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PHYSICAL REVIEW LETTERS

PRL 118, 130405 (2017)

Observation of Quantum-Limited Spin Transport in Strongly Interacting
Two-Dimensional Fermi Gases
C. Luciuk,1 S. Smale,1 F. Böttcher,2 H. Sharum,1 B. A. Olsen,1 S. Trotzky,1 T. Enss,3 and J. H. Thywissen1,4
1

Department of Physics, University of Toronto, Ontario M5S 1A7, Canada
5. Physikalisches Institut and Centre for Integrated Quantum Science and Technology,
Universität Stuttgart, D-70569 Stuttgart, Germany
3Institut für Theoretische Physik, Universität Heidelberg, D-69120 Heidelberg, Germany
4
Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada(Received 20 December 2016; published 31 March 2017)2

We measure the transport properties of two-dimensional ultracold Fermi gases during transverse
demagnetization in a magnetic field gradient. Using a phase-coherent spin-echo sequence, we are able
to distinguish bare spin diffusion from the Leggett-Rice effect, in which demagnetization is slowed by theprecession of a spin current around the local magnetization. When the two-dimensional scattering length istuned to be comparable to the inverse Fermi wave vector k−1
F , we find that the bare transverse spin diffusivity
reaches a minimum of 1.7ð6Þℏ=m, where m is the bare particle mass. The rate of demagnetization is also
reflected in the growth rate of the s-wave contact, observed using time-resolved spectroscopy. The contact
rises to 0.28ð3Þk2F per particle, which quantifies how scaling symmetry is broken by near-resonantinteractions, unlike in unitary three-dimensional systems. Our observations support the conjecture that, in
systems with strong scattering, the local relaxation rate is bounded from above by kB T=ℏ.
DOI: 10.1103/PhysRevLett.118.130405

Conjectured quantum bounds on transport appear to berespected and nearly saturated by quark-gluon plasmas[1,2], unitary Fermi gases [3–11], and bad metals [12,13].
For many modalities of transport, these bounds can berecast as an upper bound on the rate of local relaxation toequilibrium 1=τr ≲ kB T=ℏ, where kB is the Boltzmann
constant and T is the temperature [14,15]. Systems thatsaturate this “Planckian” bound do not have well-definedquasiparticles promoting transport [1,12–15]. A canonicalexample is the quantum critical regime, where one expects
diffusivity D ∼ ℏ=m, a ratio of shear viscosity to entropy
density η=s ∼ ℏ=kB, and a conductivity that is linear in T
[4,12,13]. These limiting behaviors can be understoodpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ bycombining τr with a propagation speed v ∼ kB T=m, forexample, D ∼ v2 τr . This argument applies to ultracold
three-dimensional (3D) Fermi gases, whose behavior in
the strongly interacting regime is controlled by the quantum
critical point at a divergent scattering length, zero temperature, and zero density [4,16,17]. In such systems, one
observes D ≳ 2ℏ=m [6–8] and η=s ≳ 0.4ℏ=kB [3], compatible with conjectured quantum bounds.
However, in attractive two-dimensional (2D) Fermi
gases, scale invariance is broken by the finite bound-statepair size, so the strongly interacting regime is no longer
controlled by a quantum critical point [16,18–23].
Strikingly, an extreme violation of the conjectured D ≳ℏ=m bound has been observed in the spin dynamics of an
ultracold 2D Fermi gas: an apparent diffusivity of 6.3ð8Þ ×
10−3 ℏ=m near lnðkF a2D Þ ¼ 0 [24], where kF is the Fermi
momentum and a2D is the 2D s-wave scattering length. No
0031-9007=17=118(13)=130405(6)similarly dramatic effect of dimensionality is observed incharge conductivity [12] or bulk viscosity [25], and such alow spin diffusivity is unexplained by theory [11,19].
In this work, we recreate the conditions of Ref. [24] andstudy the demagnetization dynamics of ultracold 2D Fermi
gases using both a coherent spin-echo sequence [8] andtime-resolved spectroscopy [7]. We find a modification of
the apparent diffusivity by the Leggett-Rice (LR) effect [26];however, in disagreement with Ref. [24], we find that thequantum bound for the spin diffusivity is satisfied in allconditions accessible to our apparatus. Near lnðkF a2D Þ ¼ 0,
where the minimum diffusivity is observed, we quantify thebreaking of scale invariance by measuring the contact,whose magnitude suggests that the gas is in a many-bodyexcited state during demagnetization.
Our experiments use the three lowest-energy internal
states, labeled j − zi, j þ zi, and jpri, of neutral 40 K atoms.
Interactions between j − zi and j þ zi atoms are tuned by
the s-wave Feshbach resonance [27] at 202.1 G, while jpriatoms remain weakly interacting with j  zi atoms, and any
atoms in identical spin states are noninteracting, since thegas is ultracold. An ensemble of 2D systems is prepared byloading a sympathetically cooled 3D cloud of j − zi atoms
into an optical lattice with a period of 380 nm along the x3
direction [28]. At the final lattice depth of V 0 ¼ 50ER ,where ER =ℏ ≃ 2π × 8.64 kHz, the 2D samples are isolated
from one another and in near-harmonic confinement with
ω3 ≃ 2π × 122 kHz. The transverse confinement with
ω1;2 ≃ 2π × 600 Hz is controlled by an optical dipole trap.
Immediately after loading, the 2D clouds are not rigorously

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PHYSICAL REVIEW LETTERSin thermal equilibrium, but thermalization within eachplane occurs once demagnetization commences. The effective initial temperature ðT=T F Þi (assigned assuming isentropic loading of the lattice [28]) can be varied between
0.20 and 1.20, where T F ≡ EF =kB and EFi ¼ ℏ2 k2Fi =2m is
the Fermi energy of the central 2D system in its initialpolarized state. A static magnetic field gradient B0 along x1
is set to 20.3ð2Þ G=cm unless stated otherwise.
Transport of local magnetization M ¼ hM x ; My ; M z ioccurs through a spin current J j that can be decomposed
into a longitudinal component (J∥j ∥M) and a transverse
component (J⊥j ⊥M), where bold letters indicate vectors in
Bloch space and the subscript j ∈ f1; 2; 3g denotes spatial
direction. Our measurements follow a standard spin-echo
protocol [29] that initiates a purely transverse current. In thehydrodynamic regime, J⊥
j is the sum of a dissipative term⊥
−Deff ∇j M and a reactive term −γM × D⊥eff ∇j M, where⊥ =ð1 þ γ 2 M 2 Þ is the effective transverse diffusivD⊥
¼
D
0
eff
ity and D⊥
0 is the bare diffusivity [26]. The parameter γ
quantifies the precession of the spin current about the local
magnetization, which slows demagnetization—a phenomenon known as the Leggett-Rice effect.
Dynamics are initiated by a resonant radio-frequency (rf)pulse with area θ, which creates a superposition of j − ziand j þ zi and thus a magnetization Mz ¼ − cosðθÞ andMxy ≡ M x þ iM y ¼ i sinðθÞ. The field gradient causes a
twisting of the xy magnetization into a spiral texture. The
gradient in the direction of M drives a transverse spincurrent J⊥
1 , which tends to relax M xy → 0, while M z isconserved. These dynamics are described by [26]2
∂ t Mxy ¼ −iαx1 Mxy þ D⊥
eff ð1 þ iγM z Þ∇1 M xy ;

ð1Þ

where α ¼ B0 Δμ=ℏ and Δμ is the difference in themagnetic moment between j þ zi and j − zi. The solutionsof Eq. (1) depend on a dimensionless time RM th , where th is
the total hold time between the initialization pulse and final
2 1=3
readout pulse and RM ≡ ðD⊥[28]. In our typical0α Þ−1conditions, RM is on the order of 1 ms.
We measure the vector magnetization using a spin-echo
sequence as shown in Fig. 1(a). A π pulse at time th =2
reverses all Mxy phases, so that evolution in the presence ofB0 causes an untwisting of the spiral magnetization texture.
The final π=2 pulse is applied with a variable phase lag
relative to the initialization pulse. The contrast in the final
populations in j  zi is used to determine the direction
ϕ ¼ arg ðMxy =iÞ and the magnitude jM xy j of the transverse
magnetization.
Figures 1(b) and 1(c) show an example of jMxy ðth Þj andϕðth Þ, for an initial pulse angle θ ¼ 0.25π. The solution of
Eq. (1) for γ ≠ 0 gives ϕ ¼ γM z ln jM xy =Mxy ð0Þj for all th ,and thus γ is found by linear regression on data such as
Fig. 1(c). Then, RM (and from it D⊥0 ) is determined by a

(a)

(b)

(c)

FIG. 1. Magnetization dynamics. (a) The time sequence used tomeasure the magnetization dynamics is a simple spin-echo
sequence which allows us to measure (b) the amplitude andphase (inset) of the ensemble-averaged transverse magnetization.
Populations are measured with absorption imaging after a SternGerlach separation [28]. Data are shown for θ ¼ 0.25π, which
prepares M z ¼ −0.71. (c) γ is found from the slope of ϕðth Þ
versus Mz ln jM xy =M xy ð0Þj.

nonlinear fit to jM xy ðth Þj data, again using an analytic
solution of Eq. (1). M xy ð0Þ and B0 are independently
calibrated [28].
For the data shown in Fig. 1, at lnðkFi a2D Þ ¼ 0.13ð3Þ
and ðT=T F Þi ¼ 0.36ð4Þ, we find D⊥
0 ¼ 2.3ð3Þℏ=m and
γ ¼ 0.6ð1Þ. These best-fit transport coefficients are understood as an average both over the ensemble of 2D systems
and over the dynamical changes in the cloud, discussed
below. For strong interactions when the mean free path
∼1 μm is much smaller than the Thomas-Fermi length andthe typical minimal spin-helix pitch ∼5 μm, we expectthat the trap-averaged transport coefficients are close to
the homogeneous values. In this regime, the dynamics are
essentially local [30].
We search for conditions that minimize D⊥
0 by repeating
this characterization of Mxy ðth Þ at various interaction
strengths and initial temperatures. Figure 2(a) shows thatD⊥
0 is smallest when −0.5 ≲ lnðkFi a2D Þ ≲ þ0.5, i.e., wherea2D is comparable to k−1
F . This condition can be understood byconsidering the 2D scattering amplitude in vacuum: fðkÞ ¼
2π=½− lnðka2D Þ þ iπ=2 [23,31–33], which gives a maximal(unitary) cross section 4=k at ka2D ¼ 1. Even though our
Fermi gas has a distribution of relative momenta k, the averagecross section at a low temperature can be estimated by the
replacement of k with kF , due to the logarithmic dependenceof f on the energy of the collision. In other words, corrections
to the unitary scattering cross section are only logarithmic[18–22,34], which explains the qualitative similarity ofFig. 2(a) to prior 3D measurements [8].
The lines in Fig. 2(a) show a kinetic theory both with andwithout medium scattering (solid and dashed lines, respectively) calculated in the jMj → 1 limit [11,30]. The model
also accounts for inhomogeneities in the following way.

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(a)

(b)

PHYSICAL REVIEW LETTERS
(a)

(c)

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(c)

(b)

FIG. 2. Transverse spin diffusivity. (a) D⊥
0 versus interaction
strength lnðkFi a2D Þ with ðT=T F Þi ¼ 0.31ð2Þ (black circles) andðT=T F Þi ¼ 0.21ð3Þ (open squares). Each data point corresponds
to a complete data set as shown in Fig. 1. The lines are predictions
for T=T F ¼ 0.3 by a kinetic theory, as described in the text.
(b) D⊥
0 versus the initial reduced temperature ðT=T F Þi atlnðkFi a2D Þ ¼ −0.1ð2Þ. (c) Local relaxation rate τr estimated as
2⊥D⊥
0 =vT . Shaded regions show D0 < ℏ=m in (a),(b) and τ r <ℏ=kB T in (c). Data are consistent with the conjectured quantum
bound, which would exclude the shaded areas on all plots.

First, the collision integral is solved to compute thetransverse spin diffusion time and LR parameter for a 2D
homogeneous system with the same spin density andtemperature as the trap center [11,35]. Next, these parameters are used to solve the Boltzmann equation for the
position-dependent spin density in the full trapping potential
for each 2D gas in the ensemble [30]. Finally, the averagemagnetization dynamics is analyzed using Eq. (1). This
procedure predicts a minimal D⊥
0 slightly shifted from theobserved minimum, but its results agree well with theincrease of D⊥
0 in the weakly interacting regime. This
gives us confidence that inhomogeneity effects are wellunderstood.
The lowest observed diffusivity is D⊥
0 ¼ 1.7ð6Þℏ=m, at
ðT=T F Þi ¼ 0.19ð3Þ and lnðkFi a2D Þ ¼ −0.1ð2Þ. The effectof the temperature is shown in Fig. 2(b) and by data sets in
Fig. 2(a) taken at two temperatures. In all cases, our data
support the conjectured bound D⊥
0 ≳ ℏ=m.
Assumingthatmagnetization
perturbations
propagate at
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
vT ∼ kB T=m, one can estimate the local relaxation time2
τr with D⊥
0 =vT . Figure 2(c) compares this time to the boundℏ=kB T. Another estimate of the relaxation time would use
2
the Fermi velocity vF, as τr ∼ 2D⊥
0 =vF , which is the correct
scaling for the mean free time in imbalanced Fermi liquids
at a low temperature [26,28,35]. This yields τr ∼ 20 μs at
the minimum observed diffusivity, again on the order ofℏ=kB T. In sum, a 2D Fermi gas with a2D kF ∼ 1 seems tosaturate, but not violate, the Planckian bound τ−1r ≲ kB T=ℏ
at the lowest temperatures probed here.
Figures 3(b) and 3(c) summarize measurements of γ
across a wide range of interaction strengths and temperatures. There are two implications of these data. First,
system-wide demagnetization is slowed by the spincurrent precession. At short th , the solution to Eq. (1) is

FIG. 3. Change in the sign of the interaction. (a) The fraction of
atoms remaining at th ¼ 3.5 ms. (b) γ vs the interaction strength,with ðT=T F Þi ¼ 0.31ð2Þ (black circles) and ðT=T F Þi ¼ 0.21ð3Þ
(open squares). (c) γ versus the initial reduced temperature
ðT=T F Þi , at lnðkFi a2D Þ ¼ −0.1ð2Þ. The change in the sign ofγ, at lnðkFi a2D Þ ≈ −1, is associated with the onset of a pairing
instability.
2 3
jMxy j=jM xy ð0Þj ¼ exp ð−D⊥
eff α th =12Þ, with an apparent⊥⊥
diffusivity Deff , which is ¼ D0 =ð1 þ γ 2 Þ for a fully polarized cloud. This functional form was used in Ref. [24]−3
to determine a minimum “D⊥s ” of 6.3ð8Þ × 10 ℏ=m.
⊥In similar conditions, we instead find Deff ¼ 7ð3Þ×
10−1 ℏ=m. In both works, diffusivity is observed to beminimal near lnðkFi a2D Þ ¼ 0 and to double betweenlnðkFi a2D Þ ≈ 0 and lnðkFi a2D Þ ≈ 1. However, we cannot
explain the 100-fold difference in scale.
The second implication of γ is to reveal the sign ofthe interaction between the spin current and the local
magnetization [26,36,37]. When γ < 0, as we observe
for lnðkFi a2D Þ ≲ −1.5 [see Fig. 3(b)], interactions are
repulsive, whereas when γ > 0, as we observe forlnðkFi a2D Þ ≳ −1.5, interactions are attractive. Associatedwith the sign change of γ is the onset of a pairing instability,
since both are related to the sign change of the real part ofthe low-energy scattering T matrix [8,11,38]. We findindirect evidence for this from atom loss [see Fig. 3(a)],since Feshbach dimers are a precursor to the formation of
deeply bound molecules [39], which are lost from the trap.
In 3D, this loss rate is higher on the repulsive side ofunitarity, but in 2D, we observe the strongest loss on theattractive side, at lnðkFi a2D Þ ∼ 1 [40]. We discuss thisfurther below.
One consequence of demagnetization is a cloud-wide
redistribution of energy. For a 2D harmonically trapped
Fermi gas, the virial relation is [41]1ℏ2V ¼ Eþ
C ;2
8πm 2D

ð2Þ

where V is the total potential energy, E is the total energy,
and C2D is the (extensive) 2D contact [41,42]. Even though

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PHYSICAL REVIEW LETTERS

(a)
(b)

(d)

(c)

FIG. 4. Contact dynamics at lnðkFi a2D Þ ¼ 0.35ð5Þ andðT=T F Þi ¼ 0.31ð2Þ. (a) C2D is measured after a hold time th
by a pulse detuned by ωrf from the j þ zi-to-jpri transition.
(b) Contact growth for B0 ¼ 25 G=cm. (c) Contact growth for
2 −1=3 determined
B0 ¼ 2 G=cm. (d) The best-fit RM ¼ ðD⊥
0α Þ
from contact growth (black points) versus B0 . The shaded region
corresponds to RM with D⊥
0 < ℏ=m. The open point indicates RM
from M dynamics at 20 G=cm. The dashed line shows the best-fit
diffusivity D⊥
0 ¼ 1.1ð1Þℏ=m.

the trap explicitly breaks scale invariance, an SO(2,1)
dynamical symmetry survives at the mean-field level[43] but is broken by a quantum anomaly whose expectation value is C2D [21]. E is conserved in this isolated
system; however, C2D increases from zero for the noninteracting initial state to a finite positive value for the final
state. This implies that V must also increase, which inturn dictates an increase in the rms cloud size: V=N ¼
1
2 2
2 2
2 mðω1 hx1 i þ ω2 hx2 iÞ.
Using rf spectroscopy, we measure C2D throughout thedemagnetization dynamics. The protocol is as describedin Ref. [7] and depicted in Fig. 4(a). The dynamics are
initiated with a θ ¼ π=2 pulse, and the sample is probedwith a spectroscopic pulse that couples the states j þ zi andjpri after a hold time th . The transfer rate of population tostate jpri is measured as a function of the detuning ωrf fromthe bare spin-flip resonance and is known to scale with
C2D ω−2
rf in the limit ωrf ≫ EF [34,44–46]. We compensate
for final-state interactions between the jpri atoms andj  zi atoms in our analysis [28,34,47].
At ln ðkFf a2D Þ ¼ 0.00ð5Þ, we find that the contact rises
from zero to C2D =N ¼ 0.28ð3Þk2Ff , where kFf 2 ¼ kFi 2 =2
after complete depolarization. Using Eq. (2), one finds
V − E=2 ¼ 0.022ð2ÞEFf per particle. In contrast, for a 3D
gas at unitarity, the contribution of the contact to the virial iszero: V − E=2 is proportional to C3D =a3D and goes to zero
when a−1
3D → 0.
A final thermodynamic transformation accompanying
demagnetization is a temperature rise due to the combination of increased spin entropy and decreased occupation ofweek ending
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the Fermi sea [48]. For an initial temperature of 0.3ð1ÞT Fi
and a π=2 pulse, we observe T f ¼ 0.7ð2ÞT Ff nearln ðkFf a2D Þ ¼ 0. Because of the released attractive
interaction energy, this temperature rise is larger than theT f =T Ff − T i =T Fi ≈ 0.25 one would expect from thedemagnetization of an ideal gas. However, the temperaturerise is 3 times smaller than the T f =T Ff − T i =T Fi ≈ 2.2 thatis predicted by matching the initial energy and number tothe equilibrium 2D equation of state [49–51].
One interpretation of these observations is that few or no
dimers are formed during demagnetization. This is certainly true when a2D < k−1
F , where the system is not a
dimerized superfluid as it would be in the ground state. But
even when a2D ∼ k−1
F , measurements of T and C2D suggestthat the system remains in the upper energetic branch. The
value of C2D =Nk2F we observe is roughly 20 times smaller
than the contact strength in an equilibrium mixture atln ðkF a2D Þ ¼ 0 [46,49]. The equilibrium contact is primarily due to a mean-field dimer contribution C0 ≈ 4Nk2F .
Without dimers, the contact in the upper branch would be
due to short-range correlations of unbound atoms, and infact the value we observe is comparable to C2D − C0 in thelower branch [52]. Unlike in 3D, the dimer binding energy
in 2D is greater than EF at the Feshbach resonance, so thatan attractive upper branch is energetically well defined.
Figures 4(b) and 4(c) show the typical dynamics weobserve when measuring C2D ðth Þ. Because of Pauli exclusion, we can use such data to infer magnetization dynamics:
Pairs of fermions must have a singlet wave function tointeract through an s-wave contact interaction. The singlet
fraction can be no larger than 1 − jMj and would beð1 − jMj2 Þ=4 for uncorrelated spins [7,53,54]. For the
π=2 initialization pulse performed here, jMj ¼ jM xy j, sinceMz ¼ 0. A direct comparison between M and C2D at
B0 ¼ 20 G=cm (see [28]) shows a correlation that lies
between these two limits: C2D =N is proportional to
1 − jMxy ðth Þj1.4ð2Þ . This form with γ ¼ 0.71 is used to fit
C2D data for a variety of gradients [see Figs. 4(b) and 4(c)]and extract RM .
Across the experimentally accessible gradients B0 ,Fig. 4(d) shows a range of RM from 4.4ð2Þ × 102 to
2.9ð2Þ × 103 s−1 . Throughout, RM scales with α2=3 (seethe dashed line) and can be explained by a single diffusivity⊥
D⊥
0 ¼ 1.1ð1Þℏ=m. This verifies that the microscopic D0 is0independent of B across the accessible range and, thus,
independent of the pitch of the spin helix. The comparable
magnitude of D⊥
0 determined by two measurement
techniques is also a reassuring check on the fidelity ofthe spin-echo sequence used in M measurements, since themeasurement of C2D does not rely upon the successful
rephasing of the spins at the echo time.
In summary, we observe quantum-limited spin transport
in 2D Fermi gases when a2D is tuned to be comparable to⊥k−1
F . We find that the conjectured lower bound D0 ≳ ℏ=m

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is respected for all interaction strengths, temperatures, andapplied field gradients accessible to our apparatus. This
supports the generality of the bound τ−1r ≲ kB T=ℏ beyond
quantum critical systems, since the finite C2D observed inthis system signifies a broken scaling symmetry near
unitarity.
We thank A. Georges, A. Gezerlis, M. Köhl, S. Sachdev,
E. Taylor, and Shizhong Zhang for stimulating discussions.
This work is supported by the Natural Sciences andEngineering Research Council of Canada, by the Air
Force Office of Scientific Research under FA9550-13-10063, and by the Army Research Office under W911NF15-1-0603 and is part of and supported by the Deutsche
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