with
(p ,p )2[e βhn n +eβhn n ] thebareinteraction,whiTchisreal. Atunitarity1/a 0,
1 2 − 1+ 2+ 1 2
×|T | − − becomes purely imaginary and W appears to va→nish
(1 n )(1 n )v (v v ) (26) 0
3+ 4 1j 1j 2j T
× − − − − along with Re . This is indeed observed at high tem-
0
T
with magnetic field h =(µ µ )/2 and normalization peratures,butatlowtemperaturesthemany-body T ma-
+
constant − − trix acquiresarealpartduetomediumscattering,and
T
W =0[4]. Thisisshownasthedash-dotted(blue)curve
d3p in F6 ig. 1: for weak coupling, W 4π~2a/m is constant
C⊥ = (2π~)3 vp2j(np+−np−). (27) independentofthedensityandpo≈sitioninthetrap,while
Z
atunitarityitdecreaseswithdensityintheouterregions
The T matrix (p ,p ) describes s-wave scattering be- of the trap. Note that the very similar values for W at
1 2
T
tween particle (p ,+) and particle (p , ). In order to the trap center are coincidental: at weak coupling W is
1 2
−
derive explicit expressions, the T matrix for ultracold given essentially by the bare coupling, while at unitarity
fermionswiths-wavecontactinteractionshastobeused. it is purely a many-body effect.
The two-body T matrix Spin rotation is characterized by the dimensionless
Leggett-Rice parameter
4π~2 1
(p ,p )= (28) τ Wn
T0 1 2 m a 1+ik γ =µn= D (31)
− − ~
is given in terms of the relative wavenumber k = p
1 which is plotted as the dashed (magenta) line in Fig. 1.
p /2~ andthe s-wavescatteringlength a. At weak|cou−-
2 At weak coupling,
pli|ng a 0, (p ,p ) 4π~2a/m is the regularized
0 1 2
bare c|on|t→act inTteraction. →The BCS-BEC crossover goes 1 1
τ , W a, γ , (32)
fromtheweaklyattractiveBCSregime(1/k a. 1)via D ∝ a2n4/3 ∝ ∝−an1/3
F
−
5
henceγ becomeslargeandonlyweaklydependentonthe with effective diffusivity D = D /(1+µ2m2). Since
e⊥ff 0⊥
density. At unitarity, on the other hand, γ is a purely m2 is constant in space, this is now a linear diffusion
many-body effect, much smaller, and roughly propor- equation, albeit with a complex diffusion coefficient.
tional to the density. In the spin-echo protocol, the gradient α winds up
the transverse magnetization m = m + im into a
+ x y
helix along the x direction. A transverse spin current
3
III. ANALYTICAL SOLUTIONS IN LIMITING
J ∂ m appears, which tends to smooth the helix.
3+ 3 +
CASES ∼
If J has a component perpendicular to the local mag-
3+
netizationm m zˆ(atsmalltippingangle)itprecesses
z
The evolution equations (14)–(17) leave the density around it at f≈requency Ω = Wm /~. At time t , a π
mf z π
profilelargelyinvariant,butthespindistributionchanges pulse around the y axis in Bloch space is applied; this is
dramaticallyasthemagnetizationiswoundupintoahe- equivalent to reversing the sign of α. The subsequent
lix according to the equations time evolution unwinds the helix until the echo time
t = 2t , where the transverse magnetization is again
e π
∂ m+ J +αx zˆ m=0 (33)
t j j 3 homogeneous. Inthe absenceofspinrotation,γ =0,the
∇ ×
Xj modulus of the transverse magnetization A(t)= m+(t)
∂ J +α P m+α P m+ω2x m at time t decays as a cubic exponential [5] | |
t j k k∇j ⊥ ⊥∇j j j e
W J
+(αx zˆ+ m) J = j. (34) D α2t3
3 ~ × j −τ A(t )=A exp 0⊥ e (38)
D e 0
− 12
The full numerical solution of these equations is pre- (cid:16) (cid:17)
sented in Sec. IV. In order to gain a qualitative under- where A0 = A(0). This result is approximately correct
standing of the spin evolution in a trapped gas, it is even for γ = 0 if the tipping angle θ is small, m+
6 | | ≪
instructive to consider first the approximate analytical mz ,andforshorttimes. Forfiniteγ,themagnetization
| |
solutions in the homogeneous case. decay is slowed down and is given by [4]
If the scattering time τ is much shorter than any
D
otherrelevanttimescale,forinstance,thedephasingtime 1 D α2t3
A(t )=A ηexp η 0⊥ e , (39)
τ = (D⊥α2)−1/3 due to the helix, the current reaches a e 0sη W(cid:18) (cid:20) − 6(1+γ2Mz2)(cid:21)(cid:19)
steady state and its time derivative vanishes in the ro-
A(t )
taantdinµgmfra=me,τ∂DtΩJjm=f =−γαMx3zˆo×neJfijn.dDsefining D0⊥ =α⊥τD φ(te)=γMzln(cid:18) A0e (cid:19) , (40)
−
Jj +D0⊥∇jm+τDωj2xjm−µm×Jj =0. (35) where η = γ2(A0/n)2/(1+γ2Mz2) for polarization Mz,
and (z) is the Lambert-W function. This solution for
This equation is solved by the steady-state current W
thehomogeneoussystemisusedintheanalysisoftheex-
Jj =−1+Dµ0⊥2m2 ∇jm+µm×∇jm Lpeergigmetetn-tRailcdeaptaartaomfietttehreγdfirffoumsivthiteymDe0⊥asaunrdedthmeaagpnpeatrieznat-
n tiondecayandphaseshiftφofthetrappedsystem. Inthe
+µm(µm·∇jm) −τDωj2xjm (36) following the full spin evolution in the trap is computed
explicitly to determine to what extent the homogeneous
o
where D0⊥, µ, and τD may still depend on position; the solution is still applicable to the trapped gas.
last term arises due to the trapping potential. Inserting
this current into the continuity equation for the magne-
tization (33) yields IV. RESULTS
∂ m= αx zˆ m+ ω2 (τ x m)
t − 3 × j∇j D j In the trapped gas, the assumptions which led to the
D Xj analytical solutions in Sec. III (steady-state current, ho-
+∇j1+µ0⊥2m2 ∇jm+µm×∇jm+µm(µm·∇jm) . mogeneity) are not satisfied. Instead, a full numerical
n o solution of the spin evolution equations, (33) and (34),
TheLeggettsolution[5]isrecoveredinthehomogeneous is necessary. The initial condition in the experimental
limit ωj =0, where m2 remains constant in space: protocol [4] is a fully polarized cloud of fermionic atoms
withathermalprofile,(19)and(22),fortippingangleθ.
D
∂ m= αx zˆ m+ 0⊥ 2m+µm 2m . Thisdistributionisstationaryintheabsenceofanexter-
t − 3 × 1+µ2m2 ∇ ×∇
nalmagnetic fieldgradientα. Due tothe density profile,
In this case the longitudinal mag(cid:8)netization m remai(cid:9)ns (19), also the coefficients α (x), τ (x), and W(x) de-
z
unchanged, while the transverse magnetization m = pend on the position in the⊥trap. ⊥For a small tipping
+
mx+imy evolves as angle |m+| ≪ |mz| the gas remains almost fully polar-
ized, hence the density profile and the coefficients are
∂ m = iαx m +D (1+iµm ) 2m (37)
t + − 3 + e⊥ff z ∇ + time independent evenin the presence ofthe gradientα.
6
0.8 0.8 0
te=0.2ms te=0.2ms Lambert-W γ=1.55
|=A(t)+π 00..46 |=A(t)+e 00..46 A(t)/Ae0 1 trap calc. /φπ --00..63
|m 0.2 te=1.2ms |m 0.2 t =1.2ms 0.5 -0.9
e BCS
0 0 1/k a=-2 -1.2
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 F
0 -1.5
4 2 0 0.5 1 1.5 2 -1 -0.8-0.6-0.4-0.2 0
=(t)φπ 2 te=1.2ms =(t)φe te=0.2ms Lambert-W 0 γ=0.50
m+ 0 0.2ms m+ 0 0 trap calc. -0.1
arg --42 arg -2 te=1.2ms A(t)/Ae 1 /φπ -0.2
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 0.5 -0.3
UFG
x / R x / R
3 TF 3 TF 1/k a=0 -0.4
F
0 -0.5
FIG. 2. (Color online) Transverse magnetization profiles on
0 0.5 1 1.5 2 -1 -0.8-0.6-0.4-0.2 0
the BCS side 1/kFa = −2 at temperature (T/TF)i = 0.2
and times tπ just before the π pulse (left column), and at
the echo time te =2tπ (right column). Upper panels display Lambert-W 1.5 γ=-1.61
tahrgemam+p.liStouldide Acur=ve|smi+n|;ealochweprapnaenlealrse, tfohreaphtaraseppaendglgeaφs a=t )/Ae0 1 trap calc. π 1.2
te =0.2mstote =1.2ms,incomparisonwithahomogeneous A(t /φ 0.9
gas at te=1.2ms (dashed line). 0.5 0.6
REPULS
1/k a=+2 0.3
F
0 0
Inthenumericalsolution,theexperimentalparameters
0 0.5 1 1.5 2 -1 -0.8-0.6-0.4-0.2 0
[4] are used: gradient α=1.67µm 1kHz, trap frequency
− t [ms] M ln(A/A )/π
ω =2π 750Hz, ω¯ = 2π 470Hz, roughly N 40000 e z 0
3
atomsof×40KwithaFermi×energyofE =2π~ ∼29kHz,
F
× FIG. 3. (Color online) Amplitude and phase of the trans-
and Thomas-Fermi radius R = 5.1µm. Even though
the Fermi gas is initially fullyTFpolarized, the Fermi wave verse magnetization m+ at temperature (T/TF)i = 0.2 and
vectorkF =12µm−1 andthe Fermitime τF =0.0087ms sicmalalslotliuptpioinng, (a3n9g)le(sθol=id πli/n3e0),(wMitzh=D0⊥.9a9d5j)u.stLeedftt:oafintaltyhte-
0
are taken for a hypothetical balanced gas of the same
trap calculation (circles). Right: fit of γ from the slope of
atom number in order to recover the standard relation the phase shift φ in Eq. (40). Top, 1/kFa= −2 (BCS side);
n = kF3/3π2 for the total density [4]. At the lowest ex- center, 1/kFa=0 (unitarity); bottom, 1/kFa=2 (repulsive
perimental temperature, T 280nK, the initial reduced branch).
≈
temperature is (T/T ) = 0.2 and λ = 0.52µm. The
F i
phase-spacedensityatthetrapcenterreachesλ3n=7.3,
well within the quantum degenerate regime. value γ = Wm τ /~ according to Eq. (40). As we see
z D
−
In Fig. 2, the helical state of the local magnetization below, this effect leads to a saturation of the apparent γ
in the trap at 1/k a = 2 is shown for different times for weak coupling.
F
−
t (left column). At this time, the π pulse is applied Notethatthemagnetizationprofilesarenotsymmetric
π
(equivalently, the sign of α is reversed), and the subse- with respect to x : this is due to the spin-rotation term
3
quent time evolution unwinds the helix until the echo in Eq. (37) producing an additional phase shift on top
time t = 2t (right column), which is chosen between of the gradient α. Consequently, if one reverses the sign
e π
t = 0.2ms and t = 1.2ms. For comparison, the den- of the scattering length a, the sign of γ is approximately
e e
sity profiles for a homogeneous gas with t = 1.2ms reversed, and the resulting magnetization profile is the
e
are shown as dashed lines. At time t , the φ profiles mirror image with x x . Note also that at strong
π 3 7→ − 3
of the trapped and the homogeneous gas have a simi- coupling 1/k a = 0, the phase profile at t is nearly
F e
lar slope and are shifted toward negative phase angles homogeneous at the trap center (not shown), and the
by spin rotation. At time t , instead, the phase angle kinetic coefficients for the trapped gasapproachthose in
e
of the homogeneous gas is again homogeneous, but m the homogeneous case.
+
for the trapped gas remains in a helical state. There- Figure 3 shows the decay of the trap-averaged trans-
fore, the trap-averaged transverse magnetization decays verse magnetization with time (left column), and the
quickly eventhoughthe local m is still sizable,andthe growthofthephaseangleφwiththeslopeofthemagneti-
+
apparent slope φ(t )/M ln(A(t )/A ) is lower than in zationdecay(rightcolumn). Fromtheseplots,theappar-
e z e 0
the homogeneous case, where it reaches the microscopic ent Leggett-Rice parameter γ = φ(t )/M ln(A(t )/A )
e z e 0
7
2 Trotzky et al data tothehomogeneouscase(seebelow). Figure4alsoshows
homog. calc. that the Leggett-Rice parameter changes sign, reflecting
1 trap, large θ the sign of the effective interaction between quasiparti-
trap, small θ
γ 0 cles which is attractive on the BCS side and repulsive
on the repulsive branch [4]. The sign change occurs for
-1
slightly positive values of 1/k a.
F
-2 The center panel in Fig. 4 shows the bare transverse
diffusivityD . Again,thetrapdiffusivityissignificantly
-8 -6 -4 -2 0 2 4 6 8 0⊥
lower than the homogeneous value (solid black line) at
20 weak coupling and agrees with experiment. Once γ is
known, the bare diffusivity D is found from the fit of
0⊥
15 the analytical solution (39) to the magnetization decay
~
⊥/ in Fig. 3.
0 10
D The bottom panel in Fig. 4 displays the ratio
m
5
~γ Wn
0 λ = = . (41)
0
-8 -6 -4 -2 0 2 4 6 8 −2mD0⊥ 2mα
⊥
0.4
It measures the strength of the effective interaction irre-
0.2 spective of the scattering time τ and follows the sign
D
λ0 0 change of γ since D0⊥ > 0. Again, the trap calculation
agreeswiththe experimentaldata exceptinthe instabil-
-0.2
ity region 0 < 1/k a . 1.3 [4], while the homogeneous
F
-0.4 gas has more pronounced interaction effects.
-8 -6 -4 -2 0 2 4 6 8
Spin rotation in a trapped gas is determined qual-
1/k a
F itatively by the ratio between different length scales:
the trap size R 5µm, the helix pitch ℓ =
TF helix
FIG. 4. (Color online) Interaction effect on spin transport: ≈
2π/(αt /2) 4µm for t = 2ms, and the mean free
Leggett-Rice parameter γ, bare diffusivity D0⊥, and interac- pathℓe =≈v τ ,whichreangesfrom0.5µmatunitarity
tion parameter λ0 as a function of 1/kFa at degenerate tem- mfp F D
to 3.8µm at 1/k a = 2. In simulations with a gradi-
perature (T/TF)i=0.2. Results for small (θ=π/30; circles) F −
ent α larger than the experimental value, several helix
and large (θ = π/3; triangles) tipping angles are shown and
compared to theexperimental data [4] (squares). pitches fit into the trap, RTF ℓhelix, and the trap av-
≫
eraged diffusivity and γ are equal to their homogeneous
valuesaslongasalsoℓ <ℓ ,butsaturateforlarger
mfp helix
ℓ . This is in markedcontrastto longitudinalspin dif-
can be read off as the slope of the curve through the mfp
fusion, where a scaling factor of about 5 was found to
origin. This value of γ is then used to fit the magneti-
relatethe trap-averagedandhomogeneousdiffusivityD
zation decay on the left side of Fig. 3 to the analytical
[17, 23, 24]. For weaker gradients where ℓ > R k,
decay function (39) with diffusivity D . Here, I follow helix TF
0⊥ lessthan onehelix pitch fits into the trapandthe homo-
theanalysisoftheexperimentaldatainRef.4andfitthe
geneous solution is reached not even at unitarity where
decay of the trapped gas to the homogeneous solution.
R ℓ .
TF mfp
In Fig. 4, the apparent Leggett-Rice parameter γ ex- ≫
Finally, Fig. 5 shows the temperature dependence of
tracted above is plotted versus the interaction strength,
theLeggett-Riceparameteratunitarity. γdecreaseswith
from the BCS regime (left) via unitarity (1/k a=0) to
F
temperature, which is understood as follows: the two-
the repulsive branch (right). For a trapped gas, I find
body scattering amplitude is purely imaginary at uni-
that the apparent γ saturates for weak coupling on both
tarity and would imply a vanishing γ Re . Hence,
the BCS and the repulsive sides. This behavior differs
∼ T
the observedfinite value ofγ in the quantumdegenerate
qualitatively from the homogeneous case, where γ con-
regime(T/T ) <1isamany-body effectduetomedium
tinues to increase linearly at weak coupling [cf. Eq. (32)] F i
scattering. The presence of the medium enhances both
(solid black line). The trapcalculation thus explains the
thedissipativeandthereactiveeffectsofscatteringatlow
saturationofγ observedinexperiments[4]. Theabsolute
temperatures,and more so in the homogeneousgasthan
valueofγ forthetrappedgasdependsonthetippingan-
in the trap, where only the core is strongly interacting.
gle θ of the initial polarization: for a large tipping angle
θ =π/3 as in the experiment, the Leggett-Rice parame- Similarly, the diffusivity reachesvalues of D 2~/m
0⊥ ≃
terγ saturatesatlargervaluesthaninthecaseofasmall in the quantum degenerate regime (center panel), in
tipping angle θ = π/30. The value to which γ saturates agreement with experiment, while the ratio λ becomes
0
alsodependsonthestrengthofthegradientα,incontrast large only at the lowest temperatures (lower panel).
8
2 sivity much lower than expected for a homogeneous gas
Trotzky et al. expt. data
ofthesametemperatureandinteractionstrength. These
1.5 homogeneous calculation
trap, large tipping angle resultsareobtainedwithoutanyfitparametersandagree
trap, small tipping angle verywellwiththeweak-couplingdatameasuredrecently
1
γ for a trapped gas of ultracold fermionic 40K atoms [4].
0.5 The present calculation provides an intuitive interpre-
tation of the observed saturation of γ: while the spin
0 helix in the homogeneous gas is completely unwound at
0 0.2 0.4 0.6 0.8 1 the echo time t , the trapped gas remains partially in a
e
helical state, with the average transverse magnetization
12 A(t ) strongly reduced and a smaller phase shift φ(t ).
e e
10 At weak coupling 1/k a & 2 the kinetic theory em-
F
| |
8 ployed in this study is well controlled and includes the
~
⊥/ relevant interaction effects.
0 6
D Atstrongcoupling1/k a=0(unitarity),theLeggett-
m 4 RiceeffectisabsentatthFetwo-bodylevelandarisesonly
2 due to many-body medium scattering. The trapcalcula-
0 tion at strong coupling agrees with the experimental re-
0 0.2 0.4 0.6 0.8 1 sults qualitatively;the remaining differences may be due
to reheating in a demagnetized Fermi gas [3, 4] or due
to interaction corrections to the equation of state which
0
were not included in this study; they remain a topic for
future work.
0
λ -0.2 Ultracold Fermi gases are thus ideal systems to study
the interaction and temperature dependence of the
Leggett-Rice effect and the spin transport coefficients.
-0.4
0 0.2 0.4 0.6 0.8 1 The spin dynamics of the trapped gas (Fig. 2) is much
morecomplexthaninthehomogeneouscaseandrequires
(T/T )
F i
a numerical solution. The spin evolution simplifies near
unitarity if R > ℓ > ℓ , in which case the ho-
FIG.5. (Coloronline)Temperatureeffectonspintransportat TF helix mfp
unitarity: Leggett-Riceparameterγ,barediffusivityD⊥,and mogeneoussolutionisrecoveredwithoutanytrap-related
0
interaction parameter λ0 as a function of the initial temper- scaling factors. The decay of the trap-averaged magne-
ature (T/TF)i. At unitarity the results for small (θ = π/30; tization can be analyzed by the methods developed for
circles) and large (θ = π/3; triangles) tipping angles differ the homogeneous case: the magnetization decay fits the
very little; (blue) squares are experimental data [4]. homogeneous solution surprisingly well (Fig. 3). This
study shows that the extracted transport coefficients for
the trapped gas can differ markedly, especially at weak
V. CONCLUSION
coupling, from those of the corresponding homogeneous
system (Fig. 4). Conveniently for the interpretation, the
Tosummarize,aspin-echosequenceinatrappedFermi sign of γ, which reveals the repulsive or attractive char-
gasismodeledbyanonlinearandcomplexdiffusionequa- acter of the effective interaction, does not change with
tionforthetransversemagnetization. Thespinevolution the trap average.
exhibitstheLeggett-Riceeffectofstrengthγ =µn,which The author wishes to thank E. Taylor, J. Thywissen,
appears to saturate for weak coupling, and a bare diffu- S. Trotzky, and S. Zhang for stimulating discussions.
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