Evolution of condensate fraction during rapid lattice ramps

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by  Stefan S. Natu

Evolution of condensate fraction during rapid lattice ramps
Stefan S. Natu,1,∗ David C. McKay,2 Brian DeMarco,2 and Erich J. Mueller1
1Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853, USA.
2Department of Physics, University of Illinois, 1110 W Green St., Urbana, IL 61801
Combining experiments and numerical simulations, we investigate the redistribution of quasi-
momentuminagasofatomstrappedinanopticallatticewhenthelatticedepthisrapidlyreduced.
We find that interactions lead to significant momentum redistribution on millisecond timescales,
2
thereby invalidating previous assumptions regarding adiabaticity. We show that this phenomenon
1
is driven by thepresence of low-momentum particle-hole excitations in an interacting system. Our
0
results invalidate bandmappingas an equilibrium probe in interacting gases.
2

n
a Introduction— Opticallattice experiments aremaking and1/ν settheadiabaticanddiabatictimescalesrespec-
J
importantcontributionsto ourknowledgeofequilibrium tively. Aside from its influence on E and ν [14], the
bg
6 and non-equilibrium properties of quantum many-body hopping rate t has little direct impact on bandmapping.
1
systems [1]. A prime example is that of bosons in opti- Instandardexperimentsinthe tight-bindingregime,the

callattices[2],whicharedescribedbythe BoseHubbard naturalseparationoftimescalesbetweenh/E .0.1ms
] bg
s Hamiltonian. There are many of the open questions in and 1/ν &10 ms makes it straightforwardto satisfy the
a
this system related to transport and dynamics [3]. Here adiabatic condition; a ramp time τ 1 ms is usually
g
∼
- weexploreonesuchissue: howdoesthecondensatefrac- employed.
t
n tion evolve during a lattice ramp? Complicationsariseforaninteractinggas. Interactions
a Inadditiontoitsrelevancefordevelopingparadigmsof leadto a coherentredistributionof quasi-momentumoc-
u
q nonequilibriumphysicsincoldgases,ourstudyisofprac- cupations [15, 16]. Furthermore, collisions scatter atoms
. tical value: lattice ramps are routinely used in a probe between different quasi-momenta while conserving total
t
a knownasbandmapping[4–6]. Thebandmappingprotocol quasi-momentum. The on-site interaction energy be-
m involves three steps. First, the lattice depth is reduced tween two atoms U determines the relevant dynamical
- over a carefully chosen timescale τ, with the intention timescale h/U; in most experiments, h/U . 1 ms. We
d
of mapping quasi-momentum onto momentum. Second, showthattheadditionalcriterion,τ h/U,disruptsthe
n ≪
o thegasisallowedtoballisticallyexpandforashorttime, separation of timescales that makes bandmapping suc-
c mapping momentum onto position [7]. Third, the gas is cessful in noninteracting systems.
[
imaged. Theseimageshavebeeninterpretedastheinitial Usingacombinationofexperimentandnumericalsim-

1 quasi-momentum distribution. This protocol has been ulations, we investigate the impact of interactions on
v used to measure condensate fraction [8], map Brillouin bandmapping for atoms confined in a lattice in the
4 zones [9], determine temperatures [10], and probe phase strongly correlated regime. We quantify the redistri-
5
transitions [11–13]. Here we show that bandmapping is bution of quasi-momentum during lattice ramps for a
1
an inherently non-equilibrium process. Understanding Bose-Einstein condensate of atoms in a 3D cubic opti-
3
. the timescale of evolution of quasi-momentum is there- cal lattice. The fraction of atoms in the condensate is
1
fore crucial to interpreting the data. We find that in determined after linearly ramping from lattice depth V
0 i
2 interacting systems the dynamics are much faster than (with 10ER < Vi < 14ER, spanning the superfluid and
1 previously believed, leading to systematic errors in mea- Mott-insulatorregimes)to afixedfinaldepth Vf =4ER.
: surements of quantities such as condensate fraction. Here E =(h/λ)2/2m is the recoil energy, λ is the laser
v R
i The key step in bandmapping is turning off the wavelength, and m is the atomic mass. The final depth
X
is chosen so that the atoms remain in the single-band
lattice slowly enough such that the quasi-momentum
r Bose-Hubbard limit, thereby simplifying comparison to
a states adiabatically evolve into momentum states, yet
numerical simulations. Ramps terminating at V = 0
fast enough to leave the occupations of different quasi- f
produce similar results. Our results apply to interacting
momentum states unchanged. For harmonically trapped
gases in general, including fermionic systems and mix-
non-interacting particles, there are three relevant en-
tures.
ergy scales: the bandgap E , the tunneling t, and the
bg
quantum energy of the harmonic confining potential hν ExperimentalMethod.— Ourexperimentalsetupisde-
(h=2π~isPlanck’sconstant). Providedthatthelattice scribed in detail in Ref. [10]. In summary, we create a
ramp time τ is long compared to the inverse bandgap condensatecomposedof87Rbatomsinthe F =1,mF =
|
(τ h/E ), the quasi-momentum states will adiabati- 1 stateinaharmonictrapwith(geometric)meantrap
call≫yevolvbegintomomentumstates[10]. Toavoidmotion −freqiuency ν¯0 = 35.78(6) Hz. We cool the gas until the
ofatomsinthetrap,thelatticemustbeturnedoffquickly condensate fraction exceeds 80%.
compared to the trap period (τ 1/ν). Thus, h/E We superimpose a cubic optical lattice with a d =
bg
≪
2
λ/2 = 406 nm lattice spacing on the atoms by slowly
turning on three pairs of retro-reflected laser beams.
The laser intensity determines the potential depth V.
Through Kapitza-Dirac diffraction, we calibrate V to
within 1%, but drift in the calibration results in a 6%
systematic uncertainty. The Gaussian envelope of the
latticebeamsaddstotheharmonicconfinement,andthe
overall(geometric) mean trap frequency with the lattice
on is ν¯ ν¯2+ 8Vi , where w = (120 10) µm is
≈ 0 (2π)2mw2 ±
q
the measured 1/e2 radius of the lattice laser beams.
Ten milliseconds after loading the lattice, we linearly
rampV fromV to V =4E intime τ. The lattice and
i f R
trappingpotentialsarethenremovedin10nsand0.2ms,
respectively, and the column density is imaged after the FIG. 1: Condensate fraction measured after bandmapping
gas expands for 20 ms. We extend the dynamic range from Vi = 10 ER. The insets show high optical density im-
ages where the background is resolved, but the Bragg peaks
of our measurement by imaging only a controlled frac-
are saturated. The images are shown in false color, with red
tionofatomsthataretransferredtothe F =2hyperfine
(blue) indicating regions of high (low) column density. The
state. The number of condensate atoms Nc is measured field of view is 813×813 µm. (a) τ =10 ns; (b) τ =1 ms.
using multimodalfits to “lowopticaldensity”imagesfor
which only a smallnumber of atoms are transferred. We
supplementthesewith“highopticaldensity”images,for Hamiltonian [18] to model our system:
which all are transferred and imaged. In these images,
the broad non-condensate component is resolvable, but = t a†a +h.c. + Un (n 1) µ n
the condensate peaks are saturated (see Fig. 1). The H − Xhiji(cid:16) i j (cid:17) Xi (cid:20)2 i i− − i i(cid:21)
numberofnon-condensateatomsisdeterminedbyfitting
(1)
the broadbackgroundwith the condensedpeaks masked where a and a† are bosonic annihilation and creation
and extrapolating the non-condensate component into i i
operators at lattice site i, and µ = µ V (i), where µ
themaskedregions. ThetotalnumberofparticlesN var- i − ex
is the chemical potential and V (i) is the external po-
iedfrom(103 5) 103 to(72 2) 103 forV =10E ex
± × ± × i R tential at site i [19]. The first sum in Eq. 1 is over all
to 14 E , and the condensate fraction ranged from 0.3
R nearest neighbor sites. We calculate U using the exact
to 0.05.
Wannier functions in the lowest band [20], and extract
the tunneling amplitudes t fromMathieucharacteristics.
Experimental Results.— For each V we measure the
i
We calculate dynamics using a time dependent
post-rampcondensatefractionasa functionofτ. Atyp-
Gutzwiller ansatz [18], which approximates the wave-
icaldatasetisshowninFig. 1forV =10E . Thedata
i R function by Ψ = c(i)(t)m where m is the
points follow an exponential, as illustrated by the red i m m | ii | ii
curve. The fitted time constants τrel are shown in Fig. 2 m-particleFockstaNteoPnsitei,andthecoefficientsc(mi)(t)
as a function of the initial condensate fraction (bottom are space (i) and time (t) dependent. (Note the typo-
axis)alongwiththecorrespondingV (topaxis). There- graphic distinction between the tunneling t, time t, and
i
laxation time is relatively weakly dependent on V , only ramp time τ.) This approximation leads to a simplistic
i
changing by a factor of two as the superfluid-Mott insu- quasi-momentum distribution, dividing atoms into zero
lator transition is crossed, and throughout is consistent momentum (k =0) condensed and k =0 non-condensed
6
with the simple empirical rule τrel 1/U. states. ThetotalnumberofcondensedatomsNc isgiven
∝ by N = a 2, where a = √m+1c(i) c(i).
The insets to Fig. 1 shows the high optical density Asc desPcriib|hedii|in Ref. h[21ii], SPchmr¨odinger’sme+q1uatmion
images for short (a) and moderate (b) ramps. Even for i~∂tψ = ψ for Ψ yields a set of coupled differential
rampsasshortas1ms,thequasi-momentumdistribution equationsHfor the c(i):
m
changesdramatically,asatomsaretransferredtolowmo-
mτ,enatnudmt.hMereofsotrbeadnodmnoatppminegaseuxrpeertihmeeinntistiuaslec∼on1demnssafoter i~∂tcm(i)(t)=−6t(t)(cid:16)φ∗i√m+1cm(i)+1+φi√mc(mi)−1(cid:17)+
fraction. Shorter ramps are not a solution to this prob- U(t)
m(m 1) µ m c(i), (2)
lem, as they lead to significant non-adiabatic transfer of (cid:20) 2 − − i (cid:21) m
atoms to higher energy states [17].
where the mean-fields are φ = a /6, in which the
i hjih ji
Theoretical Modeling.— We use the 3D Bose Hubbard sum is restricted to nearest neigPhbors of site i.
3
ViHERL The bottom line is obtained from initial state 1, and
14 13 11 10 the top line is obtained from initial state 2. Initial state
1 yields a higher condensate fraction compared to the
2.5 MI SF
experimental system at any given initial lattice depth.
2 It relaxes to equilibrium faster, hence providing a lower
L
ms 1.5 bound on the characteristic relaxation time τrel. Initial
Hel state2treatstheatomsasiftheywereinadeeperlattice
Τr 1 than the physical system, and therefore leads to slower
0.5 dynamics. It therefore provides an upper bound on the
characteristic relaxationtime. The dashed line indicates
0.05 0.1 0.15 0.2 0.25 0.3 the lattice depth for which a shell of unit-filling Mott-
N HΤ=0L(cid:144)N insulator emerges. Throughout, the data points lie be-
c 0
tween these two theoretical bounds.
FIG. 2: Relaxation time (τ ) for the condensate fraction
rel For V . 13 E both theoretical protocols yield sim-
for ramps from Vi to Vf = 4ER for variable ramp times τ. i R
ilar results. Here the entire system is superfluid, and
TherangeofVi spansthesuperfluid(SF)andMottinsulator
(MI) regimes (demarcated by the vertical dotted line). The mean-field theory is accurate. Throughout this regime
experimentaldataisboundedbyzerotemperatureGutzwiller τrel 0.5ms. As the Mott transition is approached, the
∼
mean-field simulations using two different initial states (see relaxationtime increases by a factor of 2, indicative of
text). The solid black line shows the relaxation time assum- slower dynamics in the insulating state.∼The simulation
ing initial state 2, while thedashed line shows therelaxation
using initial state 2 (top curve) captures this physics,
time assuming initial state 1. The error bars represent the
showing a significant increase in relaxation time; initial
uncertainty in the relaxation time from a fit to data such as
state 2 contains a Mott-insulator shell. The growth of
that shown in Fig. 1
the relaxation rate from initial state 1 is more gradual,
as the Mott-insulator transition occurs for larger values
As in the experiment, the potential depth is ramped of Vi as compared to the experiment. While the slower
V(t)=V +(V V )(t/τ), where V andV =4 E are timescales for dynamics in the Mott-insulator state are
i f i i f R
−
theinitialandfinallatticedepths,andτ istheramptime. quiteintuitive,therestorationofphasecoherencefollow-
The Hubbard parameters t and U are time-dependent ing a rapid quench is not fully understood [22, 23].
because of this ramp. In the simulations we include a Oursimulationsusedzerotemperatureinitialstates—
sphericallysymmetricexternalharmonictrappingpoten- finite temperature wouldmodify the connectionbetween
tialV ,matchedtothe(lattice-depth-dependent)exper- condensate fraction and lattice depth, effectively raising
ex
imental value ν¯. the relaxation rate obtained from initial state 1.
Wemakedirectcomparisonwiththeexperimentaldata Understanding the timescales.— Bandmapping
by studying N 75,000 87Rb atoms on a 55 55 55 timescales are too short for any significant transport in
∼ × ×
lattice with lattice spacing 406 nm. For our initial state the lattice to occur [21], and thus the observed physics
we use a local density approximation obtained by solv- is purely local. One gains insight by considering the
ing the homogenous, single-site problem. To account for homogenous case Vex = 0. Linearizing Eq. 2 about the
the overestimation of the condensate fraction in mean- homogenous Gutzwiller stationary state [24], we find
field theory, we use two different initial states to model the excitation spectrum ω(k) for the lattice gas in the
the data. Initial state 1 is the mean-field ground state shallow and deep lattice limits (Fig. 3).
obtainedbyusingthephysicallatticedepthV inthesim- Inthe shallowlattice,the onlyrelevantexcitationsare
i
ulation. This state has a largercondensatefractionthan linearlydispersingphonons. Excitationsthatchangethe
the experimental initial state. Initial state 2 is obtained overall phase of the wave-function cost no energy (c.f
byfindingthelatticedepthatwhichthecondensatefrac- Fig.3 (left)). On short timescales, the overalldensity re-
tion predicted by the theory matches the measured con- mains constantand phonons are not excited. For deeper
densate fraction at V . lattices, a particle-hole branch with a quadratic disper-
i
Time evolution, from either of these initial states, is sion, and a gap ∆ is present.
calculatedusingasplit-stepapproachwithsequentialsite The superfluid state in a deep lattice is “quantum de-
updates. We ramp the lattice down from V to 4E in pleted”: the fraction of condensed (k = 0) atoms ap-
i R
a time τ ranging from 0 to 1.5 ms, and calculate the proaches 0 near the insulating transition. Upon lower-
condensate fraction at the end of each ramp. We then ing the lattice depth quasi-momentum states coherently
fit the resulting data to an exponential curve to extract evolve from high momentum to low momentum. Full
a characteristic relaxation time τ . phase coherence is restored when these excited atoms
rel
Comparison of Theory and Experiment – In Fig. 2 we subsequently decay into particle-hole pairs of compara-
compare two theoretical curves with the experimental bleenergybutlowmomentum,whilechangingthe phase
data. of the local wave-function. The timescale for this pro-
4
lattice ramp is an indirect measure of particle-hole exci-
0.5
tations. Probes such as bandmapping are therefore in-
1
U U dispensible to developing our understanding of strongly
Ω(cid:144) Ω(cid:144) D(cid:144)U
0.5 D(cid:144)U correlated systems.
Ñ Ñ
0 0 Acknowledgements.— We thank Randy Hulet and
0 Π Π 0 Π Π
2 2 Joseph Thywissen for enlightening discussions. This
kHh(cid:144)dL kHh(cid:144)dL work was supported by a grantfrom the Army Research
FIG. 3: Typical Gutzwiller excitation spectra for deep lat- Office with funding from the DARPA OLE program.
tices. (Left): Quantum depleted superfluid near the Mott-
insulator transition (V = 13ER). Two modes are present:
a gapless phonon mode with a linear Bogoliubov (dashed)
dispersion at low k, and a gapped particle-hole mode with
quadratic dispersion. The gap ∆ ∼ U, sets the diabatic- ∗ Electronic address: [email protected]
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