Table Of ContentLongitudinal spin diffusion in a nondegenerate trapped 87Rb gas
D. Niroomand, S.D. Graham, and J.M. McGuirk
Department of Physics
Simon Fraser University, Burnaby,
British Columbia V5A 1S6, Canada
(Dated: June2, 2015)
5 Longitudinal spin diffusion of two pseudo-spin domains is studied in a trapped 87Rb sample
1
above quantum degeneracy, and the effect of coherence in the domain wall on the dynamics of the
0
system is investigated. Coherence in a domain wall leads to transverse-spin-mediated longitudinal
2
spin diffusion that is slower than classical predictions, as well as altering the domains’ oscillation
y frequency. The system also shows an instability in the longitudinal spin dynamics as longitudinal
a and transverse spin components couple, and a conversion of longitudinal spin to transverse spin is
M
observed,resulting in an increase in the total amount of coherence in thesystem.
0 PACSnumbers: 51.10.+y,51.20.+d,67.85.-d,75.40.Gb
3
] The development of ultracold trapped atomic systems In the nondegenerate case of a 1D two-domain spin
s
a offermionsandbosonshasopenedupnewpathsofstudy structure studied here, quantum modifications to dif-
g for transport phenomena. In particular, theoretical and fusion appear as a decrease in the oscillation rate of
-
experimentalstudiesofspintransporthavebeenofrecent spin domains in a harmonic potential and an increase
t
n
interest in a number of systems including strongly inter- in the longitudinal spin diffusion time, through coupling
a
acting degenerate Fermi gases near unitary [1–3], dilute to the transverse spin. This effect shows sensitivity to
u
q noncondensedbosonicgasesclosetoquantumdegeneracy the degree of coherence in the domain wall between spin
. [4, 5], and ultracold gases in optical lattices [6, 7]. domains, which highlights the quantum mechanical na-
t
a ture of the dynamics more clearly. In this Letter we
m Quantum degeneracy effects can dramatically mod-
demonstrate the alteration of longitudinal spin diffusion
ify spin transport in spin-polarized systems, for exam-
- timescales depending on the initial amount of the coher-
d pleleadingtophenomenasuchasthe Leggett-Riceeffect
ence in the domain wall and also report observation of a
n [8]. Though spin is a fundamentally quantum property,
o longitudinal spin instability, which leads to an increase
classicalBoltzmanntheorywasexpectedtodescribespin
c in the total amount of coherence in the system.
transportin dilute nondegenerategases accurately,since
[
The mechanism for coherent atom-atom scattering is
quantum degeneracy effects are typically insignificant.
2 the ISRE, where exchange scattering between indistin-
Surprisingly in 1982 Bashkin [9] and Lhuillier and Lalo¨e
v guishableparticlesleadstoprecessionofeachatom’sspin
[10] independently predicted that transport in quantum
6 about their combined spin. The theoretical description
6 gases can lead to macroscopic collective behavior even
ofspindynamicsinnondegeneratequantumsystemswas
8 innondegeneratesystemswithspin-independentinterac-
developed using semiclassical kinetic theory [13, 16–19].
3 tions. The driving process for this collective behavior is
0 We adoptthe notationof[17] anddescribethe time evo-
the identical spin rotation effect (ISRE), in which sym-
. lution of the spin density distribution, ~σ(p,z,t), with a
1 metrizationofthe two-bodywavefunctionin binarycol-
0 lisions between identical particles leads to coherent ex- 1DquantumBoltzmannequationthatincludestheISRE
5 via a spin-torque term:
change interactions when the thermal de Broglie wave-
1
length is larger than the two-body interaction length
: ∂~σ
v [10]. Examples of such collective behavior include spin ∂t~σ(p,z,t)+∂0~σ(p,z,t)−Ω~ ×~σ(p,z,t)= |1D, (1)
i ∂t
X waves [4], spin self-rephasing [11], and non-conservation
r of transverse spin [12]. where ∂0 = mp ∂∂z − ∂U∂ezxt∂∂p and Ω~ = (Udiffuˆk +gS~)/¯h.
a
Diffusion is alsostronglymodified by quantumeffects, The mean-field interaction strength for two scattering
leadingtononlinearandanisotropicspindiffusion[13,14] particles with s-wave scattering length a and mass m is
and universal spin dynamics [1]. Near unitarity, exper- g = 4πmh¯2a, and uˆk is the longitudinal unit vector on the
iments on two spin domains in a strongly interacting Blochsphere. The experimentally observable quantity is
Fermi gas showed reversal of spin currents [15], result- thespinS~(z,t)=R dp~σ(p,z,t)/2π¯h. S~ containsthe lon-
ing in quantum-limited spin diffusion. Also in unitary gitudinal spin S and the transverse spin S~ , with mag-
k ⊥
Fermi gases, further quantum modifications to diffusive nitudeS (i.e.thespincoherence). U andU arethe
⊥ ext diff
timescales were observed [1, 2]. In [6], patterned spin trapping potential and differential potential experienced
textures were used to study diffusion in various geome- between the two states (due to mean-field and Zeeman
tries, showing diffusive behavior in one dimension (1D), shifts). Lastly, the right-hand side of Eq. 1 represents
but large departures from classical transport in 2D. damping due to elastic collisions [17].
2
We create a two-domain structure of “up-down” ference is given by S (z) = N2(z) − N1(z) with Ntot =
k Ntot Ntot i
pseduo-spinstatesin anondegenerategasof87Rbatoms PNi(z), (i=1,2) summed ove2rall axia1lbins. Ramsey-
trapped in an external harmonic potential and observe z
type experiments measure the transverse spin compo-
the relaxation to equilibrium. The trap is axisymmet-
nent. We normalize transverse spin by the magnitude
ric, with trapping frequencies of 2π×(6.7,260,260) Hz.
of a fully coherent sample with no spin inhomogeneities,
Due to rapid averaging from the high radial frequency,
c(z,t) ≡ S (z,t)/Smax, and the total coherence in the
dynamics can be treated effectively as one-dimensional. ⊥ ⊥
The pseudo-spin doublet consists of two magnetically sample is ctot(t)≡R c(z,t)dz.
trapped hyperfine ground states of 87Rb (|1i ≡ |1,−1i Figure 2 shows the typical time evolution of the two-
domainspinstructure. Thelongitudinalspincomponent
and |2i ≡ |2,1i), coupled via a two-photon microwave
S exhibitsspindomainoscillationsaswellasdiffusionof
transition. (See [20, 21] for details.) k
the longitudinal spin gradient, which damps the oscilla-
We prepare spin domains with optical patterning and
tions. These measurements reveal that longitudinal do-
microwavepulsetechniques(seeFig.1)[20,21]. Magnet-
ically trapped 87Rb is evaporativelycooledto just above mains persistfor muchlongerthanexpectedfor classical
longitudinal diffusion times (∼ 25 ms [17]) and oscillate
quantumdegeneracy(temperatureT ∼650nKandpeak
density n ∼ 2.6×1013 cm−3). Initially in state |1i, a much slower than trap oscillations (Fig. 2(b)). A co-
herent domain wall has a stabilizing effect, which agrees
microwave π-pulse transfers atoms to |2i, except where
with theoreticalpredictions and previous experiments in
illuminated with an off-resonant, partially masked laser
spin-polarized gases [12, 22–24]. We study this effect in
beam that creates an optical step potential on top of
more detail by controllably tuning the amount of initial
the trapping potential. The masked laser ac Stark shifts
coherence in the domain wall.
atomic energy levels out of resonance by ∼ 50Ω . This
R
procedure creates domains of spin up and down sepa-
rated by a helical domain wall in which the spin vector
is coherently twisted, while remaining fully polarized. (a) N (b)6000 0.5
1
N
on 2 s)
(a) Rb atoms ac Stark shift laser pulati me (m S||
û|| |2〉 û|| |2〉 Po Ti
00 -0.5
-350 0 350 -350 350
Position (µm)
Ω fields on Position (µm) (d) 1
Ω (c) 0 c 0.75
|1〉 |1〉 -350 (a.u.)1
N
final state m) 00 Time (ms) 100
µ (e)
n ( (d) 1
(b) sitio (e) u.)
2 1 Po (a.1
N
350
00 335500 0
Time (ms) 0 Time (ms) 100
FIG. 1. (a) Schematic of the domain preparation, showing
FIG. 2. Time evolution of spin domains with c = 0.74.
init
the patterning laser and accompanying spin evolution in the
(a) Typical initial spin distribution of |1i (solid line) and |2i
Bloch sphere representation (longitudinal axis uˆ and trans-
k (dashed line), used to infer S . (b) Evolution of S . (c)
verseequatorialplane). Applyingaπ-pulsetoatomsinitially k k
Evolution of transverse spin amplitude, c(z,t). Coherence is
in |1i while illuminating them with the masked laser trans-
initially confined to the domain wall, in a region less than
fers atoms in the unilluminated half (left) of thedistribution
one-third the axial full-width half-maximum of the atomic
to |2i while leaving atoms in the illuminated half unchanged
distribution,∼0.3wax,butrapidlyspreadstoward theedges
(right). (b)Compositeimageofatomsin|1i(purple)and|2i
ofthecloud,coveringaregionofthesize∼0.7waxby200ms.
(green) following initialization.
Ramsey fringes at (d) the trap center z = 0 and (e) z =
0.3wax as shown by arrows in (c), with a sinusoidal fit to
The longitudinal spin projection is measured by di- guide theeye.
rectly measuring the population in states |1i and |2i
using absorption imaging. Each image is divided into The more striking behavior occurs in the time evolu-
equally spaced axial bins, and the local population dif- tion of the transverse spin component. Fig. 2(c) shows
3
heth)e)er.esnRpcaaetm)ioestxeetymrafprciotnergdaelfercovomonlturRtaiasomtnrsoaefpyitmdhleeyaarsmiusrpeeslmitwueidnthtesion(fFlSoig⊥n.g2(i(ctduo--- (a80)0(i) 0.5 nt (a.u.)00..01 (b)
dinal spin domains, where the initial fringe amplitude is ms) ome-0.1 (i)
sthmeaclllo(Fudig.ed2g(ee)s).,showingthespreadofcoherencetoward Time ( S|| pole m-0.2 ((iiii )i)
Di
Spreading ofcoherencecouldresult fromS⊥ diffusion, 00 220000 440000 660000 880000
but the total transverse spin magnitude in the gas in- 0 -0.5 TTiimmee ((mmss))
20
creases (Fig. 4(d)). This increase implies the appear- 8000(ii) z) (c)
H
ance of coherence near the cloud edges is not due to Γ ( 1155
sapsina rdeisffuultsioonf,anbuintstraabthileitrytionctohnevleornsgioitnudoifnSalkspinitnocSur⊥- me (ms) ng rate 1100
rent. This effect has been observedin spin-polarized gas Ti mpi 55
systems [22, 23] and was described as an experimental Da
0 0
mttshaimalenpitliarafaerraspt[mp2ae5tedi–toe2anr7ts]oo,.mftIhtifcehtespryahCsnytassesvimtceasripsgneoogsvssipeenrissnnsteianissbgdicltiioffhtnyeefirp[en1nhe2ted]n.exionApmletterhhniemooundegnoihs-- 4e (ms)00(iii) ency f (Hz) 43 0.2 0.4 cinit 0.6 (d)0.8
main wall, it dephases and the Sk gradient decays via Tim equ
ordinary longitudinal diffusion, since the kinetic equa- n fr 2
o
tion decouples for longitudinal and transverse spin com- 0 ati
p(io.en.elnatrsge[2L5e,g2g8e]t.t-HRoicweepvaerra,mifetthere),ISthReEloinsgliatrugdeineanlosupginh -350 Positio0n (µm)350 Oscill 1 0.2 0.4 cinit 0.6 0.8
currentbecomesunstable,couplingbetweenlongitudinal
and transverse spin dynamics occurs, and the Sk gradi- FmIaGin. w3.all(cao)heTreimnceese,vforolumtiotnopotfoSbkottfoormdciffere=ntSin/iStimalaxdo=-
ent decays via transverse diffusion across the coherent init ⊥ ⊥
0.74, 0.51, 0.28 respectively in the cloud center. (b) Dipole
domain wall. Thus transverse-spin-mediationof longitu-
moment time evolution calculated from (a), with a represen-
dinaldiffusionincreasesthetimescaleforlongitudinaldif-
tative decaying sinusoidal fit to (iii) c = 0.28. (c) Damp-
init
fusion to be comparable with slower transverse diffusion ing rate and (d) oscillation frequency of the dipole moment
times, whichareslowerdue to coherentspin interactions for different c . Error bars correspond to fit uncertainties
init
absent from longitudinal spin dynamics. for dipole moment oscillations and cinit measurements. The
To study the effect of initial coherence in the domain shaded band is theresult of numerical simulations of Eqn.1.
wall, we initialize domain walls with different S and
⊥
observe their relaxation. Coherence is controlled via the
sameoff-resonantlaserthatpreparesthespindomains. A moment; and extract f and Γ from decaying sinusoidal
shortlightpulse createsanonuniformdifferentialatomic fits to these oscillations. Figure 3(c) and (d) summa-
potential, whose inhomogeneity induces rapid dephasing rize the results. Both Γ and f decreaseas cinit increases,
ofS~ inthedomainwall. Thedegreeofcoherenceiscon- showing the stabilizing effect of a coherent domain wall.
⊥
trolled by the pulse length (≤0.6 ms) and measured via The oscillation frequency is primarily controlled by two
Ramsey spectroscopy. This forced dephasing procedure competing factors: trapping potential oscillations and
can reduce the initial coherence by over 70%. mean-field-inducedspinrotation. Intheincoherentlimit,
Figure 3(a) shows the longitudinal time evolution for f approaches the trapping frequency, as the system ap-
differentinitialdegreesofcoherence. Boththe frequency proachesamixtureofdistinguishableclassicalidealgases
oflongitudinalspindomainoscillationsaswellaslongitu- thatdiffuseaccordingtoclassicalBoltzmanntheory. The
dinal spin diffusion rates decrease as domain wall coher- largecinitlimitentrainslongitudinaldiffusionwithslower
ence is increased. Dynamics for a two-domain structure transverse diffusion.
are dominated by the dipole mode, which we isolate via We compare these measurements with transport the-
the dipole moment of the spin distribution, (cid:10)zS (z,t)(cid:11) , orybynumericallysimulatingEqn.1(see[20]fordetails).
k n
where h...i denotes a density-weighted average to com- The shaded regions in Fig. 3(c) and (d) represent one-
n
pensate for low signal-to-noise ratio at the distribution sigmaconfidencebands fromMonte Carlosimulationsof
edges. Figure 3(b) shows evolution of spin-dipole mo- Eqn.1,including statisticalfluctuationsinn, T,anddo-
ments for the coherences in Fig. 3(a). The frequency f mainwallsize,aswellasasystematicdensitycalibration
and damping rate Γ of longitudinal oscillations are ex- uncertainty. Thedataagreeswellwiththeoreticalpredic-
tracted from these moments. tions. Discrepanciesinf atlowcoherenceislikelydueto
We repeatthe experimentfor different initialamounts challenges in fitting critically damped oscillations where
of coherence, c ; calculate the time-dependent dipole the quality factor drops. Overdamping should occur for
init
4
(a) 0 Time (ms) 350 (b) 500 500 10
(a) (b)
m) -350 0.8 zz == 00..63 wwax atio
Position (µ 3500 (i) c(z) 00..46 z = 0 ax a (a) 0 + (i) a (a) 0 + coherence r
0 200 0.2 15 15 1
-350 400 T (nK) 900 15 ℓ (µm) 200
0.0
0 100 200 300
Time (ms) FIG. 5. Instability onset seen in the ratio of maximum en-
(c) semblecoherencetoinitialcoherence,cmax/cinit,fromsimula-
tot tot
(ii) (i) tionsof Eq.1, plotted logarithmically tohighlight instability
350
0 180 0.3 (ii) threshold. Wevaryaand(a)T or(b)ℓwithotherparameters
-350 (iii) set to conditions of Fig. 3(a)(ii). Since µ∝1/a, increasing a
ot0.2 diminishestheinstability,ascollisional dampinggrows faster
ct
than exchange scattering. Decreasing T or increasing ℓ sta-
0.1 bilizes thespin current as expected,through increasing adia-
(iii) baticrotationduringdomainwallcrossing. Crossesrepresent
350 0 experimental conditions used here.
0 100 200 300
0 c 0.75 Time (ms)
FIG. 4. (a) Time evolution of S for different domain wall used to simulate the domain wall spin dynamics. As the
⊥
coherences, from top to bottom cinit = 0.71, 0.51, 0.30 re- regionofexperimentalinterestliesbetweenthe collision-
spectively in the cloud center. (b) Time evolution of S⊥ in lessandhydrodynamicregimes(Knudsennumber∼1.2),
different spatial regions, denoted by z = (0, 0.3, 0.6)×wax no analytic approximationis readily available for the in-
[dotted lines in (i)]. The dashed box indicates a change in
stability criterion, such as was done in [26]. However,
sampling rate to measure both the fast initial rise of coher-
there are two primary requirements for driving the in-
enceandlongerrelaxationtoequilibrium. (c)Timeevolution
of total ensemble coherence for c shown in (a). The solid stability. First and most important is that the ratio of
init
curveisanumericalsimulationofEq.1for(i),showinggood exchange scattering to elastic scattering rates must be
qualitative agreement. Quantitative agreement is limited by approximatelygreaterthan1(quantifiedbytheLeggett-
challenges extracting Ramsey fringe amplitudes in the pres- Riceparameterµ[27],∼3forthisexperiment),lestcolli-
ence of noise, dephasing, and small signal in cloud wings.
sionsdampspincurrentsbeforetheycangrowlarge. The
secondrequirementis thatthe numberofexchangecolli-
sions while traversingthe domain wallbe small; if not, a
cinit < 0.2, but reducing coherence to this level without spin crossing the gradient will smoothly rotate from one
altering the longitudinal spin domains is challenging. domain to the other, similar to adiabatic following.
The effect of the spin instability can be seen more We extend these simulations to a broader parameter
clearly in the transverse spin dynamics (Fig. 4). The space than can be achievedin our system to explore fur-
timeevolutionofS⊥(z,t)fordifferentinitialdomainwall ther the crossover from classical to quantum diffusion,
coherence is shown in Fig. 4(a). The rise and spread of by numerically varying T, n, a, and domain wall size, ℓ.
coherence for different cinit shows similar spatial behav- Figure 5 shows the effects of T, a, and ℓ on the spin in-
ior, but S⊥ persists longer when there is more coherence stability. Because of near-critical damping and spatially
inthedomainwallinitially. Figure4(b)showstheevolu- varyingdensity(thuscollisionrates),thesystemexhibits
tion of coherence in different regions of the cloud for the a smooth transition between diffusive regimes. Further-
high cinit preparation (Fig. 4a(i)). more,sinceµisindependentofn,densityhaslittleeffect
Figure 4(c) shows total ensemble coherence, c (t), on the instability within the range between collisionless
tot
for different c , calculated by summing the coherence andhydrodynamicbehavior,andconfinementplayslittle
init
across the cloud. Coherence rapidly rises followed by a role so long as ℓ≪w .
ax
gradual decrease, but with different timescales for dif- Inconclusion,ourresultsrevealthestabilizingeffectof
ferent cinit. Maximum total coherence across the cloud coherenceinadomainwall,leadingtoanincreaseinlon-
depends on the initial coherence, and the time to reach gitudinal domain lifetimes. These effects are explained
maximum coherence also increases as cinit increases. bycouplingbetweenlongitudinalandtransversespindy-
Theseresultshighlightthetransversesourceofenhanced namics, which induces transverse-channel-mediated lon-
lifetimes of longitudinal spin domains with coherent do- gitudinal spin diffusion. While not a direct measure of
main walls. The presence of coherence in domain walls anisotropicdiffusion, alterationofdiffusivetimescalesby
links diffusion timescales and increases longitudinal dif- adjusting only the longitudinal-transverse spin coupling
fusion times to be as long as transverse diffusion times. (via the coherence) is strongly suggestive of anisotropic
Ourresults validate the quantumBoltzmannequation diffusion. An accompanying rise in the total coherence
5
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